736 lines
103 KiB
HTML
736 lines
103 KiB
HTML
<html><head><meta http-equiv="Content-Type" content="text/html; charset=utf-8"/><title>Unit 30: Induction and Inductance</title><style>
|
||
/* cspell:disable-file */
|
||
/* webkit printing magic: print all background colors */
|
||
html {
|
||
-webkit-print-color-adjust: exact;
|
||
}
|
||
* {
|
||
box-sizing: border-box;
|
||
-webkit-print-color-adjust: exact;
|
||
}
|
||
|
||
html,
|
||
body {
|
||
margin: 0;
|
||
padding: 0;
|
||
}
|
||
@media only screen {
|
||
body {
|
||
margin: 2em auto;
|
||
max-width: 900px;
|
||
color: rgb(55, 53, 47);
|
||
}
|
||
}
|
||
|
||
body {
|
||
line-height: 1.5;
|
||
white-space: pre-wrap;
|
||
}
|
||
|
||
a,
|
||
a.visited {
|
||
color: inherit;
|
||
text-decoration: underline;
|
||
}
|
||
|
||
.pdf-relative-link-path {
|
||
font-size: 80%;
|
||
color: #444;
|
||
}
|
||
|
||
h1,
|
||
h2,
|
||
h3 {
|
||
letter-spacing: -0.01em;
|
||
line-height: 1.2;
|
||
font-weight: 600;
|
||
margin-bottom: 0;
|
||
}
|
||
|
||
.page-title {
|
||
font-size: 2.5rem;
|
||
font-weight: 700;
|
||
margin-top: 0;
|
||
margin-bottom: 0.75em;
|
||
}
|
||
|
||
h1 {
|
||
font-size: 1.875rem;
|
||
margin-top: 1.875rem;
|
||
}
|
||
|
||
h2 {
|
||
font-size: 1.5rem;
|
||
margin-top: 1.5rem;
|
||
}
|
||
|
||
h3 {
|
||
font-size: 1.25rem;
|
||
margin-top: 1.25rem;
|
||
}
|
||
|
||
.source {
|
||
border: 1px solid #ddd;
|
||
border-radius: 3px;
|
||
padding: 1.5em;
|
||
word-break: break-all;
|
||
}
|
||
|
||
.callout {
|
||
border-radius: 3px;
|
||
padding: 1rem;
|
||
}
|
||
|
||
figure {
|
||
margin: 1.25em 0;
|
||
page-break-inside: avoid;
|
||
}
|
||
|
||
figcaption {
|
||
opacity: 0.5;
|
||
font-size: 85%;
|
||
margin-top: 0.5em;
|
||
}
|
||
|
||
mark {
|
||
background-color: transparent;
|
||
}
|
||
|
||
.indented {
|
||
padding-left: 1.5em;
|
||
}
|
||
|
||
hr {
|
||
background: transparent;
|
||
display: block;
|
||
width: 100%;
|
||
height: 1px;
|
||
visibility: visible;
|
||
border: none;
|
||
border-bottom: 1px solid rgba(55, 53, 47, 0.09);
|
||
}
|
||
|
||
img {
|
||
max-width: 100%;
|
||
}
|
||
|
||
@media only print {
|
||
img {
|
||
max-height: 100vh;
|
||
object-fit: contain;
|
||
}
|
||
}
|
||
|
||
@page {
|
||
margin: 1in;
|
||
}
|
||
|
||
.collection-content {
|
||
font-size: 0.875rem;
|
||
}
|
||
|
||
.column-list {
|
||
display: flex;
|
||
justify-content: space-between;
|
||
}
|
||
|
||
.column {
|
||
padding: 0 1em;
|
||
}
|
||
|
||
.column:first-child {
|
||
padding-left: 0;
|
||
}
|
||
|
||
.column:last-child {
|
||
padding-right: 0;
|
||
}
|
||
|
||
.table_of_contents-item {
|
||
display: block;
|
||
font-size: 0.875rem;
|
||
line-height: 1.3;
|
||
padding: 0.125rem;
|
||
}
|
||
|
||
.table_of_contents-indent-1 {
|
||
margin-left: 1.5rem;
|
||
}
|
||
|
||
.table_of_contents-indent-2 {
|
||
margin-left: 3rem;
|
||
}
|
||
|
||
.table_of_contents-indent-3 {
|
||
margin-left: 4.5rem;
|
||
}
|
||
|
||
.table_of_contents-link {
|
||
text-decoration: none;
|
||
opacity: 0.7;
|
||
border-bottom: 1px solid rgba(55, 53, 47, 0.18);
|
||
}
|
||
|
||
table,
|
||
th,
|
||
td {
|
||
border: 1px solid rgba(55, 53, 47, 0.09);
|
||
border-collapse: collapse;
|
||
}
|
||
|
||
table {
|
||
border-left: none;
|
||
border-right: none;
|
||
}
|
||
|
||
th,
|
||
td {
|
||
font-weight: normal;
|
||
padding: 0.25em 0.5em;
|
||
line-height: 1.5;
|
||
min-height: 1.5em;
|
||
text-align: left;
|
||
}
|
||
|
||
th {
|
||
color: rgba(55, 53, 47, 0.6);
|
||
}
|
||
|
||
ol,
|
||
ul {
|
||
margin: 0;
|
||
margin-block-start: 0.6em;
|
||
margin-block-end: 0.6em;
|
||
}
|
||
|
||
li > ol:first-child,
|
||
li > ul:first-child {
|
||
margin-block-start: 0.6em;
|
||
}
|
||
|
||
ul > li {
|
||
list-style: disc;
|
||
}
|
||
|
||
ul.to-do-list {
|
||
text-indent: -1.7em;
|
||
}
|
||
|
||
ul.to-do-list > li {
|
||
list-style: none;
|
||
}
|
||
|
||
.to-do-children-checked {
|
||
text-decoration: line-through;
|
||
opacity: 0.375;
|
||
}
|
||
|
||
ul.toggle > li {
|
||
list-style: none;
|
||
}
|
||
|
||
ul {
|
||
padding-inline-start: 1.7em;
|
||
}
|
||
|
||
ul > li {
|
||
padding-left: 0.1em;
|
||
}
|
||
|
||
ol {
|
||
padding-inline-start: 1.6em;
|
||
}
|
||
|
||
ol > li {
|
||
padding-left: 0.2em;
|
||
}
|
||
|
||
.mono ol {
|
||
padding-inline-start: 2em;
|
||
}
|
||
|
||
.mono ol > li {
|
||
text-indent: -0.4em;
|
||
}
|
||
|
||
.toggle {
|
||
padding-inline-start: 0em;
|
||
list-style-type: none;
|
||
}
|
||
|
||
/* Indent toggle children */
|
||
.toggle > li > details {
|
||
padding-left: 1.7em;
|
||
}
|
||
|
||
.toggle > li > details > summary {
|
||
margin-left: -1.1em;
|
||
}
|
||
|
||
.selected-value {
|
||
display: inline-block;
|
||
padding: 0 0.5em;
|
||
background: rgba(206, 205, 202, 0.5);
|
||
border-radius: 3px;
|
||
margin-right: 0.5em;
|
||
margin-top: 0.3em;
|
||
margin-bottom: 0.3em;
|
||
white-space: nowrap;
|
||
}
|
||
|
||
.collection-title {
|
||
display: inline-block;
|
||
margin-right: 1em;
|
||
}
|
||
|
||
.simple-table {
|
||
margin-top: 1em;
|
||
font-size: 0.875rem;
|
||
empty-cells: show;
|
||
}
|
||
.simple-table td {
|
||
height: 29px;
|
||
min-width: 120px;
|
||
}
|
||
|
||
.simple-table th {
|
||
height: 29px;
|
||
min-width: 120px;
|
||
}
|
||
|
||
.simple-table-header-color {
|
||
background: rgb(247, 246, 243);
|
||
color: black;
|
||
}
|
||
.simple-table-header {
|
||
font-weight: 500;
|
||
}
|
||
|
||
time {
|
||
opacity: 0.5;
|
||
}
|
||
|
||
.icon {
|
||
display: inline-block;
|
||
max-width: 1.2em;
|
||
max-height: 1.2em;
|
||
text-decoration: none;
|
||
vertical-align: text-bottom;
|
||
margin-right: 0.5em;
|
||
}
|
||
|
||
img.icon {
|
||
border-radius: 3px;
|
||
}
|
||
|
||
.user-icon {
|
||
width: 1.5em;
|
||
height: 1.5em;
|
||
border-radius: 100%;
|
||
margin-right: 0.5rem;
|
||
}
|
||
|
||
.user-icon-inner {
|
||
font-size: 0.8em;
|
||
}
|
||
|
||
.text-icon {
|
||
border: 1px solid #000;
|
||
text-align: center;
|
||
}
|
||
|
||
.page-cover-image {
|
||
display: block;
|
||
object-fit: cover;
|
||
width: 100%;
|
||
max-height: 30vh;
|
||
}
|
||
|
||
.page-header-icon {
|
||
font-size: 3rem;
|
||
margin-bottom: 1rem;
|
||
}
|
||
|
||
.page-header-icon-with-cover {
|
||
margin-top: -0.72em;
|
||
margin-left: 0.07em;
|
||
}
|
||
|
||
.page-header-icon img {
|
||
border-radius: 3px;
|
||
}
|
||
|
||
.link-to-page {
|
||
margin: 1em 0;
|
||
padding: 0;
|
||
border: none;
|
||
font-weight: 500;
|
||
}
|
||
|
||
p > .user {
|
||
opacity: 0.5;
|
||
}
|
||
|
||
td > .user,
|
||
td > time {
|
||
white-space: nowrap;
|
||
}
|
||
|
||
input[type="checkbox"] {
|
||
transform: scale(1.5);
|
||
margin-right: 0.6em;
|
||
vertical-align: middle;
|
||
}
|
||
|
||
p {
|
||
margin-top: 0.5em;
|
||
margin-bottom: 0.5em;
|
||
}
|
||
|
||
.image {
|
||
border: none;
|
||
margin: 1.5em 0;
|
||
padding: 0;
|
||
border-radius: 0;
|
||
text-align: center;
|
||
}
|
||
|
||
.code,
|
||
code {
|
||
background: rgba(135, 131, 120, 0.15);
|
||
border-radius: 3px;
|
||
padding: 0.2em 0.4em;
|
||
border-radius: 3px;
|
||
font-size: 85%;
|
||
tab-size: 2;
|
||
}
|
||
|
||
code {
|
||
color: #eb5757;
|
||
}
|
||
|
||
.code {
|
||
padding: 1.5em 1em;
|
||
}
|
||
|
||
.code-wrap {
|
||
white-space: pre-wrap;
|
||
word-break: break-all;
|
||
}
|
||
|
||
.code > code {
|
||
background: none;
|
||
padding: 0;
|
||
font-size: 100%;
|
||
color: inherit;
|
||
}
|
||
|
||
blockquote {
|
||
font-size: 1.25em;
|
||
margin: 1em 0;
|
||
padding-left: 1em;
|
||
border-left: 3px solid rgb(55, 53, 47);
|
||
}
|
||
|
||
.bookmark {
|
||
text-decoration: none;
|
||
max-height: 8em;
|
||
padding: 0;
|
||
display: flex;
|
||
width: 100%;
|
||
align-items: stretch;
|
||
}
|
||
|
||
.bookmark-title {
|
||
font-size: 0.85em;
|
||
overflow: hidden;
|
||
text-overflow: ellipsis;
|
||
height: 1.75em;
|
||
white-space: nowrap;
|
||
}
|
||
|
||
.bookmark-text {
|
||
display: flex;
|
||
flex-direction: column;
|
||
}
|
||
|
||
.bookmark-info {
|
||
flex: 4 1 180px;
|
||
padding: 12px 14px 14px;
|
||
display: flex;
|
||
flex-direction: column;
|
||
justify-content: space-between;
|
||
}
|
||
|
||
.bookmark-image {
|
||
width: 33%;
|
||
flex: 1 1 180px;
|
||
display: block;
|
||
position: relative;
|
||
object-fit: cover;
|
||
border-radius: 1px;
|
||
}
|
||
|
||
.bookmark-description {
|
||
color: rgba(55, 53, 47, 0.6);
|
||
font-size: 0.75em;
|
||
overflow: hidden;
|
||
max-height: 4.5em;
|
||
word-break: break-word;
|
||
}
|
||
|
||
.bookmark-href {
|
||
font-size: 0.75em;
|
||
margin-top: 0.25em;
|
||
}
|
||
|
||
.sans { font-family: ui-sans-serif, -apple-system, BlinkMacSystemFont, "Segoe UI", Helvetica, "Apple Color Emoji", Arial, sans-serif, "Segoe UI Emoji", "Segoe UI Symbol"; }
|
||
.code { font-family: "SFMono-Regular", Menlo, Consolas, "PT Mono", "Liberation Mono", Courier, monospace; }
|
||
.serif { font-family: Lyon-Text, Georgia, ui-serif, serif; }
|
||
