670 lines
68 KiB
HTML
670 lines
68 KiB
HTML
<html><head><meta http-equiv="Content-Type" content="text/html; charset=utf-8"/><title>Unit 27: Circuits</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 {
|
||
background: rgba(231, 243, 248, 1);
|
||
}
|
||
.highlight-purple_background {
|
||
background: rgba(244, 240, 247, 0.8);
|
||
}
|
||
.highlight-pink_background {
|
||
background: rgba(249, 238, 243, 0.8);
|
||
}
|
||
.highlight-red_background {
|
||
background: rgba(253, 235, 236, 1);
|
||
}
|
||
.block-color-default {
|
||
color: inherit;
|
||
fill: inherit;
|
||
}
|
||
.block-color-gray {
|
||
color: rgba(120, 119, 116, 1);
|
||
fill: rgba(120, 119, 116, 1);
|
||
}
|
||
.block-color-brown {
|
||
color: rgba(159, 107, 83, 1);
|
||
fill: rgba(159, 107, 83, 1);
|
||
}
|
||
.block-color-orange {
|
||
color: rgba(217, 115, 13, 1);
|
||
fill: rgba(217, 115, 13, 1);
|
||
}
|
||
.block-color-yellow {
|
||
color: rgba(203, 145, 47, 1);
|
||
fill: rgba(203, 145, 47, 1);
|
||
}
|
||
.block-color-teal {
|
||
color: rgba(68, 131, 97, 1);
|
||
fill: rgba(68, 131, 97, 1);
|
||
}
|
||
.block-color-blue {
|
||
color: rgba(51, 126, 169, 1);
|
||
fill: rgba(51, 126, 169, 1);
|
||
}
|
||
.block-color-purple {
|
||
color: rgba(144, 101, 176, 1);
|
||
fill: rgba(144, 101, 176, 1);
|
||
}
|
||
.block-color-pink {
|
||
color: rgba(193, 76, 138, 1);
|
||
fill: rgba(193, 76, 138, 1);
|
||
}
|
||
.block-color-red {
|
||
color: rgba(212, 76, 71, 1);
|
||
fill: rgba(212, 76, 71, 1);
|
||
}
|
||
.block-color-gray_background {
|
||
background: rgba(241, 241, 239, 1);
|
||
}
|
||
.block-color-brown_background {
|
||
background: rgba(244, 238, 238, 1);
|
||
}
|
||
.block-color-orange_background {
|
||
background: rgba(251, 236, 221, 1);
|
||
}
|
||
.block-color-yellow_background {
|
||
background: rgba(251, 243, 219, 1);
|
||
}
|
||
.block-color-teal_background {
|
||
background: rgba(237, 243, 236, 1);
|
||
}
|
||
.block-color-blue_background {
|
||
background: rgba(231, 243, 248, 1);
|
||
}
|
||
.block-color-purple_background {
|
||
background: rgba(244, 240, 247, 0.8);
|
||
}
|
||
.block-color-pink_background {
|
||
background: rgba(249, 238, 243, 0.8);
|
||
}
|
||
.block-color-red_background {
|
||
background: rgba(253, 235, 236, 1);
|
||
}
|
||
.select-value-color-pink { background-color: rgba(245, 224, 233, 1); }
|
||
.select-value-color-purple { background-color: rgba(232, 222, 238, 1); }
|
||
.select-value-color-green { background-color: rgba(219, 237, 219, 1); }
|
||
.select-value-color-gray { background-color: rgba(227, 226, 224, 1); }
|
||
.select-value-color-opaquegray { background-color: rgba(255, 255, 255, 0.0375); }
|
||
.select-value-color-orange { background-color: rgba(250, 222, 201, 1); }
|
||
.select-value-color-brown { background-color: rgba(238, 224, 218, 1); }
|
||
.select-value-color-red { background-color: rgba(255, 226, 221, 1); }
|
||
.select-value-color-yellow { background-color: rgba(253, 236, 200, 1); }
|
||
.select-value-color-blue { background-color: rgba(211, 229, 239, 1); }
|
||
|
||
.checkbox {
|
||
display: inline-flex;
|
||
vertical-align: text-bottom;
|
||
width: 16;
|
||
height: 16;
|
||
background-size: 16px;
|
||
margin-left: 2px;
|
||
margin-right: 5px;
|
||
}
|
||
|
||
.checkbox-on {
|
||
background-image: url("data:image/svg+xml;charset=UTF-8,%3Csvg%20width%3D%2216%22%20height%3D%2216%22%20viewBox%3D%220%200%2016%2016%22%20fill%3D%22none%22%20xmlns%3D%22http%3A%2F%2Fwww.w3.org%2F2000%2Fsvg%22%3E%0A%3Crect%20width%3D%2216%22%20height%3D%2216%22%20fill%3D%22%2358A9D7%22%2F%3E%0A%3Cpath%20d%3D%22M6.71429%2012.2852L14%204.9995L12.7143%203.71436L6.71429%209.71378L3.28571%206.2831L2%207.57092L6.71429%2012.2852Z%22%20fill%3D%22white%22%2F%3E%0A%3C%2Fsvg%3E");
|
||
}
|
||
|
||
.checkbox-off {
|
||
background-image: url("data:image/svg+xml;charset=UTF-8,%3Csvg%20width%3D%2216%22%20height%3D%2216%22%20viewBox%3D%220%200%2016%2016%22%20fill%3D%22none%22%20xmlns%3D%22http%3A%2F%2Fwww.w3.org%2F2000%2Fsvg%22%3E%0A%3Crect%20x%3D%220.75%22%20y%3D%220.75%22%20width%3D%2214.5%22%20height%3D%2214.5%22%20fill%3D%22white%22%20stroke%3D%22%2336352F%22%20stroke-width%3D%221.5%22%2F%3E%0A%3C%2Fsvg%3E");
|
||
}
|
||
|
||
