michael
/
notes-archive
1
0
Fork 0
Code Issues Pull Requests Packages Projects Releases Wiki Activity
notes-archive/notes/AP Physics C/Notes 7f9dac375011471f8386e.../Unit 24 Electric Potential ...

717 lines
136 KiB
HTML
Raw Permalink Normal View History

Add notes from senior year
2022-08-14 14:00:25 -04:00
<html><head><meta http-equiv="Content-Type" content="text/html; charset=utf-8"/><title>Unit 24: Electric Potential</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="b3dad431-720f-4133-a5cd-b590f4b9b103" class="page sans"><header><h1 class="page-title">Unit 24: Electric Potential</h1></header><div class="page-body"><h1 id="e38effd0-83ac-4250-9bba-41dddb57aa88" class="">24.1 - Electric Potential</h1><p id="80cd7b20-6171-400e-8f59-14576b1e0510" class="">The electric potential <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> at a point <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> in the electric field of a charged object is:</p><figure id="c3fdbaba-7162-4502-8db1-238b1baea55d" 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>V</mi><mo>=</mo><mfrac><mrow><mo>−</mo><msub><mi>W</mi><mi mathvariant="normal">∞</mi></msub></mrow><msub><mi>q</mi><mn>0</mn></msub></mfrac><mo>=</mo><mfrac><mi>U</mi><msub><mi>q</mi><mn>0</mn></msub></mfrac></mrow><annotation encoding="application/x-tex">V = \frac{-W_\infin}{q_0} = \frac{U}{q_0}</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 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:2.2407700000000004em;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:1.36033em;"><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:0.03588em;">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 style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-
1. <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 called a potential, but it should not be confused with potential energy. Yes, the two quantities are related, but they are very different and not interchangeable.
2. Electric potential is a scalar, not a vector.</div></figure><h2 id="543742c5-1567-4aa0-8bc6-bb0ae3d2b1e9" class="">Language</h2><p id="3eab1254-a719-4fc3-a0c8-e010cd5195da" class="">A potential energy is a property of a system (or configuration) of objects, but sometimes we can get away with assigning it to a single object. For example, the gravitational potential energy of a baseball hit to outfield is actually a potential energy of the baseball–Earth system (because it is associated with the force between the baseball and Earth). However, because only the baseball noticeably moves (its motion does not noticeably affect Earth), we might assign the gravitational potential energy to it alone. In a similar way, if a charged particle is placed in an electric field and has no noticeable effect on the field (or the charged object that sets up the field), we usually assign the electric potential energy to the particle alone.</p><h2 id="b30b5319-50c4-4fbb-b237-e2d4a06d726c" class="">Units</h2><p id="bff96891-eafa-4b18-93e5-7fa507700278" class="">The SI unit for potential that follows from Eq. 24-2 is the joule per coulomb. This combination occurs so often that a special unit, the volt (abbreviated <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 used to represent it. Thus,</p><figure id="8de8d89f-fd05-4bfa-881f-dd8398e30935" 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> volt</mtext><mo>=</mo><mn>1</mn><mtext> joule per coulomb</mtext><mi mathvariant="normal">.</mi></mrow><annotation encoding="application/x-tex">1 \textrm{ volt} = 1 \textrm{ joule per coulomb}.</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="mord text"><span class="mord textrm"> volt</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:0.8888799999999999em;vertical-align:-0.19444em;"></span><span class="mord">1</span><span class="mord text"><span class="mord textrm"> joule per coulomb</span></span><span class="mord">.</span></span></span></span></span></div></figure><p id="26a78fc7-531d-484c-9543-06cb75b70944" class="">With two unit conversions, we can now switch the unit for electric field from newtons per coulomb to a more conventional unit:</p><figure id="d785e687-95da-4448-b4da-e527dbac76fb" 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> N/C</mtext><mo>=</mo><mn>1</mn><mtext> V/m</mtext></mrow><annotation encoding="application/x-tex">1 \textrm{ N/C} = 1 \textrm{ V/m}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="st
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