.mono { font-family: iawriter-mono, Nitti, Menlo, Courier, monospace; }
|
||
.pdf .sans { font-family: Inter, ui-sans-serif, -apple-system, BlinkMacSystemFont, "Segoe UI", Helvetica, "Apple Color Emoji", Arial, sans-serif, "Segoe UI Emoji", "Segoe UI Symbol", 'Twemoji', 'Noto Color Emoji', 'Noto Sans CJK JP'; }
|
||
.pdf:lang(zh-CN) .sans { font-family: Inter, ui-sans-serif, -apple-system, BlinkMacSystemFont, "Segoe UI", Helvetica, "Apple Color Emoji", Arial, sans-serif, "Segoe UI Emoji", "Segoe UI Symbol", 'Twemoji', 'Noto Color Emoji', 'Noto Sans CJK SC'; }
|
||
.pdf:lang(zh-TW) .sans { font-family: Inter, ui-sans-serif, -apple-system, BlinkMacSystemFont, "Segoe UI", Helvetica, "Apple Color Emoji", Arial, sans-serif, "Segoe UI Emoji", "Segoe UI Symbol", 'Twemoji', 'Noto Color Emoji', 'Noto Sans CJK TC'; }
|
||
.pdf:lang(ko-KR) .sans { font-family: Inter, ui-sans-serif, -apple-system, BlinkMacSystemFont, "Segoe UI", Helvetica, "Apple Color Emoji", Arial, sans-serif, "Segoe UI Emoji", "Segoe UI Symbol", 'Twemoji', 'Noto Color Emoji', 'Noto Sans CJK KR'; }
|
||
.pdf .code { font-family: Source Code Pro, "SFMono-Regular", Menlo, Consolas, "PT Mono", "Liberation Mono", Courier, monospace, 'Twemoji', 'Noto Color Emoji', 'Noto Sans Mono CJK JP'; }
|
||
.pdf:lang(zh-CN) .code { font-family: Source Code Pro, "SFMono-Regular", Menlo, Consolas, "PT Mono", "Liberation Mono", Courier, monospace, 'Twemoji', 'Noto Color Emoji', 'Noto Sans Mono CJK SC'; }
|
||
.pdf:lang(zh-TW) .code { font-family: Source Code Pro, "SFMono-Regular", Menlo, Consolas, "PT Mono", "Liberation Mono", Courier, monospace, 'Twemoji', 'Noto Color Emoji', 'Noto Sans Mono CJK TC'; }
|
||
.pdf:lang(ko-KR) .code { font-family: Source Code Pro, "SFMono-Regular", Menlo, Consolas, "PT Mono", "Liberation Mono", Courier, monospace, 'Twemoji', 'Noto Color Emoji', 'Noto Sans Mono CJK KR'; }
|
||
.pdf .serif { font-family: PT Serif, Lyon-Text, Georgia, ui-serif, serif, 'Twemoji', 'Noto Color Emoji', 'Noto Serif CJK JP'; }
|
||
.pdf:lang(zh-CN) .serif { font-family: PT Serif, Lyon-Text, Georgia, ui-serif, serif, 'Twemoji', 'Noto Color Emoji', 'Noto Serif CJK SC'; }
|
||
.pdf:lang(zh-TW) .serif { font-family: PT Serif, Lyon-Text, Georgia, ui-serif, serif, 'Twemoji', 'Noto Color Emoji', 'Noto Serif CJK TC'; }
|
||
.pdf:lang(ko-KR) .serif { font-family: PT Serif, Lyon-Text, Georgia, ui-serif, serif, 'Twemoji', 'Noto Color Emoji', 'Noto Serif CJK KR'; }
|
||
.pdf .mono { font-family: PT Mono, iawriter-mono, Nitti, Menlo, Courier, monospace, 'Twemoji', 'Noto Color Emoji', 'Noto Sans Mono CJK JP'; }
|
||
.pdf:lang(zh-CN) .mono { font-family: PT Mono, iawriter-mono, Nitti, Menlo, Courier, monospace, 'Twemoji', 'Noto Color Emoji', 'Noto Sans Mono CJK SC'; }
|
||
.pdf:lang(zh-TW) .mono { font-family: PT Mono, iawriter-mono, Nitti, Menlo, Courier, monospace, 'Twemoji', 'Noto Color Emoji', 'Noto Sans Mono CJK TC'; }
|
||
.pdf:lang(ko-KR) .mono { font-family: PT Mono, iawriter-mono, Nitti, Menlo, Courier, monospace, 'Twemoji', 'Noto Color Emoji', 'Noto Sans Mono CJK KR'; }
|
||
.highlight-default {
|
||
color: rgba(55, 53, 47, 1);
|
||
}
|
||
.highlight-gray {
|
||
color: rgba(120, 119, 116, 1);
|
||
fill: rgba(120, 119, 116, 1);
|
||
}
|
||
.highlight-brown {
|
||
color: rgba(159, 107, 83, 1);
|
||
fill: rgba(159, 107, 83, 1);
|
||
}
|
||
.highlight-orange {
|
||
color: rgba(217, 115, 13, 1);
|
||
fill: rgba(217, 115, 13, 1);
|
||
}
|
||
.highlight-yellow {
|
||
color: rgba(203, 145, 47, 1);
|
||
fill: rgba(203, 145, 47, 1);
|
||
}
|
||
.highlight-teal {
|
||
color: rgba(68, 131, 97, 1);
|
||
fill: rgba(68, 131, 97, 1);
|
||
}
|
||
.highlight-blue {
|
||
color: rgba(51, 126, 169, 1);
|
||
fill: rgba(51, 126, 169, 1);
|
||
}
|
||
.highlight-purple {
|
||
color: rgba(144, 101, 176, 1);
|
||
fill: rgba(144, 101, 176, 1);
|
||
}
|
||
.highlight-pink {
|
||
color: rgba(193, 76, 138, 1);
|
||
fill: rgba(193, 76, 138, 1);
|
||
}
|
||
.highlight-red {
|
||
color: rgba(212, 76, 71, 1);
|
||
fill: rgba(212, 76, 71, 1);
|
||
}
|
||
.highlight-gray_background {
|
||
background: rgba(241, 241, 239, 1);
|
||
}
|
||
.highlight-brown_background {
|
||
background: rgba(244, 238, 238, 1);
|
||
}
|
||
.highlight-orange_background {
|
||
background: rgba(251, 236, 221, 1);
|
||
}
|
||
.highlight-yellow_background {
|
||
background: rgba(251, 243, 219, 1);
|
||
}
|
||
.highlight-teal_background {
|
||
background: rgba(237, 243, 236, 1);
|
||
}
|
||
.highlight-blue_background {
|
||
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</style></head><body><article id="c17abfc3-5a3d-4e67-83c8-356a38c7f3ed" class="page sans"><header><h1 class="page-title">Unit 30: Induction and Inductance</h1></header><div class="page-body"><h1 id="0809650e-9565-4f40-842f-e910f68e2a06" class="">30.1 - Faraday’s Law and Lenz’s Law</h1><h2 id="a271dcd2-d40c-49fd-b581-a77fe104b296" class="">Faraday’s Law of Induction</h2><p id="c24d66aa-584b-40c3-ad13-73f79c10038c" class="">Faraday's law of induction describes how an electric current produces a magnetic field and, conversely, how a changing magnetic field generates an electric current in a conductor.</p><h3 id="463f1a8e-8114-422d-95c4-d9235f724c43" class="">Magnetic Flux</h3><p id="2e667766-6645-4130-99ba-b7d51575b149" class="">To put Faraday’s law to work, we need a way to calculate the <em>amount of magnetic field</em> that passes through a loop. Suppose a loop enclosing an area <style>@import url('https://cdnjs.cloudflare.com/ajax/libs/KaTeX/0.13.2/katex.min.css')</style><span data-token-index="0" contenteditable="false" class="notion-text-equation-token" style="user-select:all;-webkit-user-select:all;-moz-user-select:all"><span></span><span><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord mathnormal">A</span></span></span></span></span><span></span></span> is placed in a magnetic field <style>@import url('https://cdnjs.cloudflare.com/ajax/libs/KaTeX/0.13.2/katex.min.css')</style><span data-token-index="0" contenteditable="false" class="notion-text-equation-token" style="user-select:all;-webkit-user-select:all;-moz-user-select:all"><span></span><span><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mover accent="true"><mi>B</mi><mo>⃗</mo></mover></mrow><annotation encoding="application/x-tex">\vec B</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9663299999999999em;vertical-align:0em;"></span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9663299999999999em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span></span><span style="top:-3.25233em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.15216em;"><span class="overlay" style="height:0.714em;width:0.471em;"><svg width='0.471em' height='0.714em' style='width:0.471em' viewBox='0 0 471 714' preserveAspectRatio='xMinYMin'><path d='M377 20c0-5.333 1.833-10 5.5-14S391 0 397 0c4.667 0 8.667 1.667 12 5
|
||
3.333 2.667 6.667 9 10 19 6.667 24.667 20.333 43.667 41 57 7.333 4.667 11
|
||
10.667 11 18 0 6-1 10-3 12s-6.667 5-14 9c-28.667 14.667-53.667 35.667-75 63
|
||
-1.333 1.333-3.167 3.5-5.5 6.5s-4 4.833-5 5.5c-1 .667-2.5 1.333-4.5 2s-4.333 1
|
||
-7 1c-4.667 0-9.167-1.833-13.5-5.5S337 184 337 178c0-12.667 15.667-32.333 47-59
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H213l-171-1c-8.667-6-13-12.333-13-19 0-4.667 4.333-11.333 13-20h359
|
||
c-16-25.333-24-45-24-59z'/></svg></span></span></span></span></span></span></span></span></span></span></span><span></span></span>. Then the <strong>magnetic flux</strong> through the loop is:</p><figure id="5a23eca1-d747-44ee-89e4-7189cde3d472" class="equation"><style>@import url('https://cdnjs.cloudflare.com/ajax/libs/KaTeX/0.13.2/katex.min.css')</style><div class="equation-container"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><msub><mi mathvariant="normal">Φ</mi><mi>B</mi></msub><mo>=</mo><mo>∫</mo><mover accent="true"><mi>B</mi><mo>⃗</mo></mover><mo>⋅</mo><mi>d</mi><mi>A</mi></mrow><annotation encoding="application/x-tex">\Phi_B = \int \vec B \cdot dA</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Φ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.32833099999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.05017em;">B</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:2.22225em;vertical-align:-0.86225em;"></span><span class="mop op-symbol large-op" style="margin-right:0.44445em;position:relative;top:-0.0011249999999999316em;">∫</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9663299999999999em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span></span><span style="top:-3.25233em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.15216em;"><span class="overlay" style="height:0.714em;width:0.471em;"><svg width='0.471em' height='0.714em' style='width:0.471em' viewBox='0 0 471 714' preserveAspectRatio='xMinYMin'><path d='M377 20c0-5.333 1.833-10 5.5-14S391 0 397 0c4.667 0 8.667 1.667 12 5
|
||
