{"id":985,"date":"2017-10-27T16:31:11","date_gmt":"2017-10-27T16:31:11","guid":{"rendered":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/chapter\/electric-field-concept-of-a-field-revisited\/"},"modified":"2017-11-08T03:25:59","modified_gmt":"2017-11-08T03:25:59","slug":"electric-field-concept-of-a-field-revisited","status":"publish","type":"chapter","link":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/chapter\/electric-field-concept-of-a-field-revisited\/","title":{"raw":"Electric Field: Concept of a Field Revisited","rendered":"Electric Field: Concept of a Field Revisited"},"content":{"raw":"\n<div class=\"textbox learning-objectives\">\n<h3 itemprop=\"educationalUse\">Learning Objectives<\/h3>\n<ul>\n<li>Describe a force field and calculate the strength of an electric field due to a point charge.<\/li>\n<li>Calculate the force exerted on a test charge by an electric field.<\/li>\n<li>Explain the relationship between electrical force (F) on a test charge and electrical field strength (E).<\/li>\n<\/ul>\n<\/div>\n<p id=\"import-auto-id2688112\">Contact forces, such as between a baseball and a bat, are explained on the small scale by the interaction of the charges in atoms and molecules in close proximity. They interact through forces that include the <span data-type=\"term\" id=\"import-auto-id2648325\">Coulomb force<\/span>. Action at a distance is a force between objects that are not close enough for their atoms to \u201ctouch.\u201d That is, they are separated by more than a few atomic diameters.<\/p>\n<p id=\"import-auto-id2590313\">For example, a charged rubber comb attracts neutral bits of paper from a distance via the Coulomb force. It is very useful to think of an object being surrounded in space by a <span data-type=\"term\" id=\"import-auto-id3145066\">force field<\/span>. The force field carries the force to another object (called a test object) some distance away.<\/p>\n<div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id2662859\">\n<h1 data-type=\"title\">Concept of a Field<\/h1>\n<p id=\"import-auto-id1954099\">A field is a way of conceptualizing and mapping the force that surrounds any object and acts on another object at a distance without apparent physical connection. For example, the gravitational field surrounding the earth (and all other masses) represents the gravitational force that would be experienced if another mass were placed at a given point within the field.<\/p>\n<p id=\"import-auto-id3408279\">In the same way, the Coulomb force field surrounding any charge extends throughout space. Using Coulomb\u2019s law, [latex]F=k|{q}_{1}{q}_{2}|\/{r}^{2}[\/latex], its magnitude is given by the equation <\/p>\n<p>[latex]F=k|\\mathrm{qQ}|\/{r}^{2}[\/latex], for a <span data-type=\"term\" id=\"import-auto-id1916842\">point charge<\/span> (a particle having a charge <em data-effect=\"italics\">[latex]Q[\/latex]<\/em>) acting on a <span data-type=\"term\" id=\"import-auto-id2383341\">test charge <\/span>[latex]q[\/latex] at a distance [latex]r[\/latex] (see <a href=\"#import-auto-id2408057\" class=\"autogenerated-content\">(Figure)<\/a>). Both the magnitude and direction of the Coulomb force field depend on <em data-effect=\"italics\">[latex]Q[\/latex]<\/em> and the test charge [latex]q[\/latex].<\/p>\n<div class=\"bc-figure figure\" id=\"import-auto-id2408057\">\n<div class=\"bc-figcaption figcaption\">The Coulomb force field due to a positive charge [latex]Q[\/latex] is shown acting on two different charges. Both charges are the same distance from [latex]Q[\/latex]. (a) Since [latex]{q}_{1}[\/latex] is positive, the force [latex]{F}_{1}[\/latex] acting on it is repulsive. (b) The charge [latex]{q}_{2}[\/latex] is negative and greater in magnitude than [latex]{q}_{1}[\/latex], and so the force [latex]{F}_{2}[\/latex] acting on it is attractive and stronger than [latex]{F}_{1}[\/latex]. The Coulomb force field is thus not unique at any point in space, because it depends on the test charges [latex]{q}_{1}[\/latex] and [latex]{q}_{2}[\/latex] as well as the charge [latex]Q[\/latex].<\/div>\n<p><span data-type=\"media\" id=\"import-auto-id2584308\" data-alt=\"In part a, two charges Q and q one are placed at a distance r. The force vector F one on charge q one is shown by an arrow pointing toward right away from Q. In part b, two charges Q and q two are placed at a distance r. The force vector F two on charge q two is shown by an arrow pointing toward left toward Q.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/clalonde\/wp-content\/uploads\/sites\/280\/2017\/10\/Figure_19_04_02a.jpg\" data-media-type=\"image\/jpg\" alt=\"In part a, two charges Q and q one are placed at a distance r. The force vector F one on charge q one is shown by an arrow pointing toward right away from Q. In part b, two charges Q and q two are placed at a distance r. The force vector F two on charge q two is shown by an arrow pointing toward left toward Q.\" width=\"200\"><\/span><\/p><\/div>\n<p>To simplify things, we would prefer to have a field that depends only on <em data-effect=\"italics\">[latex]Q[\/latex]<\/em> and not on the test charge [latex]q[\/latex]. The electric field is defined in such a manner that it represents only the charge creating it and is unique at every point in space. Specifically, the electric field [latex]E[\/latex] is defined to be the ratio of the Coulomb force to the test charge:\n<\/p>\n<div data-type=\"equation\" class=\"equation\">[latex]\\mathbf{\\text{E}}=\\frac{\\mathbf{\\text{F}}}{q},[\/latex]<\/div>\n<p id=\"import-auto-id1576569\">where [latex]\\mathbf{\\text{F}}[\/latex] is the electrostatic force (or Coulomb force) exerted on a positive test charge<br>\n[latex]q[\/latex]. It is understood that<br>\n[latex]\\mathbf{\\text{E}}[\/latex] is in the same direction as<br>\n[latex]\\mathbf{\\text{F}}[\/latex]. It is also assumed that [latex]q[\/latex] is so small that it does not alter the charge distribution creating the electric field. The units of electric field are newtons per coulomb (N\/C). If the electric field is known, then the electrostatic force on any charge [latex]q[\/latex] is simply obtained by multiplying charge times electric field, or [latex]\\mathbf{\\text{F}}=q\\mathbf{\\text{E}}[\/latex]. Consider the electric field due to a point charge [latex]Q[\/latex]. According to Coulomb\u2019s law, the force it exerts on a test charge <\/p>\n<p>[latex]q[\/latex] is <\/p>\n<p>[latex]F=k|\\mathrm{qQ}|\/{r}^{2}[\/latex]. Thus the magnitude of the electric field,<br>\n[latex]E[\/latex], for a point charge is<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-588\">[latex]E=|\\frac{F}{q}|=k|\\frac{\\text{qQ}}{{\\mathrm{qr}}^{2}}|=k\\frac{|Q|}{{r}^{2}}.[\/latex]<\/div>\n<p id=\"import-auto-id3008249\">Since the test charge cancels, we see that<\/p>\n<div data-type=\"equation\" class=\"equation\">[latex]E=k\\frac{|Q|}{{r}^{2}}.[\/latex]<\/div>\n<p>The electric field is thus seen to depend only on the charge <em data-effect=\"italics\">[latex]Q[\/latex]<\/em> and the distance [latex]r[\/latex]; it is completely independent of the test charge [latex]q[\/latex].<\/p>\n<div data-type=\"example\" class=\"textbox examples\">\n<div data-type=\"title\" class=\"title\">Calculating the Electric Field of a Point Charge<\/div>\n<p id=\"import-auto-id3028516\">Calculate the strength and direction of the electric field [latex]E[\/latex] due to a point charge of 2.00 nC (nano-Coulombs) at a distance of 5.00 mm from the charge.