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 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
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
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></div></figure><p id="8ab5be01-8afc-49bf-9dd4-88347c205d7c" class="">where the integral is taken over any path connecting the points. If the integration is difficult along a particular path, we can choose a different path along which the integration might be easier.</p><p id="111258e9-5aae-448c-9aa7-86d2e1263cde" class="">If we choose <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>V</mi><mi>i</mi></msub><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">V_i = 0</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:0.22222em;">V</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:-0.22222em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</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:0.64444em;vertical-align:0em;"></span><span class="mord">0</span></span></span></span></span><span></span></span>, we have, for the potential at a particular point</p><figure id="4c31c95a-436e-4a04-9d1d-2f3dd9629a10" 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>V</mi><mo>=</mo><mo>−</mo><msubsup><mo>∫</mo><mi>i</mi><mi>f</mi></msubsup><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">V = -\int_i^f \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:0.68333em;vertical-align:0em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">V</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:2.510958em;vertical-align:-0.9119499999999999em;"></span><span class="mord">−</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mop"><span class="mop op-symbol large-op" style="margin-right:0.44445em;position:relative;top:-0.0011249999999999316em;">∫</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.5990080000000002em;"><span style="top:-1.7880500000000001em;margin-left:-0.44445em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.812900000000001em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord
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
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 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
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
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></div></figure><p id="0c1cb1a5-2a48-4375-a2f9-ff0b7f8ebf93" class="">In a uniform field of magnitude <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>E</mi></mrow><annotation encoding="application/x-tex">E</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:0.05764em;">E</span></span></span></span></span><span></span></span>, the change in potential from a higher equipotential surface to a lower one, separated by distance <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="normal">Δ</mi><mi>x</mi></mrow><annotation encoding="application/x-tex">\Delta x</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">Δ</span><span class="mord mathnormal">x</span></span></span></span></span><span></span></span>, is</p><figure id="fb233903-086f-4d2f-bddd-4c163c215951" 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="normal">Δ</mi><mi>V</mi><mo>=</mo><mo>−</mo><mi>E</mi><mi mathvariant="normal">Δ</mi><mi>x</mi></mrow><annotation encoding="application/x-tex">\Delta V = -E \Delta x</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">Δ</span><span class="mord mathnormal" style="margin-right:0.22222em;">V</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:0.76666em;vertical-align:-0.08333em;"></span><span class="mord">−</span><span class="mord mathnormal" style="margin-right:0.05764em;">E</span><span class="mord">Δ</span><span class="mord mathnormal">x</span></span></span></span></span></div></figure><h1 id="0530206e-9fd3-4372-934c-2df92f837350" class="">24.3 - Potential due to a Charged Particle</h1><div id="309fab74-188e-428d-a2e7-4d0fc63131c8" class="column-list"><div id="d2443d34-5517-43e6-bc55-84e5a932b712" style="width:75%" class="column"><p id="bd95d755-df5c-46b7-b48f-bc333cd597fe" class="">The electric potential due to a single charged particle at a distance <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
</p></div></div><div id="abfaf615-9d7d-42f2-902a-2d5908e5e2e9" class="column-list"><div id="f4c36be0-c4c5-4dd5-9175-90e1aed770e5" style="width:31.25%" class="column"><p id="9aaadbfc-ed50-4452-8f51-40491ae1909e" class="">