3.333 2.667 6.667 9 10 19 6.667 24.667 20.333 43.667 41 57 7.333 4.667 11
|
||
10.667 11 18 0 6-1 10-3 12s-6.667 5-14 9c-28.667 14.667-53.667 35.667-75 63
|
||
-1.333 1.333-3.167 3.5-5.5 6.5s-4 4.833-5 5.5c-1 .667-2.5 1.333-4.5 2s-4.333 1
|
||
-7 1c-4.667 0-9.167-1.833-13.5-5.5S337 184 337 178c0-12.667 15.667-32.333 47-59
|
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H213l-171-1c-8.667-6-13-12.333-13-19 0-4.667 4.333-11.333 13-20h359
|
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c-16-25.333-24-45-24-59z'/></svg></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">⋅</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.69444em;vertical-align:0em;"></span><span class="mord mathnormal">d</span><span class="mord mathnormal">A</span></span></span></span></span></div></figure><figure class="block-color-gray_background callout" style="white-space:pre-wrap;display:flex" id="ebf4ea83-eb84-4df9-8e1f-b5784c0a9f7b"><div style="font-size:1.5em"><span class="icon">💡</span></div><div style="width:100%">As with electric flux, we want the component of the field that pierces the surface (not skims along it).</div></figure><h3 id="aa255260-7c8e-4336-920c-c0459feb0ac3" class="">Uniform, Parallel Magnetic Field</h3><p id="679c47b6-19a5-4dac-bac4-cd9edaa004c0" class="">Suppose that the loop lies in a plane and that the magnetic field is perpendicular to the plane of the loop. This simplifies the above equation to:</p><figure id="10f705a8-969e-4154-bc18-67600ab0bc9b" class="equation"><style>@import url('https://cdnjs.cloudflare.com/ajax/libs/KaTeX/0.13.2/katex.min.css')</style><div class="equation-container"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><msub><mi mathvariant="normal">Φ</mi><mi>B</mi></msub><mo>=</mo><mi>B</mi><mi>A</mi><mspace width="2em"/><mo stretchy="false">(</mo><mover accent="true"><mi>B</mi><mo>⃗</mo></mover><mo>⊥</mo><mtext>area</mtext><mo separator="true">,</mo><mover accent="true"><mi>B</mi><mo>⃗</mo></mover><mtext> uniform</mtext><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\Phi_B = BA \hspace{2em} (\vec B \perp \textrm{area}, \vec B \textrm{ uniform})</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Φ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.32833099999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.05017em;">B</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1.21633em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mord mathnormal">A</span><span class="mspace" style="margin-right:2em;"></span><span class="mopen">(</span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9663299999999999em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span></span><span style="top:-3.25233em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.15216em;"><span class="overlay" style="height:0.714em;width:0.471em;"><svg width='0.471em' height='0.714em' style='width:0.471em' viewBox='0 0 471 714' preserveAspectRatio='xMinYMin'><path d='M377 20c0-5.333 1.833-10 5.5-14S391 0 397 0c4.667 0 8.667 1.667 12 5
|
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|
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10.667 11 18 0 6-1 10-3 12s-6.667 5-14 9c-28.667 14.667-53.667 35.667-75 63
|
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-1.333 1.333-3.167 3.5-5.5 6.5s-4 4.833-5 5.5c-1 .667-2.5 1.333-4.5 2s-4.333 1
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-7 1c-4.667 0-9.167-1.833-13.5-5.5S337 184 337 178c0-12.667 15.667-32.333 47-59
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H213l-171-1c-8.667-6-13-12.333-13-19 0-4.667 4.333-11.333 13-20h359
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c-16-25.333-24-45-24-59z'/></svg></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">⊥</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1.21633em;vertical-align:-0.25em;"></span><span class="mord text"><span class="mord textrm">area</span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9663299999999999em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span></span><span style="top:-3.25233em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.15216em;"><span class="overlay" style="height:0.714em;width:0.471em;"><svg width='0.471em' height='0.714em' style='width:0.471em' viewBox='0 0 471 714' preserveAspectRatio='xMinYMin'><path d='M377 20c0-5.333 1.833-10 5.5-14S391 0 397 0c4.667 0 8.667 1.667 12 5
|
||
3.333 2.667 6.667 9 10 19 6.667 24.667 20.333 43.667 41 57 7.333 4.667 11
|
||
10.667 11 18 0 6-1 10-3 12s-6.667 5-14 9c-28.667 14.667-53.667 35.667-75 63
|
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-1.333 1.333-3.167 3.5-5.5 6.5s-4 4.833-5 5.5c-1 .667-2.5 1.333-4.5 2s-4.333 1
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-7 1c-4.667 0-9.167-1.833-13.5-5.5S337 184 337 178c0-12.667 15.667-32.333 47-59
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H213l-171-1c-8.667-6-13-12.333-13-19 0-4.667 4.333-11.333 13-20h359
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c-16-25.333-24-45-24-59z'/></svg></span></span></span></span></span></span></span><span class="mord text"><span class="mord textrm"> uniform</span></span><span class="mclose">)</span></span></span></span></span></div></figure><p id="c219b5c0-9700-4881-8ddb-8af1b3f2e3ea" class="">Magnetic flux is measured in <strong>Webers:</strong></p><figure id="b79c3803-f744-4770-97d6-81f54ea1bec7" class="equation"><style>@import url('https://cdnjs.cloudflare.com/ajax/libs/KaTeX/0.13.2/katex.min.css')</style><div class="equation-container"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mn>1</mn><mtext> weber</mtext><mo>=</mo><mn>1</mn><mtext> Wb</mtext><mo>=</mo><mn>1</mn><mtext> T</mtext><mo>⋅</mo><msup><mtext>m</mtext><mn>2</mn></msup></mrow><annotation encoding="application/x-tex">1 \space \textrm{weber} = 1 \space \textrm{Wb} = 1 \space \textrm T \cdot \textrm m^2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.69444em;vertical-align:0em;"></span><span class="mord">1</span><span class="mspace"> </span><span class="mord text"><span class="mord textrm">weber</span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.69444em;vertical-align:0em;"></span><span class="mord">1</span><span class="mspace"> </span><span class="mord text"><span class="mord textrm">Wb</span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord">1</span><span class="mspace"> </span><span class="mord text"><span class="mord textrm">T</span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">⋅</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.8641079999999999em;vertical-align:0em;"></span><span class="mord"><span class="mord text"><span class="mord textrm">m</span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641079999999999em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span></span></div></figure><h3 id="39fdfc4c-2e20-4d79-8ca7-4a82328817f7" class="">Faraday’s Law</h3><p id="1de265ec-025a-43f9-8ec8-8967eae40ffa" class="">The magnitude of the emf <style>@import url('https://cdnjs.cloudflare.com/ajax/libs/KaTeX/0.13.2/katex.min.css')</style><span data-token-index="0" contenteditable="false" class="notion-text-equation-token" style="user-select:all;-webkit-user-select:all;-moz-user-select:all"><span></span><span><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="script">E</mi></mrow><annotation encoding="application/x-tex">\mathscr E</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7em;vertical-align:0em;"></span><span class="mord mathscr" style="margin-right:0.18583em;">E</span></span></span></span></span><span></span></span> induced in a conducting loop is equal to the rate at which the magnetic flux <style>@import url('https://cdnjs.cloudflare.com/ajax/libs/KaTeX/0.13.2/katex.min.css')</style><span data-token-index="0" contenteditable="false" class="notion-text-equation-token" style="user-select:all;-webkit-user-select:all;-moz-user-select:all"><span></span><span><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mover accent="true"><mi>B</mi><mo>⃗</mo></mover></mrow><annotation encoding="application/x-tex">\vec B</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9663299999999999em;vertical-align:0em;"></span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9663299999999999em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span></span><span style="top:-3.25233em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.15216em;"><span class="overlay" style="height:0.714em;width:0.471em;"><svg width='0.471em' height='0.714em' style='width:0.471em' viewBox='0 0 471 714' preserveAspectRatio='xMinYMin'><path d='M377 20c0-5.333 1.833-10 5.5-14S391 0 397 0c4.667 0 8.667 1.667 12 5
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-1.333 1.333-3.167 3.5-5.5 6.5s-4 4.833-5 5.5c-1 .667-2.5 1.333-4.5 2s-4.333 1
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-7 1c-4.667 0-9.167-1.833-13.5-5.5S337 184 337 178c0-12.667 15.667-32.333 47-59