<\/p>\n<p id=\"import-auto-id1427922\"><strong>Strategy<\/strong><\/p>\n<p id=\"import-auto-id1448833\">We can find the electric field created by a point charge by using the equation [latex]E=\\text{kQ}\/{r}^{2}[\/latex].<\/p>\n<p id=\"import-auto-id408742\"><strong>Solution<\/strong><\/p>\n<p id=\"import-auto-id2661548\">Here [latex]Q=2\\text{.}\\text{00}\u00d7{\\text{10}}^{-9}[\/latex] C and [latex]r=5\\text{.}\\text{00}\u00d7{\\text{10}}^{-3}[\/latex] m. Entering those values into the above equation gives<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-380\">[latex]\\begin{array}{lll}E&amp; =&amp; k\\frac{Q}{{r}^{2}}\\\\ &amp; =&amp; \\left(\\text{8.99}\u00d7{\\text{10}}^{9}\\phantom{\\rule{0.25em}{0ex}}\\text{N}\\cdot {\\text{m}}^{2}{\\text{\/C}}^{2}\\right)\u00d7\\frac{\\left(\\text{2.00}\u00d7{\\text{10}}^{-9}\\phantom{\\rule{0.25em}{0ex}}\\text{C}\\right)}{\\left(\\text{5.00}\u00d7{\\text{10}}^{-3}\\phantom{\\rule{0.25em}{0ex}}\\text{m}{\\right)}^{2}}\\\\ &amp; =&amp; \\text{7.19}\u00d7{\\text{10}}^{5}\\phantom{\\rule{0.25em}{0ex}}\\text{N\/C.}\\end{array}[\/latex]<\/div>\n<p id=\"import-auto-id2452411\"><strong>Discussion<\/strong><\/p>\n<p id=\"import-auto-id2442802\">This <span data-type=\"term\" id=\"import-auto-id3085589\">electric field strength<\/span> is the same at any point 5.00 mm away from the charge <em data-effect=\"italics\">[latex]Q[\/latex]<\/em> that creates the field. It is positive, meaning that it has a direction pointing away from the charge <em data-effect=\"italics\">[latex]Q[\/latex]<\/em>.<\/p>\n<\/div>\n<div data-type=\"example\" class=\"textbox examples\" id=\"fs-id2429320\">\n<div data-type=\"title\" class=\"title\">Calculating the Force Exerted on a Point Charge by an Electric Field<\/div>\n<p id=\"import-auto-id2598572\">What force does the electric field found in the previous example exert on a point charge of [latex]\u20130.250\\phantom{\\rule{0.25em}{0ex}}\\mu \\text{C}[\/latex]?<\/p>\n<p id=\"import-auto-id2672938\"><strong>Strategy<\/strong><\/p>\n<p id=\"import-auto-id2424249\">Since we know the electric field strength and the charge in the field, the force on that charge can be calculated using the definition of electric field [latex]\\mathbf{\\text{E}}=\\mathbf{\\text{F}}\/q[\/latex] rearranged to [latex]\\mathbf{\\text{F}}=q\\mathbf{\\text{E}}[\/latex].<\/p>\n<p id=\"import-auto-id2055367\"><strong>Solution<\/strong><\/p>\n<p id=\"import-auto-id3356261\">The magnitude of the force on a charge [latex]q=-0\\text{.}\\text{250}\\phantom{\\rule{0.25em}{0ex}}\\text{\u03bcC}[\/latex] exerted by a field of strength [latex]E=7\\text{.}\\text{20}\u00d7{\\text{10}}^{5}[\/latex] N\/C is thus,<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-502\">[latex]\\begin{array}{lll}F&amp; =&amp; -\\text{qE}\\\\ &amp; =&amp; \\left(\\text{0.250}\u00d7{\\text{10}}^{\\text{\u20136}}\\phantom{\\rule{0.25em}{0ex}}\\text{C}\\right)\\left(7.20\u00d7{\\text{10}}^{5}\\phantom{\\rule{0.25em}{0ex}}\\text{N\/C}\\right)\\\\ &amp; =&amp; \\text{0.180 N.}\\end{array}[\/latex]<\/div>\n<p>Because  [latex]q[\/latex] is negative, the force is directed opposite to the direction of the field.<\/p>\n<p id=\"import-auto-id2979684\"><strong>Discussion<\/strong><\/p>\n<p id=\"import-auto-id1997901\">The force is attractive, as expected for unlike charges. (The field was created by a positive charge and here acts on a negative charge.) The charges in this example are typical of common static electricity, and the modest attractive force obtained is similar to forces experienced in static cling and similar situations.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"note\" class=\"note\" data-has-label=\"true\" data-label=\"\">\n<div data-type=\"title\" class=\"title\">PhET Explorations: Electric Field of Dreams<\/div>\n<p id=\"eip-161\">Play ball! Add charges to the Field of Dreams and see how they react to the electric field. Turn on a background electric field and adjust the direction and magnitude.<\/p>\n<div class=\"bc-figure figure\" id=\"eip-id1694688\">\n<div class=\"bc-figcaption figcaption\"><a href=\"\/resources\/5cbe37ce1e93b892ccf08efaf855a23fdd3227b2\/efield_en.jar\">Electric Field of Dreams<\/a><\/div>\n<p><span data-type=\"media\" id=\"Phet_module_19.4\" data-alt=\"\"><a href=\"\/resources\/5cbe37ce1e93b892ccf08efaf855a23fdd3227b2\/efield_en.jar\" data-type=\"image\"><img src=\"https:\/\/pressbooks.bccampus.ca\/clalonde\/wp-content\/uploads\/sites\/280\/2017\/10\/PhET_Icon.png\" data-media-type=\"image\/png\" alt=\"\" data-print=\"false\" width=\"450\"><\/a><span data-media-type=\"image\/png\" data-print=\"true\" data-src=\"\/resources\/075500ad9f71890a85fe3f7a4137ac08e2b7907c\/PhET_Icon.png\" data-type=\"image\"><\/span><\/span><\/p><\/div>\n<\/div>\n<div class=\"section-summary\" data-depth=\"1\" id=\"fs-id2684001\">\n<h1 data-type=\"title\">Section Summary<\/h1>\n<ul id=\"fs-id2957147\">\n<li id=\"import-auto-id3389125\">The electrostatic force field surrounding a charged object extends out into space in all directions.<\/li>\n<li id=\"import-auto-id3054736\">The electrostatic force exerted by a point charge on a test charge at a distance [latex]r[\/latex] depends on the charge of both charges, as well as the distance between the two.<\/li>\n<li id=\"import-auto-id1427738\">The electric field [latex]\\mathbf{\\text{E}}[\/latex] is defined to be\n<div data-type=\"equation\" class=\"equation\">[latex]\\mathbf{\\text{E}}=\\frac{\\mathbf{\\text{F}}}{q,}[\/latex]<\/div>\n<p>where [latex]\\mathbf{\\text{F}}[\/latex] is the Coulomb or electrostatic force exerted on a small positive test charge [latex]q[\/latex]. [latex]\\mathbf{\\text{E}}[\/latex] has units of N\/C.<\/p>\n<\/li>\n<li id=\"import-auto-id1486899\">The magnitude of the electric field [latex]\\mathbf{\\text{E}}[\/latex] created by a point charge <em data-effect=\"italics\">[latex]Q[\/latex]<\/em> is\n<div data-type=\"equation\" class=\"equation\">[latex]\\mathbf{\\text{E}}=k\\frac{|Q|}{{r}^{2}}.[\/latex]<\/div>\n<p id=\"import-auto-id2437723\">where [latex]r[\/latex] is the distance from <em data-effect=\"italics\">[latex]Q[\/latex]<\/em>. The electric field [latex]\\mathbf{\\text{E}}[\/latex] is a vector and fields due to multiple charges add like vectors.<\/p>\n<\/li>\n<\/ul>\n<\/div>\n<div class=\"conceptual-questions\" data-depth=\"1\" data-element-type=\"conceptual-questions\">\n<h1 data-type=\"title\">Conceptual Questions<\/h1>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id3047767\" data-element-type=\"conceptual-questions\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1958513\">\n<p>Why must the test charge [latex]q[\/latex] in the definition of the electric field be vanishingly small?<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id2668714\" data-element-type=\"conceptual-questions\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id2972867\">\n<p id=\"import-auto-id2450132\">Are the direction and magnitude of the Coulomb force unique at a given point in space? What about the electric field?<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"problems-exercises\" data-depth=\"1\" data-element-type=\"problems-exercises\">\n<h1 data-type=\"title\">Problem Exercises<\/h1>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1586790\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id3037335\">\n<p id=\"import-auto-id1908117\">What is the magnitude and direction of an electric field that exerts a [latex]2\\text{.