</style></head><body><article id="7d91d88b-ad90-4957-aa72-b19e800e751d" class="page sans"><header><h1 class="page-title">Unit 27: Circuits</h1></header><div class="page-body"><h1 id="46cf4c96-e92c-4397-9059-2e77363a9c3b" class="">27.1 - Single Loop Circuits</h1><p id="c18b8cf5-84a6-4f70-8546-a8516e1a2343" class="">To produce a steady flow of charge, you need a “charge pump,” a device that—by doing work on the charge carriers—maintains a potential difference between a pair of terminals. We call such a device an <strong>emf device</strong>, and the device is said to provide an <strong>emf</strong> <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>, which means that it does work on charge carriers. </p><p id="60f8c6b0-feb3-497c-82fa-3c81524e6de3" class="">An emf device is sometimes called a seat of emf. The term emf comes from the outdated phrase electromotive force, which was adopted before scientists clearly understood the function of an emf device.</p><p id="fee95fe2-71ba-4c75-97d1-edf22c0aa144" class="">An emf device does work on charges to maintain a potential difference between its output terminals. 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>d</mi><mi>W</mi></mrow><annotation encoding="application/x-tex">dW</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 mathnormal">d</span><span class="mord mathnormal" style="margin-right:0.13889em;">W</span></span></span></span></span><span></span></span> is the work the device does to force positive charge <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>d</mi><mi>q</mi></mrow><annotation encoding="application/x-tex">dq</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 mathnormal">d</span><span class="mord mathnormal" style="margin-right:0.03588em;">q</span></span></span></span></span><span></span></span> from the negative to the positive terminal, then the emf (work per unit charge) of the device is:</p><figure id="c9fd0d04-7cac-41ad-8cf8-d203838e97ef" 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><mfrac><mrow><mi>d</mi><mi>W</mi></mrow><mrow><mi>d</mi><mi>q</mi></mrow></mfrac></mrow><annotation encoding="application/x-tex">\mathscr{E} = \frac{dW}{dq}</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:2.25188em;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:1.37144em;"><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" style="margin-right:0.03588em;">q</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" style="margin-right:0.13889em;">W</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><h2 id="991e8ddc-dc7f-4ed3-b548-66522e616526" class="">Ideal vs. Real EMF Devices</h2><p id="3d525f2a-18de-4281-afae-96893a6ed0e5" class="">An ideal emf device is one that lacks any internal resistance. The potential difference between its terminals is equal to the emf.</p><p id="05fc18eb-563e-4c5e-8178-3df60a858e22" class="">A real emf device has internal resistance. The potential difference between its terminals is equal to the emf only if there is no current through the device.</p><h2 id="02feb228-215b-4639-a574-43eb0212d52d" class="">Resistance Rule</h2><p id="4f7ff121-8f59-42ff-9d6f-b4faa88be9f2" class="">The change in potential in traversing 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> in the direction of the current is <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><mo>−</mo><mi>i</mi><mi>R</mi></mrow><annotation encoding="application/x-tex">-iR</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.76666em;vertical-align:-0.08333em;"></span><span class="mord">−</span><span class="mord mathnormal">i</span><span class="mord mathnormal" style="margin-right:0.00773em;">R</span></span></span></span></span><span></span></span>; in the opposite direction it is <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><mo>+</mo><mi>i</mi><mi>R</mi></mrow><annotation encoding="application/x-tex">+iR</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.76666em;vertical-align:-0.08333em;"></span><span class="mord">+</span><span class="mord mathnormal">i</span><span class="mord mathnormal" style="margin-right:0.00773em;">R</span></span></span></span></span><span></span></span>. </p><h2 id="ff0eb93d-5bb4-42b0-ad2d-9dda25ecd6cf" class="">EMF Rule</h2><p id="6134d4a7-798e-4204-bd8b-d6731cc62ca2" class="">The change in potential in traversing an ideal emf device in the direction of the emf arrow is <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><mo>+</mo><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:0.78333em;vertical-align:-0.08333em;"></span><span class="mord">+</span><span class="mord mathscr" style="margin-right:0.18583em;">E</span></span></span></span></span><span></span></span>; in the opposite direction it is <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><mo>−</mo><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:0.78333em;vertical-align:-0.08333em;"></span><span class="mord">−</span><span class="mord mathscr" style="margin-right:0.18583em;">E</span></span></span></span></span><span></span></span>.