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c-16-25.333-24-45-24-59z'/></svg></span></span></span></span></span></span></span></span></span></span></span><span></span></span> through that loop changes with time.</p><figure id="0ed27dbe-14f4-463f-bc24-1894b84b194e" class="equation"><style>@import url('https://cdnjs.cloudflare.com/ajax/libs/KaTeX/0.13.2/katex.min.css')</style><div class="equation-container"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi mathvariant="script">E</mi><mo>=</mo><mo>−</mo><mfrac><mrow><mi>d</mi><msub><mi mathvariant="normal">Φ</mi><mi>B</mi></msub></mrow><mrow><mi>d</mi><mi>t</mi></mrow></mfrac></mrow><annotation encoding="application/x-tex">\mathscr E = -\frac{d\Phi_B}{dt}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7em;vertical-align:0em;"></span><span class="mord mathscr" style="margin-right:0.18583em;">E</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:2.05744em;vertical-align:-0.686em;"></span><span class="mord">−</span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.37144em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">d</span><span class="mord mathnormal">t</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">d</span><span class="mord"><span class="mord">Φ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.32833099999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.05017em;">B</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></div></figure><p id="a31f1361-9ac8-4dc5-acd4-8bb07a05e298" class="">If we change the magnetic flux through a coil of <style>@import url('https://cdnjs.cloudflare.com/ajax/libs/KaTeX/0.13.2/katex.min.css')</style><span data-token-index="0" contenteditable="false" class="notion-text-equation-token" style="user-select:all;-webkit-user-select:all;-moz-user-select:all"><span></span><span><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>N</mi></mrow><annotation encoding="application/x-tex">N</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord mathnormal" style="margin-right:0.10903em;">N</span></span></span></span></span><span></span></span> turns, an induced emf appears in every turn and the total emf induced in the coil is the sum of these individual induced emfs. If the coil is tightly wound (closely packed), so that the same magnetic flux <style>@import url('https://cdnjs.cloudflare.com/ajax/libs/KaTeX/0.13.2/katex.min.css')</style><span data-token-index="0" contenteditable="false" class="notion-text-equation-token" style="user-select:all;-webkit-user-select:all;-moz-user-select:all"><span></span><span><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi mathvariant="normal">Φ</mi><mi>B</mi></msub></mrow><annotation encoding="application/x-tex">\Phi_B</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Φ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.32833099999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.05017em;">B</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></span><span></span></span> passes through all the turns, the total emf induced in the coil is:</p><figure id="dda5850f-b8c2-4883-8737-40d5f48974df" class="equation"><style>@import url('https://cdnjs.cloudflare.com/ajax/libs/KaTeX/0.13.2/katex.min.css')</style><div class="equation-container"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi mathvariant="script">E</mi><mo>=</mo><mo>−</mo><mi>N</mi><mfrac><mrow><mi>d</mi><msub><mi mathvariant="normal">Φ</mi><mi>B</mi></msub></mrow><mrow><mi>d</mi><mi>t</mi></mrow></mfrac><mspace width="2em"/><mo stretchy="false">(</mo><mtext>coil of </mtext><mi>N</mi><mtext> turns</mtext><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathscr E = -N \frac{d\Phi_B}{dt} \hspace{2em} (\textrm{coil of } N \textrm{ turns})</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7em;vertical-align:0em;"></span><span class="mord mathscr" style="margin-right:0.18583em;">E</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:2.05744em;vertical-align:-0.686em;"></span><span class="mord">−</span><span class="mord mathnormal" style="margin-right:0.10903em;">N</span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.37144em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">d</span><span class="mord mathnormal">t</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">d</span><span class="mord"><span class="mord">Φ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.32833099999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.05017em;">B</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:2em;"></span><span class="mopen">(</span><span class="mord text"><span class="mord textrm">coil of </span></span><span class="mord mathnormal" style="margin-right:0.10903em;">N</span><span class="mord text"><span class="mord textrm"> turns</span></span><span class="mclose">)</span></span></span></span></span></div></figure><h2 id="fdd82fa5-7b75-469b-b261-deab57d35053" class="">Lenz’s Law</h2><p id="72585f5f-f8e8-4813-877b-f929f0a09293" class="">Soon after Faraday proposed his law of induction, Heinrich Friedrich Lenz devised a rule for determining the direction of an induced current in a loop:</p><figure class="block-color-gray_background callout" style="white-space:pre-wrap;display:flex" id="cbd8d432-9d75-408b-ba49-a86d2c85a98d"><div style="font-size:1.5em"><span class="icon">⭐</span></div><div style="width:100%">An induced current has a direction such that the magnetic field due to <em>the current</em> opposes the change in the magnetic flux that induces the current.</div></figure><h1 id="215eb239-c4c4-4323-86af-0c2c7a47ca4a" class="">30.2 - Induction and Energy Transfers</h1><p id="c3931004-1058-4b46-9210-47a28269c28e" class="">The induction of a current by a changing flux means that energy is being transferred to that current. The energy can then be transferred to other forms, such as thermal energy.</p><h2 id="545a400c-20b2-42c6-97c1-3d8af15cdb27" class="">Eddy Currents</h2><figure id="e8dcb5bf-7170-4331-9497-7947b8615e63" class="image"><a href="Unit%2030%20Induction%20and%20Inductance%20c17abfc35a3d4e6783c8356a38c7f3ed/Untitled.png"><img style="width:288px" src="Unit%2030%20Induction%20and%20Inductance%20c17abfc35a3d4e6783c8356a38c7f3ed/Untitled.png"/></a></figure><p id="5f4f87e0-b13d-45b2-ad24-ea0cc7f8bef8" class="">Suppose we move a solid conducting plate out of a magnetic field, as shown above. The relative motion of the field and the conductor again induces a current in the conductor. Thus, we again encounter an opposing force and must do work because of the induced current. With the plate, however, the conduction electrons making up the induced current do not follow one path as they do with the loop. Instead, the electrons swirl about within the plate as if they were caught in an eddy (whirlpool) of water. Such a current is called an <strong>eddy current</strong>, and can be represented <em>as if</em> it followed a single path as shown above.</p><h1 id="aeb26302-c4ea-423e-bebd-7f02a3352141" class="">30.3 - Induced Electric Fields</h1><p id="597145e4-3bf6-4f6b-86c3-64d7fbbcb180" class="">An emf is induced by a changing magnetic flux even if the loop through which the flux is changing is not a physical conductor but an imaginary line. The changing magnetic field induces an electric field <style>@import url('https://cdnjs.cloudflare.com/ajax/libs/KaTeX/0.13.2/katex.min.css')</style><span data-token-index="0" contenteditable="false" class="notion-text-equation-token" style="user-select:all;-webkit-user-select:all;-moz-user-select:all"><span></span><span><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mover accent="true"><mi>E</mi><mo>⃗</mo></mover></mrow><annotation encoding="application/x-tex">\vec E</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9663299999999999em;vertical-align:0em;"></span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9663299999999999em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathnormal" style="margin-right:0.05764em;">E</span></span><span style="top:-3.25233em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.15216em;"><span class="overlay" style="height:0.714em;width:0.471em;"><svg width='0.471em' height='0.714em' style='width:0.471em' viewBox='0 0 471 714' preserveAspectRatio='xMinYMin'><path d='M377 20c0-5.333 1.833-10 5.5-14S391 0 397 0c4.667 0 8.667 1.667 12 5