}\\text{00}\u00d7{\\text{10}}^{-5}\\phantom{\\rule{0.25em}{0ex}}\\text{N}[\/latex] upward force on a [latex]\u20131.75\\phantom{\\rule{0.25em}{0ex}}\\mu \\text{C}[\/latex] charge?<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id2989956\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id2684116\">\n<p id=\"import-auto-id2669656\">What is the magnitude and direction of the force exerted on a [latex]3.50\\phantom{\\rule{0.25em}{0ex}}\\mu \\text{C}[\/latex] charge by a 250 N\/C electric field that points due east?<\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\">\n<p id=\"import-auto-id1386271\">[latex]8\\text{.}\\text{75}\u00d7{\\text{10}}^{-4}[\/latex] N<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id2588568\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id3042804\">\n<p id=\"import-auto-id1888472\">Calculate the magnitude of the electric field 2.00 m from a point charge of 5.00 mC (such as found on the terminal of a Van de Graaff).<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1537546\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id3073177\">\n<p>(a) What magnitude point charge creates a 10,000 N\/C electric field at a distance of 0.250 m? (b) How large is the field at 10.0 m?<\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"fs-id2410613\">\n<p id=\"import-auto-id3398477\">(a) [latex]6\\text{.}\\text{94}\u00d7{\\text{10}}^{-8}\\phantom{\\rule{0.25em}{0ex}}\\text{C}[\/latex]<\/p>\n<p id=\"import-auto-id3122616\">(b) [latex]6\\text{.}\\text{25}\\phantom{\\rule{0.25em}{0ex}}\\text{N\/C}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\">\n<p>Calculate the initial (from rest) acceleration of a proton in a [latex]5\\text{.}\\text{00}\u00d7{\\text{10}}^{6}\\phantom{\\rule{0.25em}{0ex}}\\text{N\/C}[\/latex] electric field (such as created by a research Van de Graaff). Explicitly show how you follow the steps in the Problem-Solving Strategy for electrostatics.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id2687821\">\n<p id=\"import-auto-id3042498\">(a) Find the direction and magnitude of an electric field that exerts a [latex]4\\text{.}\\text{80}\u00d7{\\text{10}}^{-\\text{17}}\\phantom{\\rule{0.25em}{0ex}}\\text{N}[\/latex] westward force on an electron. (b) What magnitude and direction force does this field exert on a proton?<\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"fs-id1599293\">\n<p id=\"import-auto-id1845690\">(a) [latex]\\text{300}\\phantom{\\rule{0.25em}{0ex}}\\text{N\/C}\\phantom{\\rule{0.25em}{0ex}}\\left(\\text{east}\\right)[\/latex]<\/p>\n<p id=\"import-auto-id2401155\">(b) [latex]4\\text{.}\\text{80}\u00d7{\\text{10}}^{-\\text{17}}\\phantom{\\rule{0.25em}{0ex}}\\text{N}\\phantom{\\rule{0.25em}{0ex}}\\left(\\text{east}\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"glossary\" class=\"textbox shaded\">\n<h2 data-type=\"glossary-title\">Glossary<\/h2>\n<dl class=\"definition\" id=\"import-auto-id2057931\">\n<dt>field<\/dt>\n<dd id=\"fs-id1899657\">a map of the amount and direction of a force acting on other objects, extending out into space<\/dd>\n<\/dl>\n<dl class=\"definition\" id=\"import-auto-id2017072\">\n<dt>point charge<\/dt>\n<dd id=\"fs-id3077230\">A charged particle, designated <em data-effect=\"italics\">[latex]Q[\/latex],<\/em> generating an electric field<\/dd>\n<\/dl>\n<dl class=\"definition\">\n<dt>test charge<\/dt>\n<dd id=\"fs-id742532\">A particle (designated [latex]q[\/latex]) with either a positive or negative charge set down within an electric field generated by a point charge<\/dd>\n<\/dl>\n<\/div>\n\n","rendered":"<div class=\"textbox learning-objectives\">\n<h3 itemprop=\"educationalUse\">Learning Objectives<\/h3>\n<ul>\n<li>Describe a force field and calculate the strength of an electric field due to a point charge.<\/li>\n<li>Calculate the force exerted on a test charge by an electric field.<\/li>\n<li>Explain the relationship between electrical force (F) on a test charge and electrical field strength (E).<\/li>\n<\/ul>\n<\/div>\n<p id=\"import-auto-id2688112\">Contact forces, such as between a baseball and a bat, are explained on the small scale by the interaction of the charges in atoms and molecules in close proximity. They interact through forces that include the <span data-type=\"term\" id=\"import-auto-id2648325\">Coulomb force<\/span>. Action at a distance is a force between objects that are not close enough for their atoms to \u201ctouch.\u201d That is, they are separated by more than a few atomic diameters.<\/p>\n<p id=\"import-auto-id2590313\">For example, a charged rubber comb attracts neutral bits of paper from a distance via the Coulomb force. It is very useful to think of an object being surrounded in space by a <span data-type=\"term\" id=\"import-auto-id3145066\">force field<\/span>. The force field carries the force to another object (called a test object) some distance away.<\/p>\n<div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id2662859\">\n<h1 data-type=\"title\">Concept of a Field<\/h1>\n<p id=\"import-auto-id1954099\">A field is a way of conceptualizing and mapping the force that surrounds any object and acts on another object at a distance without apparent physical connection. For example, the gravitational field surrounding the earth (and all other masses) represents the gravitational force that would be experienced if another mass were placed at a given point within the field.<\/p>\n<p id=\"import-auto-id3408279\">In the same way, the Coulomb force field surrounding any charge extends throughout space. Using Coulomb\u2019s law, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-fdf122727037317511e7ec0794f25ddc_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#70;&#61;&#107;&#124;&#123;&#113;&#125;&#95;&#123;&#49;&#125;&#123;&#113;&#125;&#95;&#123;&#50;&#125;&#124;&#47;&#123;&#114;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"113\" style=\"vertical-align: -5px;\" \/>, its magnitude is given by the equation <\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-6df39685c06225f3a902145fbe087453_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#70;&#61;&#107;&#124;&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#113;&#81;&#125;&#124;&#47;&#123;&#114;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"105\" style=\"vertical-align: -5px;\" \/>, for a <span data-type=\"term\" id=\"import-auto-id1916842\">point charge<\/span> (a particle having a charge <em data-effect=\"italics\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-2c758bec4c272382411b95fc0e7ee250_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#81;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"14\" style=\"vertical-align: -4px;\" \/><\/em>) acting on a <span data-type=\"term\" id=\"import-auto-id2383341\">test charge <\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-ac7da57d7f507262338bb5168feb3e06_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#113;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"8\" style=\"vertical-align: -4px;\" \/> at a distance <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-c409433a9e2dfcdb83360a974d243f18_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#114;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"8\" style=\"vertical-align: 0px;\" \/> (see <a href=\"#import-auto-id2408057\" class=\"autogenerated-content\">(Figure)<\/a>). Both the magnitude and direction of the Coulomb force field depend on <em data-effect=\"italics\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-2c758bec4c272382411b95fc0e7ee250_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#81;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"14\" style=\"vertical-align: -4px;\" \/><\/em> and the test charge <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-ac7da57d7f507262338bb5168feb3e06_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#113;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"8\" style=\"vertical-align: -4px;\" \/>.