</p><figure id="88c2bd29-bbba-43b9-ba5a-61f619723d96" class="image"><a href="Unit%2024%20Electric%20Potential%20b3dad431720f4133a5cdb590f4b9b103/Untitled%203.png"><img style="width:572px" src="Unit%2024%20Electric%20Potential%20b3dad431720f4133a5cdb590f4b9b103/Untitled%203.png"/></a></figure></div><div id="71d03635-675e-4125-abde-9c755fc84f78" style="width:68.75%" class="column"><h2 id="c076e708-961c-4bc6-ba14-ac762f763d89" class="">Induced Dipole Moment</h2><p id="92f46f57-1e1c-45a0-97b3-ef0c2233af9e" class="">Many molecules, such as water, have <em>permanent </em>electric dipole moments. In other molecules (called nonpolar molecules) and in every isolated atom, the centers of the positive and negative charges coincide (Fig. a) and thus no dipole moment is set up. However, if we place an atom or a nonpolar molecule in an external electric field, the field distorts the electron orbits and separates the centers of positive and negative charge (Fig. b). Because the electrons are negatively charged, they tend to be shifted in a direction opposite the field. This shift sets up a dipole moment that points in the direction of the field. This dipole moment is said to be induced by the field, and the atom or molecule is then said to be polarized by the field (that is, it has a positive side and a negative side). When the field is removed, the induced dipole moment and the polarization disappear.</p></div></div><h1 id="4db5ee87-850e-41d3-bb93-d673c7fae77a" class="">24.5 - Potential due to a Continuous Charge Distribution</h1><p id="0df286c1-d5dd-4c0c-961d-cb90f7dbd3a2" class="">For a continuous distribution of charge (over an extended object), the potential is found by (1) dividing the distribution into charge elements <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> that can be treated as particles and then (2) summing the potential due to each element by integrating over the full distribution:</p><figure id="315ec1b8-23dd-432c-8932-071cd5574e89" 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>V</mi><mo>=</mo><mfrac><mn>1</mn><mrow><mn>4</mn><mi>π</mi><msub><mi>ϵ</mi><mn>0</mn></msub></mrow></mfrac><mo>∫</mo><mfrac><mrow><mi>d</mi><mi>q</mi></mrow><mi>r</mi></mfrac></mrow><annotation encoding="application/x-tex">V = \frac{1}{4\pi\epsilon_0} \int \frac{dq}{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:0.22222em;">V</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:2.2336899999999997em;vertical-align:-0.86225em;"></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:-2.3139999999999996em;"><span class="pstrut" style="height:3em;"
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
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> in any direction is the negative of the rate at which the potential changes with distance in that direction:</p><figure id="07226cc4-3fbd-43f0-bba5-74311f1d83de" 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>E</mi><mi>S</mi></msub><mo>=</mo><mo>−</mo><mfrac><mrow><mi mathvariant="normal">∂</mi><mi>V</mi></mrow><mrow><mi mathvariant="normal">∂</mi><mi>s</mi></mrow></mfrac></mrow><annotation encoding="application/x-tex">E_S = -\frac{\partial V}{\partial s}</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:0.05764em;">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:-0.05764em;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:0.05764em;">S</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"><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" style="margin-right:0.05556em;">∂</span><span class="mord mathnormal">s</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" style="margin-right:0.05556em;">∂</span><span class="mord mathnormal" style="margin-right:0.22222em;">V</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="8596c769-f685-4ae3-8e7b-bb33e3702e93" class="">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>x</mi></mrow><annotation encoding="application/x-tex">x</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">x</span></span></span></span></span><span></span></span>, <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">
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