</p><h2 id="44c261a5-fbf3-4ea9-9ea3-52a0c6c0d9de" class="">Loop Rule</h2><p id="49545297-b985-4b3d-9382-20ff7c463df4" class="">Conservation of energy leads to the <strong>Loop Rule</strong>: The algebraic sum of the changes in potential encountered in a complete traversal of any loop of a circuit must be zero.</p><h2 id="ed266e06-29ec-4331-8a68-9f7705c82ccd" class="">Junction Rule</h2><p id="fed555bd-d362-497b-b620-0239f52e13ac" class="">Conservation of charge leads to the <strong>Junction Rule:</strong> The sum of the currents entering any junction must be equal to the sum of the currents leaving that junction.</p><h2 id="5d17ac42-6aad-4563-8ab4-da3e8defc5f6" class="">Internal Resistance (Real Batteries)</h2><p id="3023b613-9c78-4a1a-95aa-5598144d3d0a" class="">When a real battery of 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> and internal 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:0.43056em;vertical-align:0em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">r</span></span></span></span></span><span></span></span> does work on the charge carriers in an 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> through the battery, the rate <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>P</mi></mrow><annotation encoding="application/x-tex">P</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:0.13889em;">P</span></span></span></span></span><span></span></span> of energy transfer to the charge carriers is:</p><figure id="f8b1504a-987e-403f-9604-c054e01589bb" 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>P</mi><mo>=</mo><mi>i</mi><mi>V</mi></mrow><annotation encoding="application/x-tex">P = iV</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:0.13889em;">P</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 mathnormal" style="margin-right:0.22222em;">iV</span></span></span></span></span></div></figure><p id="2b881d4f-2518-4f81-abc4-e70c02195fe8" 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>V</mi></mrow><annotation encoding="application/x-tex">V</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:0.22222em;">V</span></span></span></span></span><span></span></span> is the potential across the terminals of the battery.</p><p id="04694682-8c77-4f60-903f-e929a27b773c" class="">The rate <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>P</mi><mi>r</mi></msub></mrow><annotation encoding="application/x-tex">P_r</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:0.13889em;">P</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em;"><span style="top:-2.5500000000000003em;margin-left:-0.13889em;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:0.02778em;">r</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> at which energy is dissipated as thermal energy in the battery is:</p><figure id="8d6e5ac6-20a7-454f-9b11-8ab731461198" 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>P</mi><mi>r</mi></msub><mo>=</mo><msup><mi>i</mi><mn>2</mn></msup><mi>r</mi></mrow><annotation encoding="application/x-tex">P_r = i^2r</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:0.13889em;">P</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em;"><span style="top:-2.5500000000000003em;margin-left:-0.13889em;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:0.02778em;">r</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:0.8641079999999999em;vertical-align:0em;"></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 class="mord mathnormal" style="margin-right:0.02778em;">r</span></span></span></span></span></div></figure><p id="53f68300-a533-4c9d-8a41-64ab68597da4" class="">The rate <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>P</mi><mtext>emf</mtext></msub></mrow><annotation encoding="application/x-tex">P_{\textrm{emf}}</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:0.13889em;">P</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:-0.13889em;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 text mtight"><span class="mord textrm mtight">emf</span></span></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> at which the chemical energy in the battery changes is:</p><figure id="34ca0ef3-64cb-4dec-bf1b-4699a51d51cd" 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>P</mi><mtext>emf</mtext></msub><mo>=</mo><mi>i</mi><mi mathvariant="script">E</mi></mrow><annotation encoding="application/x-tex">P_{\textrm{emf}} = i\mathscr{E}</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:0.13889em;">P</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:-0.13889em;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 text mtight"><span class="mord textrm mtight">emf</span></span></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:0.7em;vertical-align:0em;"></span><span