<\/p>\n<div class=\"bc-figure figure\" id=\"import-auto-id2408057\">\n<div class=\"bc-figcaption figcaption\">The Coulomb force field due to a positive charge <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-2c758bec4c272382411b95fc0e7ee250_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#81;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"14\" style=\"vertical-align: -4px;\" \/> is shown acting on two different charges. Both charges are the same distance from <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-2c758bec4c272382411b95fc0e7ee250_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#81;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"14\" style=\"vertical-align: -4px;\" \/>. (a) Since <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-fdaff7da50dc5681038d2dd80303f36a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#113;&#125;&#95;&#123;&#49;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"14\" style=\"vertical-align: -4px;\" \/> is positive, the force <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-0bcc807371c5b58f028eea711c3c5d73_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#70;&#125;&#95;&#123;&#49;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"17\" style=\"vertical-align: -4px;\" \/> acting on it is repulsive. (b) The charge <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-77eb47258ca86c69e883fdf19176e199_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#113;&#125;&#95;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"15\" style=\"vertical-align: -4px;\" \/> is negative and greater in magnitude than <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-fdaff7da50dc5681038d2dd80303f36a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#113;&#125;&#95;&#123;&#49;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"14\" style=\"vertical-align: -4px;\" \/>, and so the force <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-1199024cc1ce9d55a90e7fd3e350fe2f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#70;&#125;&#95;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"18\" style=\"vertical-align: -3px;\" \/> acting on it is attractive and stronger than <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-0bcc807371c5b58f028eea711c3c5d73_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#70;&#125;&#95;&#123;&#49;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"17\" style=\"vertical-align: -4px;\" \/>. The Coulomb force field is thus not unique at any point in space, because it depends on the test charges <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-fdaff7da50dc5681038d2dd80303f36a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#113;&#125;&#95;&#123;&#49;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"14\" style=\"vertical-align: -4px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-77eb47258ca86c69e883fdf19176e199_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#113;&#125;&#95;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"15\" style=\"vertical-align: -4px;\" \/> as well as the charge <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-2c758bec4c272382411b95fc0e7ee250_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#81;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"14\" style=\"vertical-align: -4px;\" \/>.<\/div>\n<p><span data-type=\"media\" id=\"import-auto-id2584308\" data-alt=\"In part a, two charges Q and q one are placed at a distance r. The force vector F one on charge q one is shown by an arrow pointing toward right away from Q. In part b, two charges Q and q two are placed at a distance r. The force vector F two on charge q two is shown by an arrow pointing toward left toward Q.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/clalonde\/wp-content\/uploads\/sites\/280\/2017\/10\/Figure_19_04_02a.jpg\" data-media-type=\"image\/jpg\" alt=\"In part a, two charges Q and q one are placed at a distance r. The force vector F one on charge q one is shown by an arrow pointing toward right away from Q. In part b, two charges Q and q two are placed at a distance r. The force vector F two on charge q two is shown by an arrow pointing toward left toward Q.\" width=\"200\" \/><\/span><\/p>\n<\/div>\n<p>To simplify things, we would prefer to have a field that depends only on <em data-effect=\"italics\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-2c758bec4c272382411b95fc0e7ee250_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#81;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"14\" style=\"vertical-align: -4px;\" \/><\/em> and not on the test charge <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-ac7da57d7f507262338bb5168feb3e06_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#113;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"8\" style=\"vertical-align: -4px;\" \/>. The electric field is defined in such a manner that it represents only the charge creating it and is unique at every point in space. Specifically, the electric field <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-764e1c770271f92700e1a4fbce46c668_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#69;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"14\" style=\"vertical-align: 0px;\" \/> is defined to be the ratio of the Coulomb force to the test charge:\n<\/p>\n<div data-type=\"equation\" class=\"equation\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-3b3e49d334afd38d5b9302428f81579c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#69;&#125;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#70;&#125;&#125;&#125;&#123;&#113;&#125;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"53\" style=\"vertical-align: -9px;\" \/><\/div>\n<p id=\"import-auto-id1576569\">where <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-c5cd16916557096d431ff3a1af0c9119_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#70;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"11\" style=\"vertical-align: -1px;\" \/> is the electrostatic force (or Coulomb force) exerted on a positive test charge<br \/>\n<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-ac7da57d7f507262338bb5168feb3e06_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#113;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"8\" style=\"vertical-align: -4px;\" \/>. It is understood that<br \/>\n<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-9fc16303eb82a65e9d2ecd5cb242a595_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#69;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"12\" style=\"vertical-align: 0px;\" \/> is in the same direction as<br \/>\n<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-c5cd16916557096d431ff3a1af0c9119_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#70;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"11\" style=\"vertical-align: -1px;\" \/>. It is also assumed that <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-ac7da57d7f507262338bb5168feb3e06_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#113;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"8\" style=\"vertical-align: -4px;\" \/> is so small that it does not alter the charge distribution creating the electric field. The units of electric field are newtons per coulomb (N\/C). If the electric field is known, then the electrostatic force on any charge <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-ac7da57d7f507262338bb5168feb3e06_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#113;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"8\" style=\"vertical-align: -4px;\" \/> is simply obtained by multiplying charge times electric field, or <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-b5d7d35a7fb5937538bf16a99e179e2d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#70;&#125;&#125;&#61;&#113;&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#69;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"56\" style=\"vertical-align: -4px;\" \/>. Consider the electric field due to a point charge <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-2c758bec4c272382411b95fc0e7ee250_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#81;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"14\" style=\"vertical-align: -4px;\" \/>. According to Coulomb\u2019s law, the force it exerts on a test charge <\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-ac7da57d7f507262338bb5168feb3e06_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#113;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"8\" style=\"vertical-align: -4px;\" \/> is <\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-6df39685c06225f3a902145fbe087453_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#70;&#61;&#107;&#124;&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#113;&#81;&#125;&#124;&#47;&#123;&#114;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"105\" style=\"vertical-align: -5px;\" \/>. Thus the magnitude of the electric field,<br \/>\n<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-764e1c770271f92700e1a4fbce46c668_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#69;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"14\" style=\"vertical-align: 0px;\" \/>, for a point charge is<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-588\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-40fc4ae9c161ff135f49d5b6a48ee8bf_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#69;&#61;&#124;&#92;&#102;&#114;&#97;&#99;&#123;&#70;&#125;&#123;&#113;&#125;&#124;&#61;&#107;&#124;&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#113;&#81;&#125;&#125;&#123;&#123;&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#113;&#114;&#125;&#125;&#94;&#123;&#50;&#125;&#125;&#124;&#61;&#107;&#92;&#102;&#114;&#97;&#99;&#123;&#124;&#81;&#124;&#125;&#123;&#123;&#114;&#125;&#94;&#123;&#50;&#125;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"29\" width=\"188\" style=\"vertical-align: -10px;\" \/><\/div>\n<p id=\"import-auto-id3008249\">Since the test charge cancels, we see that<\/p>\n<div data-type=\"equation\" class=\"equation\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-a67efa117055e5352b4e16c7be9f8de8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#69;&#61;&#107;&#92;&#102;&#114;&#97;&#99;&#123;&#124;&#81;&#124;&#125;&#123;&#123;&#114;&#125;&#94;&#123;&#50;&#125;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"74\" style=\"vertical-align: -7px;\" \/><\/div>\n<p>The electric field is thus seen to depend only on the charge <em data-effect=\"italics\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-2c758bec4c272382411b95fc0e7ee250_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#81;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"14\" style=\"vertical-align: -4px;\" \/><\/em> and the distance <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-c409433a9e2dfcdb83360a974d243f18_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#114;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"8\" style=\"vertical-align: 0px;\" \/>; it is completely independent of the test charge <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-ac7da57d7f507262338bb5168feb3e06_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#113;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"8\" style=\"vertical-align: -4px;\" \/>.<\/p>\n<div data-type=\"example\" class=\"textbox examples\">\n<div data-type=\"title\" class=\"title\">Calculating the Electric Field of a Point Charge<\/div>\n<p id=\"import-auto-id3028516\">Calculate the strength and direction of the electric field <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-764e1c770271f92700e1a4fbce46c668_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#69;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"14\" style=\"vertical-align: 0px;\" \/> due to a point charge of 2.00 nC (nano-Coulombs) at a distance of 5.00 mm from the charge.<\/p>\n<p id=\"import-auto-id1427922\"><strong>Strategy<\/strong><\/p>\n<p id=\"import-auto-id1448833\">We can find the electric field created by a point charge by using the equation <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-f207f8aa5df78102a360583e5a549166_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#69;&#61;&#92;&#116;&#101;&#120;&#116;&#123;&#107;&#81;&#125;&#47;&#123;&#114;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"85\" style=\"vertical-align: -5px;\" \/>.<\/p>\n<p id=\"import-auto-id408742\"><strong>Solution<\/strong><\/p>\n<p id=\"import-auto-id2661548\">Here <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-bdbeb8f52da06a7f642e14010bbf1d90_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#81;&#61;&#50;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#48;&#48;&#125;&times;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#125;&#125;&#94;&#123;&#45;&#57;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"105\" style=\"vertical-align: -4px;\" \/> C and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-a02780b1dadc05766eb34ba55ba015df_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#114;&#61;&#53;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#48;&#48;&#125;&times;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#125;&#125;&#94;&#123;&#45;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"99\" style=\"vertical-align: -1px;\" \/> m. Entering those values into the above equation