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> may be found from:</p><div id="f6e58ece-2b88-4032-a33b-a49f075219a9" class="column-list"><div id="49b0036c-c628-4cdb-9e65-50a4be57d9a2" style="width:33.333333333333336%" class="column"><figure id="0664ca34-b8df-476b-aace-43ea70147f8c" 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>E</mi><mi>x</mi></msub><mo>=</mo><mo>−</mo><mfrac><mrow><mi mathvariant="normal">∂</mi><mi>V</mi></mrow><mrow><mi mathvariant="normal">∂</mi><mi>x</mi></mrow></mfrac></mrow><annotation encoding="application/x-tex">E_x = -\frac{\partial V}{\partial x}</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:0.05764em;">E</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:-0.05764em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">x</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"><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" style="margin-right:0.05556em;">∂</span><span class="mord mathnormal">x</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" style="margin-right:0.05556em;">∂</span><span class="mord mathnormal" style="margin-right:0.22222em;">V</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></div><div id="e3ed0d97-d86a-4be8-886b-59db4ccd21df" style="width:33.333333333333336%" class="column"><figure id="fe07b2e2-d178-4ab6-8ec7-8b7c3702454f" 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>E</mi><mi>y</mi></msub><mo>=</mo><mo>−</mo><mfrac><mrow><mi mathvariant="normal">∂</mi><mi>V</mi></mrow><mrow><mi mathvariant="normal">∂</mi><mi>y</mi></mrow></mfrac></mrow><annotation encoding="application/x-tex">E_y = -\frac{\partial V}{\partial y}</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:0.05764em;">E</span><span class="msupsub"><span class="vlist-t vlist-t2"><span
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
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> is uniform, this all reduces to:</p><figure id="466b0c2e-f31c-494b-97ee-04f874287c58" 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>E</mi><mo>=</mo><mo>−</mo><mfrac><mrow><mi mathvariant="normal">Δ</mi><mi>V</mi></mrow><mrow><mi mathvariant="normal">Δ</mi><mi>s</mi></mrow></mfrac></mrow><annotation encoding="application/x-tex">E = -\frac{\Delta V}{\Delta s}</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:0.05764em;">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:2.04633em;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:1.36033em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">Δ</span><span class="mord mathnormal">s</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">Δ</span><span class="mord mathnormal" style="margin-right:0.22222em;">V</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="9ff987f7-3450-4011-8ac1-4e1b1edcf107" 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>s</mi></mrow><annotation encoding="application/x-tex">s</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">s</span></span></span></span></span><span></span></span> is perpendicular to the equipotential surfaces.</p><p id="68efe23c-1431-4168-a925-83acd578e375" class="">The electric field is zero parallel to an equipotential surface, as the field points directly perpendicular to the surface of the equipotential surface.</p><h1 id="46f0f126-4c13-4702-96bd-7751a872f1dc" class="">24.7 - Electric Potential Energy of a System of Charged Particles</h1><p id="5dab236a-4c5f-40cd-b440-767be0d94770" class="">The electric potential energy of a system of charged particles is equal to the work needed to assemble the system with the particles initially at rest and infinitely distant from each other. For two particles at separation <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
An excess charge placed on a conductor will, in the equilibrium state, be located entirely on the outer surface of the conductor.</div></figure><div id="7f7d20d6-0f5f-480e-8677-97e6dd6389f5" class="column-list"><div id="987d6c69-6878-4580-94ca-450c832d31cc" style="width:75%" class="column"><p id="5a29ae8a-2ea8-499a-bd33-0967fc94a4f7" class="">The entire conductor, including interior points, is at a uniform potential.</p><p id="bea086b9-77fd-41f3-9f85-4fbe9bd711b3" class="">If an isolated charged conductor is placed in an external electric field, then at every internal point, the electric field due to the charge cancels the external electric field that otherwise would have been there.</p><p id="40039190-64d0-45ad-8ebc-eec77de9fd37" class="">Also, the net electric field at every point on the surface is perpendicular to the surface.</p></div><div id="7e1726bc-869b-4332-b970-8d562d069958" style="width:25%" class="column"><figure id="91e03659-81b1-43b2-9803-0c15146e9020" class="image"><a href="Unit%2024%20Electric%20Potential%20b3dad431720f4133a5cdb590f4b9b103/Untitled%204.png"><img style="width:527px" src="Unit%2024%20Electric%20Potential%20b3dad431720f4133a5cdb590f4b9b103/Untitled%204.png"/></a></figure></div></div><p id="4145d40b-232c-4b10-b456-3d4b944970fc" class="">
</p></div></article></body></html>