class="mord mathnormal">i</span><span class="mord mathscr" style="margin-right:0.18583em;">E</span></span></span></span></span></div></figure><p id="b93a47ff-ba3d-439c-b669-8e6f338929dd" class="">When resistances are in series, they have the same current. The equivalent resistance that can replace a series combination of resistances is:</p><figure id="f0cda328-ac50-4e92-a151-2cb75b536214" 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>R</mi><mrow><mi>e</mi><mi>q</mi></mrow></msub><mo>=</mo><munderover><mo>∑</mo><mrow><mi>j</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></munderover><msub><mi>R</mi><mi>j</mi></msub></mrow><annotation encoding="application/x-tex">R_{eq} = \sum_{j=1}^n R_j</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.969438em;vertical-align:-0.286108em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.00773em;">R</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.15139200000000003em;"><span style="top:-2.5500000000000003em;margin-left:-0.00773em;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 mathnormal mtight">e</span><span class="mord mathnormal mtight" style="margin-right:0.03588em;">q</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em;"><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:3.0651740000000007em;vertical-align:-1.4137769999999998em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.6513970000000007em;"><span style="top:-1.872331em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span><span class="mrel mtight">=</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.050005em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">∑</span></span></span><span style="top:-4.3000050000000005em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:1.4137769999999998em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.00773em;">R</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.311664em;"><span style="top:-2.5500000000000003em;margin-left:-0.00773em;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:0.05724em;">j</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em;"><span></span></span></span></span></span></span></span></span></span></span></div></figure><p id="49792239-e25c-4814-81fe-3e6ce53f107b" class="">(this is from last unit)</p><h1 id="d0c50e94-e2fe-494b-8b66-18a73449e3a1" class="">27.2 - Multiloop Circuits</h1><p id="deb25301-44b5-46ae-b425-30c2f94fb108" class="">When resistances are in parallel, they have the same potential difference (<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>V</mi></mrow><annotation encoding="application/x-tex">V</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:0.22222em;">V</span></span></span></span></span><span></span></span>). The equivalent resistance that can replace a parallel combination of resistances is given by:</p><figure id="6621e626-e5a8-405e-83aa-f8444914f4e0" 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><mn>1</mn><msub><mi>R</mi><mrow><mi>e</mi><mi>q</mi></mrow></msub></mfrac><mo>=</mo><munderover><mo>∑</mo><mrow><mi>j</mi><mo>=</mo><mi>i</mi></mrow><mi>n</mi></munderover><mfrac><mn>1</mn><msub><mi>R</mi><mi>j</mi></msub></mfrac></mrow><annotation encoding="application/x-tex">\frac{1}{R_{eq}} = \sum_{j=i}^n \frac{1}{R_j}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.293548em;vertical-align:-0.972108em;"></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"><span class="mord mathnormal" style="margin-right:0.00773em;">R</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.15139200000000003em;"><span style="top:-2.5500000000000003em;margin-left:-0.00773em;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 mathnormal mtight">e</span><span class="mord mathnormal mtight" style="margin-right:0.03588em;">q</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em;"><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">1</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.972108em;"><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:3.0651740000000007em;vertical-align:-1.4137769999999998em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.6513970000000007em;"><span style="top:-1.872331em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span><span class="mrel mtight">=</span><span class="mord mathnormal mtight">i</span></span></span></span><span style="top:-3.050005em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">∑</span></span></span><span style="top:-4.3000050000000005em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:1.4137769999999998em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></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"><span