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c-16-25.333-24-45-24-59z'/></svg></span></span></span></span></span></span></span></span></span></span></span><span></span></span> at every point of such a loop; the induced emf is related to <style>@import url('https://cdnjs.cloudflare.com/ajax/libs/KaTeX/0.13.2/katex.min.css')</style><span data-token-index="0" contenteditable="false" class="notion-text-equation-token" style="user-select:all;-webkit-user-select:all;-moz-user-select:all"><span></span><span><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mover accent="true"><mi>E</mi><mo>⃗</mo></mover></mrow><annotation encoding="application/x-tex">\vec E</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9663299999999999em;vertical-align:0em;"></span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9663299999999999em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathnormal" style="margin-right:0.05764em;">E</span></span><span style="top:-3.25233em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.15216em;"><span class="overlay" style="height:0.714em;width:0.471em;"><svg width='0.471em' height='0.714em' style='width:0.471em' viewBox='0 0 471 714' preserveAspectRatio='xMinYMin'><path d='M377 20c0-5.333 1.833-10 5.5-14S391 0 397 0c4.667 0 8.667 1.667 12 5
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3.333 2.667 6.667 9 10 19 6.667 24.667 20.333 43.667 41 57 7.333 4.667 11
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10.667 11 18 0 6-1 10-3 12s-6.667 5-14 9c-28.667 14.667-53.667 35.667-75 63
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-1.333 1.333-3.167 3.5-5.5 6.5s-4 4.833-5 5.5c-1 .667-2.5 1.333-4.5 2s-4.333 1
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-7 1c-4.667 0-9.167-1.833-13.5-5.5S337 184 337 178c0-12.667 15.667-32.333 47-59
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c-16-25.333-24-45-24-59z'/></svg></span></span></span></span></span></span></span></span></span></span></span><span></span></span> by:</p><figure id="7ee119a8-1b44-4a9d-b3de-3a56d0ee5c09" class="equation"><style>@import url('https://cdnjs.cloudflare.com/ajax/libs/KaTeX/0.13.2/katex.min.css')</style><div class="equation-container"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi mathvariant="script">E</mi><mo>=</mo><mo>∮</mo><mover accent="true"><mi>E</mi><mo>⃗</mo></mover><mo>⋅</mo><mi>d</mi><mover accent="true"><mi>s</mi><mo>⃗</mo></mover></mrow><annotation encoding="application/x-tex">\mathscr{E} = \oint \vec E \cdot d\vec s</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7em;vertical-align:0em;"></span><span class="mord mathscr" style="margin-right:0.18583em;">E</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:2.22225em;vertical-align:-0.86225em;"></span><span class="mop op-symbol large-op" style="margin-right:0.44445em;position:relative;top:-0.0011249999999999316em;">∮</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9663299999999999em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathnormal" style="margin-right:0.05764em;">E</span></span><span style="top:-3.25233em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.15216em;"><span class="overlay" style="height:0.714em;width:0.471em;"><svg width='0.471em' height='0.714em' style='width:0.471em' viewBox='0 0 471 714' preserveAspectRatio='xMinYMin'><path d='M377 20c0-5.333 1.833-10 5.5-14S391 0 397 0c4.667 0 8.667 1.667 12 5
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3.333 2.667 6.667 9 10 19 6.667 24.667 20.333 43.667 41 57 7.333 4.667 11
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10.667 11 18 0 6-1 10-3 12s-6.667 5-14 9c-28.667 14.667-53.667 35.667-75 63
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-1.333 1.333-3.167 3.5-5.5 6.5s-4 4.833-5 5.5c-1 .667-2.5 1.333-4.5 2s-4.333 1
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-7 1c-4.667 0-9.167-1.833-13.5-5.5S337 184 337 178c0-12.667 15.667-32.333 47-59
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H213l-171-1c-8.667-6-13-12.333-13-19 0-4.667 4.333-11.333 13-20h359
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c-16-25.333-24-45-24-59z'/></svg></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">⋅</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.714em;vertical-align:0em;"></span><span class="mord mathnormal">d</span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.714em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathnormal">s</span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.17994em;"><span class="overlay" style="height:0.714em;width:0.471em;"><svg width='0.471em' height='0.714em' style='width:0.471em' viewBox='0 0 471 714' preserveAspectRatio='xMinYMin'><path d='M377 20c0-5.333 1.833-10 5.5-14S391 0 397 0c4.667 0 8.667 1.667 12 5
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3.333 2.667 6.667 9 10 19 6.667 24.667 20.333 43.667 41 57 7.333 4.667 11
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10.667 11 18 0 6-1 10-3 12s-6.667 5-14 9c-28.667 14.667-53.667 35.667-75 63
|
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-1.333 1.333-3.167 3.5-5.5 6.5s-4 4.833-5 5.5c-1 .667-2.5 1.333-4.5 2s-4.333 1
|
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-7 1c-4.667 0-9.167-1.833-13.5-5.5S337 184 337 178c0-12.667 15.667-32.333 47-59
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H213l-171-1c-8.667-6-13-12.333-13-19 0-4.667 4.333-11.333 13-20h359
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c-16-25.333-24-45-24-59z'/></svg></span></span></span></span></span></span></span></span></span></span></span></div></figure><p id="708933f4-998c-4514-bb34-cf988e85827c" class="">Using the induced electric field, we can write Faraday’s law in its most general form as:</p><figure id="40bb3778-483b-4c4f-81a3-8207e8185a81" class="equation"><style>@import url('https://cdnjs.cloudflare.com/ajax/libs/KaTeX/0.13.2/katex.min.css')</style><div class="equation-container"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mo>∮</mo><mover accent="true"><mi>E</mi><mo>⃗</mo></mover><mo>⋅</mo><mi>d</mi><mover accent="true"><mi>s</mi><mo>⃗</mo></mover><mo>=</mo><mo>−</mo><mfrac><mrow><mi>d</mi><msub><mi mathvariant="normal">Φ</mi><mi>b</mi></msub></mrow><mrow><mi>d</mi><mi>t</mi></mrow></mfrac></mrow><annotation encoding="application/x-tex">\oint \vec E \cdot d\vec s = -\frac{d\Phi_b}{dt}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.22225em;vertical-align:-0.86225em;"></span><span class="mop op-symbol large-op" style="margin-right:0.44445em;position:relative;top:-0.0011249999999999316em;">∮</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9663299999999999em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathnormal" style="margin-right:0.05764em;">E</span></span><span style="top:-3.25233em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.15216em;"><span class="overlay" style="height:0.714em;width:0.471em;"><svg width='0.471em' height='0.714em' style='width:0.471em' viewBox='0 0 471 714' preserveAspectRatio='xMinYMin'><path d='M377 20c0-5.333 1.833-10 5.5-14S391 0 397 0c4.667 0 8.667 1.667 12 5
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3.333 2.667 6.667 9 10 19 6.667 24.667 20.333 43.667 41 57 7.333 4.667 11
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10.667 11 18 0 6-1 10-3 12s-6.667 5-14 9c-28.667 14.667-53.667 35.667-75 63
|
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-1.333 1.333-3.167 3.5-5.5 6.5s-4 4.833-5 5.5c-1 .667-2.5 1.333-4.5 2s-4.333 1
|
||
-7 1c-4.667 0-9.167-1.833-13.5-5.5S337 184 337 178c0-12.667 15.667-32.333 47-59
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H213l-171-1c-8.667-6-13-12.333-13-19 0-4.667 4.333-11.333 13-20h359
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c-16-25.333-24-45-24-59z'/></svg></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">⋅</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.714em;vertical-align:0em;"></span><span class="mord mathnormal">d</span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.714em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathnormal">s</span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.17994em;"><span class="overlay" style="height:0.714em;width:0.471em;"><svg width='0.471em' height='0.714em' style='width:0.471em' viewBox='0 0 471 714' preserveAspectRatio='xMinYMin'><path d='M377 20c0-5.333 1.833-10 5.5-14S391 0 397 0c4.667 0 8.667 1.667 12 5
|
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3.333 2.667 6.667 9 10 19 6.667 24.667 20.333 43.667 41 57 7.333 4.667 11
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10.667 11 18 0 6-1 10-3 12s-6.667 5-14 9c-28.667 14.667-53.667 35.667-75 63
|
||
-1.333 1.333-3.167 3.5-5.5 6.5s-4 4.833-5 5.5c-1 .667-2.5 1.333-4.5 2s-4.333 1
|
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-7 1c-4.667 0-9.167-1.833-13.5-5.5S337 184 337 178c0-12.667 15.667-32.333 47-59
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H213l-171-1c-8.667-6-13-12.333-13-19 0-4.667 4.333-11.333 13-20h359