gives<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-380\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-3bff9b57b84bad7f4cd3ccce1a601ebd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#108;&#108;&#108;&#125;&#69;&#38;&#32;&#61;&#38;&#32;&#107;&#92;&#102;&#114;&#97;&#99;&#123;&#81;&#125;&#123;&#123;&#114;&#125;&#94;&#123;&#50;&#125;&#125;&#92;&#92;&#32;&#38;&#32;&#61;&#38;&#32;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#116;&#101;&#120;&#116;&#123;&#56;&#46;&#57;&#57;&#125;&times;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#125;&#125;&#94;&#123;&#57;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#78;&#125;&#92;&#99;&#100;&#111;&#116;&#32;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#109;&#125;&#125;&#94;&#123;&#50;&#125;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#47;&#67;&#125;&#125;&#94;&#123;&#50;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&times;&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#116;&#101;&#120;&#116;&#123;&#50;&#46;&#48;&#48;&#125;&times;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#125;&#125;&#94;&#123;&#45;&#57;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#67;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#116;&#101;&#120;&#116;&#123;&#53;&#46;&#48;&#48;&#125;&times;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#125;&#125;&#94;&#123;&#45;&#51;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#109;&#125;&#123;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#125;&#92;&#92;&#32;&#38;&#32;&#61;&#38;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#55;&#46;&#49;&#57;&#125;&times;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#125;&#125;&#94;&#123;&#53;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#78;&#47;&#67;&#46;&#125;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"84\" width=\"319\" style=\"vertical-align: -36px;\" \/><\/div>\n<p id=\"import-auto-id2452411\"><strong>Discussion<\/strong><\/p>\n<p id=\"import-auto-id2442802\">This <span data-type=\"term\" id=\"import-auto-id3085589\">electric field strength<\/span> is the same at any point 5.00 mm away from the charge <em data-effect=\"italics\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-2c758bec4c272382411b95fc0e7ee250_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#81;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"14\" style=\"vertical-align: -4px;\" \/><\/em> that creates the field. It is positive, meaning that it has a direction pointing away from the charge <em data-effect=\"italics\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-2c758bec4c272382411b95fc0e7ee250_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#81;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"14\" style=\"vertical-align: -4px;\" \/><\/em>.<\/p>\n<\/div>\n<div data-type=\"example\" class=\"textbox examples\" id=\"fs-id2429320\">\n<div data-type=\"title\" class=\"title\">Calculating the Force Exerted on a Point Charge by an Electric Field<\/div>\n<p id=\"import-auto-id2598572\">What force does the electric field found in the previous example exert on a point charge of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-6c5fa17e47779f469568453594b96509_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#48;&#46;&#50;&#53;&#48;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#109;&#117;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#67;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"81\" style=\"vertical-align: -4px;\" \/>?<\/p>\n<p id=\"import-auto-id2672938\"><strong>Strategy<\/strong><\/p>\n<p id=\"import-auto-id2424249\">Since we know the electric field strength and the charge in the field, the force on that charge can be calculated using the definition of electric field <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-1fb593f60512741f83aaac7e65ca5e4e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#69;&#125;&#125;&#61;&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#70;&#125;&#125;&#47;&#113;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"64\" style=\"vertical-align: -5px;\" \/> rearranged to <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-b5d7d35a7fb5937538bf16a99e179e2d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#70;&#125;&#125;&#61;&#113;&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#69;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"56\" style=\"vertical-align: -4px;\" \/>.<\/p>\n<p id=\"import-auto-id2055367\"><strong>Solution<\/strong><\/p>\n<p id=\"import-auto-id3356261\">The magnitude of the force on a charge <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-c3d49e19faca65fe9f877d85b6098782_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#113;&#61;&#45;&#48;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#50;&#53;&#48;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&mu;&#67;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"103\" style=\"vertical-align: -4px;\" \/> exerted by a field of strength <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-0f9e3f420b4090abd305ffbca4ef97bc_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#69;&#61;&#55;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#50;&#48;&#125;&times;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#125;&#125;&#94;&#123;&#53;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"94\" style=\"vertical-align: -1px;\" \/> N\/C is thus,<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-502\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-ada1c6aa596bc602ab0d84698cb70c5b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#108;&#108;&#108;&#125;&#70;&#38;&#32;&#61;&#38;&#32;&#45;&#92;&#116;&#101;&#120;&#116;&#123;&#113;&#69;&#125;&#92;&#92;&#32;&#38;&#32;&#61;&#38;&#32;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#116;&#101;&#120;&#116;&#123;&#48;&#46;&#50;&#53;&#48;&#125;&times;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#125;&#125;&#94;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#45;&#54;&#125;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#67;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#55;&#46;&#50;&#48;&times;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#125;&#125;&#94;&#123;&#53;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#78;&#47;&#67;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#92;&#32;&#38;&#32;&#61;&#38;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#48;&#46;&#49;&#56;&#48;&#32;&#78;&#46;&#125;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"57\" width=\"277\" style=\"vertical-align: -23px;\" \/><\/div>\n<p>Because  <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-ac7da57d7f507262338bb5168feb3e06_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#113;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"8\" style=\"vertical-align: -4px;\" \/> is negative, the force is directed opposite to the direction of the field.<\/p>\n<p id=\"import-auto-id2979684\"><strong>Discussion<\/strong><\/p>\n<p id=\"import-auto-id1997901\">The force is attractive, as expected for unlike charges. (The field was created by a positive charge and here acts on a negative charge.) The charges in this example are typical of common static electricity, and the modest attractive force obtained is similar to forces experienced in static cling and similar situations.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"note\" class=\"note\" data-has-label=\"true\" data-label=\"\">\n<div data-type=\"title\" class=\"title\">PhET Explorations: Electric Field of Dreams<\/div>\n<p id=\"eip-161\">Play ball! Add charges to the Field of Dreams and see how they react to the electric field. Turn on a background electric field and adjust the direction and magnitude.