class="mord mathnormal" style="margin-right:0.00773em;">R</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.311664em;"><span style="top:-2.5500000000000003em;margin-left:-0.00773em;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:0.05724em;">j</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em;"><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">1</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.972108em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></div></figure><h1 id="9c2eda6e-f92a-42fd-b212-f437a875f842" class="">27.3 - The Ammeter and the Voltmeter</h1><p id="7f7a0df7-8fd4-428e-ab98-6df96944d62c" class="">An instrument used to measure currents is called an <strong>ammeter</strong>. To measure the current in a wire, you usually have to break or cut the wire and insert the ammeter.</p><p id="23498aa8-fe62-49a1-a45d-74e8465fc111" class="">A meter used to measure potential differences is called a <strong>voltmeter</strong>. To find the potential difference between any two points in the circuit, the voltmeter terminals are connected between those points without breaking or cutting the wire.</p><p id="b0a8272c-8de2-4932-9303-f02c230b0528" class="">Often a single meter is packaged so that, by means of a switch, it can be made to serve as either an ammeter or a voltmeter—and usually also as an ohmmeter, designed to measure the resistance of any element connected between its terminals. Such a versatile unit is called a <strong>multimeter</strong>.</p><h1 id="e493ee9c-bfad-4230-b2a1-58d06432a6c7" class="">27.4 - RC Circuits</h1><p id="560d6a11-591a-470a-8037-1cbaa33ece56" class="">When an 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 applied to 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 capacitance <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>C</mi></mrow><annotation encoding="application/x-tex">C</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:0.07153em;">C</span></span></span></span></span><span></span></span> in series, the charge on the capacitor increases according to:</p><figure id="8f6626a4-e1cd-469b-8a3a-685ed6f303a2" 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>q</mi><mo>=</mo><mi>C</mi><mi mathvariant="script">E</mi><mrow><mo fence="true">(</mo><mn>1</mn><mo>−</mo><msup><mi>e</mi><mrow><mo>−</mo><mi>t</mi><mi mathvariant="normal">/</mi><mi>R</mi><mi>C</mi></mrow></msup><mo fence="true">)</mo></mrow></mrow><annotation encoding="application/x-tex">q = C\mathscr{E}\left(1 - e^{-t/RC}\right)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.19444em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">q</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:1.80002em;vertical-align:-0.65002em;"></span><span class="mord mathnormal" style="margin-right:0.07153em;">C</span><span class="mord mathscr" style="margin-right:0.18583em;">E</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size2">(</span></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 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 mathnormal mtight" style="margin-right:0.07153em;">RC</span></span></span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em;"><span class="delimsizing size2">)</span></span></span></span></span></span></span></div></figure><p id="b90bc9ec-6755-441d-b96d-1a766384070a" class="">in which <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>C</mi><mi mathvariant="script">E</mi><mo>=</mo><msub><mi>q</mi><mn>0</mn></msub></mrow><annotation encoding="application/x-tex">C\mathscr{E} = q_0</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 mathnormal" style="margin-right:0.07153em;">C</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:0.625em;vertical-align:-0.19444em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">q</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:-0.03588em;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> is the equilibrium (final) charge 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>R</mi><mi>C</mi><mo>=</mo><mi>τ</mi></mrow><annotation encoding="application/x-tex">RC = \tau</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:0.07153em;">RC</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:0.43056em;vertical-align:0em;"></span><span class="mord mathnormal" style="margin-right:0.1132em;">τ</span></span></span></span></span><span></span></span> is the capacitive time constant of the circuit.