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c-16-25.333-24-45-24-59z'/></svg></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:2.05744em;vertical-align:-0.686em;"></span><span class="mord">−</span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.37144em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">d</span><span class="mord mathnormal">t</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">d</span><span class="mord"><span class="mord">Φ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">b</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></div></figure><h1 id="e4411540-d1f1-4e00-9427-9709242b05ed" class="">30.4 - Inductors and Inductance</h1><p id="c92eb858-ef55-4682-bab1-730114e5e76d" class="">An <strong>inductor</strong> is a device (coil, toroid, or solenoid) that can be used to produce a known magnetic field in a specified region. If a current <style>@import url('https://cdnjs.cloudflare.com/ajax/libs/KaTeX/0.13.2/katex.min.css')</style><span data-token-index="0" contenteditable="false" class="notion-text-equation-token" style="user-select:all;-webkit-user-select:all;-moz-user-select:all"><span></span><span><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>i</mi></mrow><annotation encoding="application/x-tex">i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.65952em;vertical-align:0em;"></span><span class="mord mathnormal">i</span></span></span></span></span><span></span></span> is established through each of the <style>@import url('https://cdnjs.cloudflare.com/ajax/libs/KaTeX/0.13.2/katex.min.css')</style><span data-token-index="0" contenteditable="false" class="notion-text-equation-token" style="user-select:all;-webkit-user-select:all;-moz-user-select:all"><span></span><span><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>N</mi></mrow><annotation encoding="application/x-tex">N</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord mathnormal" style="margin-right:0.10903em;">N</span></span></span></span></span><span></span></span> windings of an inductor, a magnetic flux <style>@import url('https://cdnjs.cloudflare.com/ajax/libs/KaTeX/0.13.2/katex.min.css')</style><span data-token-index="0" contenteditable="false" class="notion-text-equation-token" style="user-select:all;-webkit-user-select:all;-moz-user-select:all"><span></span><span><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi mathvariant="normal">Φ</mi><mi>B</mi></msub></mrow><annotation encoding="application/x-tex">\Phi_B</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">Φ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.32833099999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.05017em;">B</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></span><span></span></span> links those windings. The inductance <style>@import url('https://cdnjs.cloudflare.com/ajax/libs/KaTeX/0.13.2/katex.min.css')</style><span data-token-index="0" contenteditable="false" class="notion-text-equation-token" style="user-select:all;-webkit-user-select:all;-moz-user-select:all"><span></span><span><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>L</mi></mrow><annotation encoding="application/x-tex">L</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord mathnormal">L</span></span></span></span></span><span></span></span> of the inductor is:</p><figure id="68d73a96-19c2-4d16-b9ab-3cf07d2ff3ca" class="equation"><style>@import url('https://cdnjs.cloudflare.com/ajax/libs/KaTeX/0.13.2/katex.min.css')</style><div class="equation-container"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>L</mi><mo>=</mo><mfrac><mrow><mi>N</mi><msub><mi mathvariant="normal">Φ</mi><mi>B</mi></msub></mrow><mi>i</mi></mfrac></mrow><annotation encoding="application/x-tex">L = \frac{N\Phi_B}{i}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord mathnormal">L</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:2.04633em;vertical-align:-0.686em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.36033em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">i</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.10903em;">N</span><span class="mord"><span class="mord">Φ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.32833099999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.05017em;">B</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></div></figure><p id="4699a5e3-6f33-4aca-95df-4fa70ac10477" class="">The SI unit of inductance is the henry (H), where:</p><figure id="3f272775-b209-4257-a602-a8743246ec71" class="equation"><style>@import url('https://cdnjs.cloudflare.com/ajax/libs/KaTeX/0.13.2/katex.min.css')</style><div class="equation-container"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mn>1</mn><mtext> henry</mtext><mo>=</mo><mn>1</mn><mtext> H</mtext><mo>=</mo><mn>1</mn><mtext> T</mtext><mo>⋅</mo><msup><mtext>m</mtext><mn>2</mn></msup><mi mathvariant="normal">/</mi><mtext>A</mtext></mrow><annotation encoding="application/x-tex">1 \space \textrm{henry} = 1 \space \textrm{H} = 1 \space \textrm{T} \cdot \textrm m ^2 / \textrm A</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8888799999999999em;vertical-align:-0.19444em;"></span><span class="mord">1</span><span class="mspace"> </span><span class="mord text"><span class="mord textrm">henry</span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord">1</span><span class="mspace"> </span><span class="mord text"><span class="mord textrm">H</span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord">1</span><span class="mspace"> </span><span class="mord text"><span class="mord textrm">T</span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">⋅</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:1.1141079999999999em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord text"><span class="mord textrm">m</span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641079999999999em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mord">/</span><span class="mord text"><span class="mord textrm">A</span></span></span></span></span></span></div></figure><p id="29e4c4b4-91f4-4493-aa68-0d7169f1bf4e" class="">The inductance per unit length near the middle of a long solenoid of cross-sectional area <style>@import url('https://cdnjs.cloudflare.com/ajax/libs/KaTeX/0.13.2/katex.min.css')</style><span data-token-index="0" contenteditable="false" class="notion-text-equation-token" style="user-select:all;-webkit-user-select:all;-moz-user-select:all"><span></span><span><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord mathnormal">A</span></span></span></span></span><span></span></span> and <style>@import url('https://cdnjs.cloudflare.com/ajax/libs/KaTeX/0.13.2/katex.min.css')</style><span data-token-index="0" contenteditable="false" class="notion-text-equation-token" style="user-select:all;-webkit-user-select:all;-moz-user-select:all"><span></span><span><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord mathnormal">n</span></span></span></span></span><span></span></span> turns per unit length is:</p><figure id="0a87284e-9727-4f2f-b2dc-e61247f3f5b4" class="equation"><style>@import url('https://cdnjs.cloudflare.com/ajax/libs/KaTeX/0.13.2/katex.min.css')</style><div class="equation-container"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mfrac><mi>L</mi><mi>l</mi></mfrac><mo>=</mo><msub><mi>μ</mi><mn>0</mn></msub><msup><mi>n</mi><mn>2</mn></msup><mi>A</mi></mrow><annotation encoding="application/x-tex">\frac{L}{l} = \mu_0n^2A</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.04633em;vertical-align:-0.686em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.36033em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.01968em;">l</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">L</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1.0585479999999998em;vertical-align:-0.19444em;"></span><span class="mord"><span class="mord mathnormal">μ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal">n</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641079999999999em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mord mathnormal">A</span></span></span></span></span></div></figure><h1 id="f8310f8a-c9df-4ead-a0d2-a02c472b0e39" class="">30.5 - Self-Induction</h1><p id="94169587-ef2b-44c0-b2f4-496066563e38" class="">If a current <style>@import url('https://cdnjs.cloudflare.com/ajax/libs/KaTeX/0.13.2/katex.min.css')</style><span data-token-index="0" contenteditable="false" class="notion-text-equation-token" style="user-select:all;-webkit-user-select:all;-moz-user-select:all"><span></span><span><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>i</mi></mrow><annotation encoding="application/x-tex">i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.65952em;vertical-align:0em;"></span><span class="mord mathnormal">i</span></span></span></span></span><span></span></span> in a coil changes with time, an emf is induced in the coil. This self-induced emf is:</p><figure id="1a154414-85f1-4e5c-bf7d-a44bc60fd20f" class="equation"><style>@import url('https://cdnjs.cloudflare.com/ajax/libs/KaTeX/0.13.2/katex.min.css')</style><div class="equation-container"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><msub><mi mathvariant="script">E</mi><mi>L</mi></msub><mo>=</mo><mo>−</mo><mi>L</mi><mfrac><mrow><mi>d</mi><mi>i</mi></mrow><mrow><mi>d</mi><mi>t</mi></mrow></mfrac></mrow><annotation