<\/p>\n<div class=\"bc-figure figure\" id=\"eip-id1694688\">\n<div class=\"bc-figcaption figcaption\"><a href=\"\/resources\/5cbe37ce1e93b892ccf08efaf855a23fdd3227b2\/efield_en.jar\">Electric Field of Dreams<\/a><\/div>\n<p><span data-type=\"media\" id=\"Phet_module_19.4\" data-alt=\"\"><a href=\"\/resources\/5cbe37ce1e93b892ccf08efaf855a23fdd3227b2\/efield_en.jar\" data-type=\"image\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/clalonde\/wp-content\/uploads\/sites\/280\/2017\/10\/PhET_Icon.png\" data-media-type=\"image\/png\" alt=\"\" data-print=\"false\" width=\"450\" \/><\/a><span data-media-type=\"image\/png\" data-print=\"true\" data-src=\"\/resources\/075500ad9f71890a85fe3f7a4137ac08e2b7907c\/PhET_Icon.png\" data-type=\"image\"><\/span><\/span><\/p>\n<\/div>\n<\/div>\n<div class=\"section-summary\" data-depth=\"1\" id=\"fs-id2684001\">\n<h1 data-type=\"title\">Section Summary<\/h1>\n<ul id=\"fs-id2957147\">\n<li id=\"import-auto-id3389125\">The electrostatic force field surrounding a charged object extends out into space in all directions.<\/li>\n<li id=\"import-auto-id3054736\">The electrostatic force exerted by a point charge on a test charge at a distance <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-c409433a9e2dfcdb83360a974d243f18_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#114;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"8\" style=\"vertical-align: 0px;\" \/> depends on the charge of both charges, as well as the distance between the two.<\/li>\n<li id=\"import-auto-id1427738\">The electric field <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-9fc16303eb82a65e9d2ecd5cb242a595_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#69;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"12\" style=\"vertical-align: 0px;\" \/> is defined to be\n<div data-type=\"equation\" class=\"equation\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-00b517b46e0663330bc977c949f5d988_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#69;&#125;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#70;&#125;&#125;&#125;&#123;&#113;&#44;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"49\" style=\"vertical-align: -9px;\" \/><\/div>\n<p>where <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-c5cd16916557096d431ff3a1af0c9119_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#70;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"11\" style=\"vertical-align: -1px;\" \/> is the Coulomb or electrostatic force exerted on a small positive test charge <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-ac7da57d7f507262338bb5168feb3e06_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#113;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"8\" style=\"vertical-align: -4px;\" \/>. <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-9fc16303eb82a65e9d2ecd5cb242a595_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#69;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"12\" style=\"vertical-align: 0px;\" \/> has units of N\/C.<\/p>\n<\/li>\n<li id=\"import-auto-id1486899\">The magnitude of the electric field <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-9fc16303eb82a65e9d2ecd5cb242a595_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#69;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"12\" style=\"vertical-align: 0px;\" \/> created by a point charge <em data-effect=\"italics\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-2c758bec4c272382411b95fc0e7ee250_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#81;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"14\" style=\"vertical-align: -4px;\" \/><\/em> is\n<div data-type=\"equation\" class=\"equation\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-158373f5e71cfb39c566306cb45b56e9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#69;&#125;&#125;&#61;&#107;&#92;&#102;&#114;&#97;&#99;&#123;&#124;&#81;&#124;&#125;&#123;&#123;&#114;&#125;&#94;&#123;&#50;&#125;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"72\" style=\"vertical-align: -7px;\" \/><\/div>\n<p id=\"import-auto-id2437723\">where <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-c409433a9e2dfcdb83360a974d243f18_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#114;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"8\" style=\"vertical-align: 0px;\" \/> is the distance from <em data-effect=\"italics\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-2c758bec4c272382411b95fc0e7ee250_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#81;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"14\" style=\"vertical-align: -4px;\" \/><\/em>. The electric field <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-9fc16303eb82a65e9d2ecd5cb242a595_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#69;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"12\" style=\"vertical-align: 0px;\" \/> is a vector and fields due to multiple charges add like vectors.<\/p>\n<\/li>\n<\/ul>\n<\/div>\n<div class=\"conceptual-questions\" data-depth=\"1\" data-element-type=\"conceptual-questions\">\n<h1 data-type=\"title\">Conceptual Questions<\/h1>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id3047767\" data-element-type=\"conceptual-questions\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1958513\">\n<p>Why must the test charge <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-ac7da57d7f507262338bb5168feb3e06_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#113;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"8\" style=\"vertical-align: -4px;\" \/> in the definition of the electric field be vanishingly small?<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id2668714\" data-element-type=\"conceptual-questions\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id2972867\">\n<p id=\"import-auto-id2450132\">Are the direction and magnitude of the Coulomb force unique at a given point in space? What about the electric field?<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"problems-exercises\" data-depth=\"1\" data-element-type=\"problems-exercises\">\n<h1 data-type=\"title\">Problem Exercises<\/h1>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1586790\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id3037335\">\n<p id=\"import-auto-id1908117\">What is the magnitude and direction of an electric field that exerts a <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-2fa71beccb38302800b5a42d6c7fd05a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#48;&#48;&#125;&times;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#125;&#125;&#94;&#123;&#45;&#53;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#78;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"85\" style=\"vertical-align: -1px;\" \/> upward force on a <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-30836969caabc8ce9987457eec7d5243_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#49;&#46;&#55;&#53;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#109;&#117;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#67;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"72\" style=\"vertical-align: -4px;\" \/> charge?<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id2989956\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id2684116\">\n<p id=\"import-auto-id2669656\">What is the magnitude and direction of the force exerted on a <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-c9e6b3b3050d8bd9d3562859f0e21800_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#46;&#53;&#48;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#109;&#117;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#67;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"59\" style=\"vertical-align: -4px;\" \/> charge by a 250 N\/C electric field that points due east?