</p><figure class="block-color-gray_background callout" style="white-space:pre-wrap;display:flex" id="526297ca-f99a-4518-a4ba-ef99d6f65f8b"><div style="font-size:1.5em"><span class="icon">💡</span></div><div style="width:100%">A capacitor that is being charged initially acts like ordinary connecting wire relative to the charging current. A long time later, it acts like a broken wire.</div></figure><p id="45707868-3330-4689-8c91-bab9f52d79f8" class="">During charging, the current is:</p><figure id="911dc401-94ef-4f63-89c4-c0caf808b998" 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><mrow><mi>d</mi><mi>q</mi></mrow><mrow><mi>d</mi><mi>t</mi></mrow></mfrac><mo>=</mo><mrow><mo fence="true">(</mo><mfrac><mi mathvariant="script">E</mi><mi>R</mi></mfrac><mo fence="true">)</mo></mrow><msup><mi>e</mi><mrow><mo>−</mo><mi>t</mi><mi mathvariant="normal">/</mi><mi>R</mi><mi>C</mi></mrow></msup></mrow><annotation encoding="application/x-tex">i = \frac{dq}{dt} = \left(\frac{\mathscr{E}}{R}\right) e^{-t/RC}</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:2.0574399999999997em;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:1.3714399999999998em;"><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" style="margin-right:0.03588em;">q</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:2.40003em;vertical-align:-0.95003em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size3">(</span></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="mclose delimcenter" style="top:0em;"><span class="delimsizing size3">)</span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></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 mathnormal mtight" style="margin-right:0.07153em;">RC</span></span></span></span></span></span></span></span></span></span></span></span></span></div></figure><p id="97617a58-95d6-49b1-9565-85043f37750b" class="">When a capacitor discharges through 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>, the charge on the capacitor decays according to:</p><figure id="c5a1f531-cf36-4a77-a9ea-dbaf575d7a4f" 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>q</mi><mo>=</mo><msub><mi>q</mi><mn>0</mn></msub><msup><mi>e</mi><mrow><mo>−</mo><mi>t</mi><mi mathvariant="normal">/</mi><mi>R</mi><mi>C</mi></mrow></msup></mrow><annotation encoding="application/x-tex">q = q_0 e^{-t/RC}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.19444em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">q</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:1.13244em;vertical-align:-0.19444em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">q</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:-0.03588em;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 mathnormal mtight" style="margin-right:0.07153em;">RC</span></span></span></span></span></span></span></span></span></span></span></span></span></div></figure><p id="1910b9af-17a9-467c-9422-8e998c3d7849" class="">During the discharging, the current is equal to:</p><figure id="39c223bd-1d8a-494a-9abe-efc5060d2cfe" 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><mrow><mi>d</mi><mi>q</mi></mrow><mrow><mi>d</mi><mi>t</mi></mrow></mfrac><mo>=</mo><mo>−</mo><mrow><mo fence="true">(</mo><mfrac><msub><mi>q</mi><mn>0</mn></msub><mrow><mi>R</mi><mi>C</mi></mrow></mfrac><mo fence="true">)</mo></mrow><msup><mi>e</mi><mrow><mo>−</mo><mi>t</mi><mi mathvariant="normal">/</mi><mi>R</mi><mi>C</mi></mrow></msup></mrow><annotation encoding="application/x-tex">i = \frac{dq}{dt} = -\left(\frac{q_0}{RC}\right)e^{-t/RC}</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:2.0574399999999997em;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:1.3714399999999998em;"><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" style="margin-right:0.03588em;">q</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:1.8359999999999999em;vertical-align:-0.686em;"></span><span class="mord">−</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size2">(</span></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:1.1075599999999999em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07153em;">RC</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:0.03588em;">q</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:-0.03588em;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 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="mclose delimcenter" style="top:0em;"><span class="delimsizing size2">)</span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></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 mathnormal mtight" style="margin-right:0.07153em;">RC</span></span></span></span></span></span></span></span></span></span></span></span></span></div></figure><p id="04332650-3761-4afb-acad-69372d39680c" class="">
|
||
</p></div></article></body></html> |