encoding="application/x-tex">\mathscr E_L = -L\frac{di}{dt}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.85em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathscr" style="margin-right:0.18583em;">E</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.32833099999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.18583em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">L</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:2.05744em;vertical-align:-0.686em;"></span><span class="mord">−</span><span class="mord mathnormal">L</span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.37144em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">d</span><span class="mord mathnormal">t</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">d</span><span class="mord mathnormal">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></div></figure><p id="c9948fbc-134c-41a8-87ff-32623e5a4b14" class="">The direction of <style>@import url('https://cdnjs.cloudflare.com/ajax/libs/KaTeX/0.13.2/katex.min.css')</style><span data-token-index="0" contenteditable="false" class="notion-text-equation-token" style="user-select:all;-webkit-user-select:all;-moz-user-select:all"><span></span><span><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi mathvariant="script">E</mi><mi>L</mi></msub></mrow><annotation encoding="application/x-tex">\mathscr E_L</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.85em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathscr" style="margin-right:0.18583em;">E</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.32833099999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.18583em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">L</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></span><span></span></span> is found from Lenz’s law: The self-induced emf acts to oppose the charge that produces it. </p><h1 id="c9c5d1ae-fd26-4e7c-9cb3-cb78b31d9916" class="">30.6 - RL Circuits</h1><p id="6fd52c98-e073-4a37-a51b-b974287e1f98" class="">If a constant emf <style>@import url('https://cdnjs.cloudflare.com/ajax/libs/KaTeX/0.13.2/katex.min.css')</style><span data-token-index="0" contenteditable="false" class="notion-text-equation-token" style="user-select:all;-webkit-user-select:all;-moz-user-select:all"><span></span><span><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="script">E</mi></mrow><annotation encoding="application/x-tex">\mathscr E</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7em;vertical-align:0em;"></span><span class="mord mathscr" style="margin-right:0.18583em;">E</span></span></span></span></span><span></span></span> is introduced into a single-loop circuit containing a resistance <style>@import url('https://cdnjs.cloudflare.com/ajax/libs/KaTeX/0.13.2/katex.min.css')</style><span data-token-index="0" contenteditable="false" class="notion-text-equation-token" style="user-select:all;-webkit-user-select:all;-moz-user-select:all"><span></span><span><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord mathnormal" style="margin-right:0.00773em;">R</span></span></span></span></span><span></span></span> and an inductance <style>@import url('https://cdnjs.cloudflare.com/ajax/libs/KaTeX/0.13.2/katex.min.css')</style><span data-token-index="0" contenteditable="false" class="notion-text-equation-token" style="user-select:all;-webkit-user-select:all;-moz-user-select:all"><span></span><span><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>L</mi></mrow><annotation encoding="application/x-tex">L</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord mathnormal">L</span></span></span></span></span><span></span></span>, the current rises to an equilibrium value of <style>@import url('https://cdnjs.cloudflare.com/ajax/libs/KaTeX/0.13.2/katex.min.css')</style><span data-token-index="0" contenteditable="false" class="notion-text-equation-token" style="user-select:all;-webkit-user-select:all;-moz-user-select:all"><span></span><span><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="script">E</mi><mi mathvariant="normal">/</mi><mi>R</mi></mrow><annotation encoding="application/x-tex">\mathscr E / R</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathscr" style="margin-right:0.18583em;">E</span><span class="mord">/</span><span class="mord mathnormal" style="margin-right:0.00773em;">R</span></span></span></span></span><span></span></span> according to:</p><figure id="8fd33153-708b-420e-8714-b7a329a0ca1c" class="equation"><style>@import url('https://cdnjs.cloudflare.com/ajax/libs/KaTeX/0.13.2/katex.min.css')</style><div class="equation-container"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>i</mi><mo>=</mo><mfrac><mi mathvariant="script">E</mi><mi>R</mi></mfrac><mo stretchy="false">(</mo><mn>1</mn><mo>−</mo><msup><mi>e</mi><mrow><mo>−</mo><mi>t</mi><mi mathvariant="normal">/</mi><msub><mi>τ</mi><mi>L</mi></msub></mrow></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">i = \frac{\mathscr E}{R}(1-e^{-t/\tau_L})</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.65952em;vertical-align:0em;"></span><span class="mord mathnormal">i</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:2.063em;vertical-align:-0.686em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.377em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.00773em;">R</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathscr" style="margin-right:0.18583em;">E</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mopen">(</span><span class="mord">1</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:1.188em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal">e</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.938em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mathnormal mtight">t</span><span class="mord mtight">/</span><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.1132em;">τ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3448em;"><span style="top:-2.3567071428571427em;margin-left:-0.1132em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mathnormal mtight">L</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.14329285714285717em;"><span></span></span></span></span></span></span></span></span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span></span></div></figure><p id="d049e65c-1f0c-46c8-a7ad-2b603494fb44" class="">Here <style>@import url('https://cdnjs.cloudflare.com/ajax/libs/KaTeX/0.13.2/katex.min.css')</style><span data-token-index="0" contenteditable="false" class="notion-text-equation-token" style="user-select:all;-webkit-user-select:all;-moz-user-select:all"><span></span><span><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>τ</mi><mi>L</mi></msub><mo stretchy="false">(</mo><mo>=</mo><mi>L</mi><mi mathvariant="normal">/</mi><mi>R</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\tau_L (=L/R)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.1132em;">τ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.32833099999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.1132em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">L</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">L</span><span class="mord">/</span><span class="mord mathnormal" style="margin-right:0.00773em;">R</span><span class="mclose">)</span></span></span></span></span><span></span></span> governs the rate of rise of the current and is called the inductive time constant of the circuit. </p><p id="698460c2-c936-4f6f-9c14-07aec9cdd83a" class="">When the course of constant emf is removed, the current decays from a value <style>@import url('https://cdnjs.cloudflare.com/ajax/libs/KaTeX/0.13.2/katex.min.css')</style><span data-token-index="0" contenteditable="false" class="notion-text-equation-token" style="user-select:all;-webkit-user-select:all;-moz-user-select:all"><span></span><span><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>i</mi><mn>0</mn></msub></mrow><annotation encoding="application/x-tex">i_0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.80952em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">i</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></span><span></span></span> according to:</p><figure id="5977066c-ef21-4e5f-bbce-e19deb8d0abd" class="equation"><style>@import url('https://cdnjs.cloudflare.com/ajax/libs/KaTeX/0.13.2/katex.min.css')</style><div class="equation-container"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>i</mi><mo>=</mo><msub><mi>i</mi><mn>0</mn></msub><msup><mi>e</mi><mrow><mo>−</mo><mi>t</mi><mi mathvariant="normal">/</mi><msub><mi>τ</mi><mi>L</mi></msub></mrow></msup></mrow><annotation encoding="application/x-tex">i = i_0 e^{-t/\tau_L}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.65952em;vertical-align:0em;"></span><span class="mord