<\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\">\n<p id=\"import-auto-id1386271\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-49aa452519d5fa1208bc7399d1bc7bf0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#56;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#55;&#53;&#125;&times;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#125;&#125;&#94;&#123;&#45;&#52;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"67\" style=\"vertical-align: -1px;\" \/> N<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id2588568\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id3042804\">\n<p id=\"import-auto-id1888472\">Calculate the magnitude of the electric field 2.00 m from a point charge of 5.00 mC (such as found on the terminal of a Van de Graaff).<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1537546\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id3073177\">\n<p>(a) What magnitude point charge creates a 10,000 N\/C electric field at a distance of 0.250 m? (b) How large is the field at 10.0 m?<\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"fs-id2410613\">\n<p id=\"import-auto-id3398477\">(a) <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-0c23c1c6c2419346812e464b963ffe43_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#54;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#57;&#52;&#125;&times;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#125;&#125;&#94;&#123;&#45;&#56;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#67;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"84\" style=\"vertical-align: -1px;\" \/><\/p>\n<p id=\"import-auto-id3122616\">(b) <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-9dffb417639aa67ee3777e5b432ca455_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#54;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#50;&#53;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#78;&#47;&#67;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"70\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\">\n<p>Calculate the initial (from rest) acceleration of a proton in a <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-f3b0a68793521f05188118b9c026db33_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#53;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#48;&#48;&#125;&times;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#125;&#125;&#94;&#123;&#54;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#78;&#47;&#67;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"95\" style=\"vertical-align: -4px;\" \/> electric field (such as created by a research Van de Graaff). Explicitly show how you follow the steps in the Problem-Solving Strategy for electrostatics.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id2687821\">\n<p id=\"import-auto-id3042498\">(a) Find the direction and magnitude of an electric field that exerts a <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-a794eff1336e5930911ad8f34139676c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#52;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#56;&#48;&#125;&times;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#125;&#125;&#94;&#123;&#45;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#55;&#125;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#78;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"92\" style=\"vertical-align: -1px;\" \/> westward force on an electron. (b) What magnitude and direction force does this field exert on a proton?<\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"fs-id1599293\">\n<p id=\"import-auto-id1845690\">(a) <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-e9bd4486580316a7f53f602943b5743d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#51;&#48;&#48;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#78;&#47;&#67;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#116;&#101;&#120;&#116;&#123;&#101;&#97;&#115;&#116;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"117\" style=\"vertical-align: -4px;\" \/><\/p>\n<p id=\"import-auto-id2401155\">(b) <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-a64b107b0fecc706bdbc5c969bba317e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#52;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#56;&#48;&#125;&times;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#125;&#125;&#94;&#123;&#45;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#55;&#125;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#78;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#116;&#101;&#120;&#116;&#123;&#101;&#97;&#115;&#116;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"143\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"glossary\" class=\"textbox shaded\">\n<h2 data-type=\"glossary-title\">Glossary<\/h2>\n<dl class=\"definition\" id=\"import-auto-id2057931\">\n<dt>field<\/dt>\n<dd id=\"fs-id1899657\">a map of the amount and direction of a force acting on other objects, extending out into space<\/dd>\n<\/dl>\n<dl class=\"definition\" id=\"import-auto-id2017072\">\n<dt>point charge<\/dt>\n<dd id=\"fs-id3077230\">A charged particle, designated <em data-effect=\"italics\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-2c758bec4c272382411b95fc0e7ee250_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#81;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"14\" style=\"vertical-align: -4px;\" \/>,<\/em> generating an electric field<\/dd>\n<\/dl>\n<dl class=\"definition\">\n<dt>test charge<\/dt>\n<dd id=\"fs-id742532\">A particle (designated <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-ac7da57d7f507262338bb5168feb3e06_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#113;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"8\" style=\"vertical-align: -4px;\" \/>) with either a positive or negative charge set down within an electric field generated by a point charge<\/dd>\n<\/dl>\n<\/div>\n","protected":false},"author":211,"menu_order":1,"template":"","meta":{"pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":"all-rights-reserved"},"chapter-type":[],"contributor":[],"license":[56],"class_list":["post-985","chapter","type-chapter","status-publish","hentry","license-all-rights-reserved"],"part":957,"_links":{"self":[{"href":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-json\/pressbooks\/v2\/chapters\/985","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-json\/wp\/v2\/users\/211"}],"version-history":[{"count":1,"href":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-json\/pressbooks\/v2\/chapters\/985\/revisions"}],"predecessor-version":[{"id":986,"href":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-json\/pressbooks\/v2\/chapters\/985\/revisions\/986"}],"part":[{"href":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-json\/pressbooks\/v2\/parts\/957"}],"metadata":[{"href":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-json\/pressbooks\/v2\/chapters\/985\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-json\/wp\/v2\/media?parent=985"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-json\/pressbooks\/v2\/chapter-type?post=985"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-json\/wp\/v2\/contributor?post=985"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-json\/wp\/v2\/license?post=985"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}