mathnormal">i</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1.0879999999999999em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">i</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal">e</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.938em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mathnormal mtight">t</span><span class="mord mtight">/</span><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.1132em;">τ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3448em;"><span style="top:-2.3567071428571427em;margin-left:-0.1132em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mathnormal mtight">L</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.14329285714285717em;"><span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></div></figure><h1 id="2426ce92-2090-4478-9ea2-671c911063d6" class="">30.7 - Energy Stored in a Magnetic Field</h1><p id="a1c3d154-6e5f-4718-88a7-1549aecfa8ae" class="">If an inductor <style>@import url('https://cdnjs.cloudflare.com/ajax/libs/KaTeX/0.13.2/katex.min.css')</style><span data-token-index="0" contenteditable="false" class="notion-text-equation-token" style="user-select:all;-webkit-user-select:all;-moz-user-select:all"><span></span><span><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>L</mi></mrow><annotation encoding="application/x-tex">L</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord mathnormal">L</span></span></span></span></span><span></span></span> carries a current <style>@import url('https://cdnjs.cloudflare.com/ajax/libs/KaTeX/0.13.2/katex.min.css')</style><span data-token-index="0" contenteditable="false" class="notion-text-equation-token" style="user-select:all;-webkit-user-select:all;-moz-user-select:all"><span></span><span><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>i</mi></mrow><annotation encoding="application/x-tex">i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.65952em;vertical-align:0em;"></span><span class="mord mathnormal">i</span></span></span></span></span><span></span></span>, the inductor’s magnetic field stores an energy given by:</p><figure id="78232644-aba5-4906-8e29-3d8e7ca82dd9" class="equation"><style>@import url('https://cdnjs.cloudflare.com/ajax/libs/KaTeX/0.13.2/katex.min.css')</style><div class="equation-container"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><msub><mi>U</mi><mi>B</mi></msub><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mi>L</mi><msup><mi>i</mi><mn>2</mn></msup></mrow><annotation encoding="application/x-tex">U_B = \frac{1}{2} Li^2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.10903em;">U</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.32833099999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.10903em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.05017em;">B</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:2.00744em;vertical-align:-0.686em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.32144em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">2</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mord mathnormal">L</span><span class="mord"><span class="mord mathnormal">i</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641079999999999em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span></span></div></figure><h1 id="69cfef57-f4ad-4d85-937b-28ed19f24ffa" class="">30.8 - Energy Density of a Magnetic Field</h1><p id="c37ff498-e297-4604-8659-6e32b42c1e28" class="">If <style>@import url('https://cdnjs.cloudflare.com/ajax/libs/KaTeX/0.13.2/katex.min.css')</style><span data-token-index="0" contenteditable="false" class="notion-text-equation-token" style="user-select:all;-webkit-user-select:all;-moz-user-select:all"><span></span><span><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span></span></span></span></span><span></span></span> is the magnitude of a magnetic field at any point (in an inductor or anywhere else), the density of stored magnetic energy at that point is:</p><figure id="5270495f-5f9b-42b2-829a-58dc06bccd94" class="equation"><style>@import url('https://cdnjs.cloudflare.com/ajax/libs/KaTeX/0.13.2/katex.min.css')</style><div class="equation-container"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><msub><mi>u</mi><mi>B</mi></msub><mo>=</mo><mfrac><msup><mi>B</mi><mn>2</mn></msup><mrow><mn>2</mn><msub><mi>μ</mi><mn>0</mn></msub></mrow></mfrac></mrow><annotation encoding="application/x-tex">u_B = \frac{B^2}{2\mu_0}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.58056em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">u</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.32833099999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.05017em;">B</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:2.371548em;vertical-align:-0.8804400000000001em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.491108em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">2</span><span class="mord"><span class="mord mathnormal">μ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.8804400000000001em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></div></figure><h1 id="60a0c595-9dee-419a-a95d-40ef75226b12" class="">30.9 - Mutual Induction</h1><p id="ef9b2db5-ec1c-413a-8949-ac2d83bbda19" class="">If coils 1 and 2 are near each other, a changing current in either coil can induce an emf in the other. This mutual induction is described by:</p><figure id="2f077ba3-fe75-4cb0-82c4-9b044de8d51a" class="equation"><style>@import url('https://cdnjs.cloudflare.com/ajax/libs/KaTeX/0.13.2/katex.min.css')</style><div class="equation-container"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><msub><mi mathvariant="script">E</mi><mn>2</mn></msub><mo>=</mo><mo>−</mo><mi>M</mi><mfrac><mrow><mi>d</mi><msub><mi>i</mi><mn>1</mn></msub></mrow><mrow><mi>d</mi><mi>t</mi></mrow></mfrac></mrow><annotation encoding="application/x-tex">\mathscr E_2 = -M \frac{di_1}{dt}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.85em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathscr" style="margin-right:0.18583em;">E</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.18583em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:2.05744em;vertical-align:-0.686em;"></span><span class="mord">−</span><span class="mord mathnormal" style="margin-right:0.10903em;">M</span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.37144em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">d</span><span class="mord mathnormal">t</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">d</span><span class="mord"><span class="mord mathnormal">i</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></div></figure><figure id="89949827-bb9c-4070-981e-11000da9bf1d" class="equation"><style>@import url('https://cdnjs.cloudflare.com/ajax/libs/KaTeX/0.13.2/katex.min.css')</style><div class="equation-container"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><msub><mi mathvariant="script">E</mi><mn>1</mn></msub><mo>=</mo><mo>−</mo><mi>M</mi><mfrac><mrow><mi>d</mi><msubsup><mi>i</mi><mn>2</mn><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msubsup></mrow><mrow><mi>d</mi><mi>t</mi></mrow></mfrac></mrow><annotation encoding="application/x-tex">\mathscr E_1 = -M \frac{di_2'}{dt}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.85em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathscr" style="margin-right:0.18583em;">E</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.18583em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:2.114892em;vertical-align:-0.686em;"></span><span class="mord">−</span><span class="mord mathnormal" style="margin-right:0.10903em;">M</span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.428892em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">d</span><span class="mord mathnormal">t</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">d</span><span class="mord"><span class="mord mathnormal">i</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.751892em;"><span style="top:-2.4518920000000004em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.24810799999999997em;"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></div></figure><p id="4ddd416f-74fb-44d4-8391-6ea81b7a99dc" class="">where <style>@import url('https://cdnjs.cloudflare.com/ajax/libs/KaTeX/0.13.2/katex.min.css')</style><span data-token-index="0" contenteditable="false" class="notion-text-equation-token" style="user-select:all;-webkit-user-select:all;-moz-user-select:all"><span></span><span><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>M</mi></mrow><annotation encoding="application/x-tex">M</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord mathnormal" style="margin-right:0.10903em;">M</span></span></span></span></span><span></span></span> (measured in henries) is the <strong>mutual inductance</strong>.</p><p id="f56889bd-e288-42cc-a1a5-c186d0777ce7" class="">
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