{"id":1377,"date":"2017-10-27T16:32:02","date_gmt":"2017-10-27T16:32:02","guid":{"rendered":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/chapter\/maxwells-equations-electromagnetic-waves-predicted-and-observed\/"},"modified":"2017-11-08T03:27:00","modified_gmt":"2017-11-08T03:27:00","slug":"maxwells-equations-electromagnetic-waves-predicted-and-observed","status":"publish","type":"chapter","link":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/chapter\/maxwells-equations-electromagnetic-waves-predicted-and-observed\/","title":{"raw":"Maxwell\u2019s Equations: Electromagnetic Waves Predicted and Observed","rendered":"Maxwell\u2019s Equations: Electromagnetic Waves Predicted and Observed"},"content":{"raw":"\n<div class=\"textbox learning-objectives\">\n<h3 itemprop=\"educationalUse\">Learning Objectives<\/h3>\n<ul>\n<li>Restate Maxwell\u2019s equations.<\/li>\n<\/ul>\n<\/div>\n<p id=\"import-auto-id1169736621559\">The Scotsman James Clerk Maxwell (1831\u20131879) is regarded as the greatest theoretical physicist of the 19th century. (See <a href=\"#import-auto-id1169738037078\" class=\"autogenerated-content\">(Figure)<\/a>.) Although he died young, Maxwell not only formulated a complete electromagnetic theory, represented by <span data-type=\"term\" id=\"import-auto-id1169738036336\">Maxwell\u2019s equations<\/span>, he also developed the kinetic theory of gases and made significant contributions to the understanding of color vision and the nature of Saturn\u2019s rings.<\/p>\n<div class=\"bc-figure figure\" id=\"import-auto-id1169738037078\">\n<div class=\"bc-figcaption figcaption\">James Clerk Maxwell, a 19th-century physicist, developed a theory that explained the relationship between electricity and magnetism and correctly predicted that visible light is caused by electromagnetic waves. (credit: G. J. Stodart)<\/div>\n<p><span data-type=\"media\" id=\"import-auto-id1169736583866\" data-alt=\"This black and white engraving shows physicist James Clerk Maxwell as a Victorian era gentleman dressed in bowtie, vest, and jacket, and sporting a full, graying beard and moustache.\"><img src=\"\/resources\/ef86d1c63370fbe012f632b601c96463dd3ee059\/Figure 25_01_01a.jpg#fixme#fixme\" data-media-type=\"image\/jpg\" alt=\"This black and white engraving shows physicist James Clerk Maxwell as a Victorian era gentleman dressed in bowtie, vest, and jacket, and sporting a full, graying beard and moustache.\" width=\"175\"><\/span><\/p><\/div>\n<p id=\"import-auto-id1169737968947\">Maxwell brought together all the work that had been done by brilliant physicists such as Oersted, Coulomb, Gauss, and Faraday, and added his own insights to develop the overarching theory of electromagnetism. Maxwell\u2019s equations are paraphrased here in words because their mathematical statement is beyond the level of this text. However, the equations illustrate how apparently simple mathematical statements can elegantly unite and express a multitude of concepts\u2014why mathematics is the language of science.<\/p>\n<div data-type=\"note\" class=\"note\" data-has-label=\"true\" id=\"fs-id1169738061393\" data-label=\"\">\n<div data-type=\"title\" class=\"title\">Maxwell\u2019s Equations<\/div>\n<ol id=\"fs-id1169737923940\" data-number-style=\"arabic\">\n<li id=\"import-auto-id1169736584314\"><span data-type=\"term\" id=\"import-auto-id1169736584124\">Electric field lines<\/span> originate on positive charges and terminate on negative charges. The electric field is defined as the force per unit charge on a test charge, and the strength of the force is related to the electric constant [latex]{\\epsilon }_{0}[\/latex], also known as the permittivity of free space. From Maxwell\u2019s first equation we obtain a special form of Coulomb\u2019s law known as Gauss\u2019s law for electricity. <\/li>\n<li id=\"import-auto-id1169738090192\"><span data-type=\"term\" id=\"import-auto-id1169737924198\">Magnetic field lines<\/span> are continuous, having no beginning or end. No magnetic monopoles are known to exist. The strength of the magnetic force is related to the magnetic constant [latex]{\\mu }_{0}[\/latex], also known as the permeability of free space. This second of Maxwell\u2019s equations is known as Gauss\u2019s law for magnetism. <\/li>\n<li id=\"import-auto-id1169738047114\">A changing magnetic field induces an electromotive force (emf) and, hence, an electric field. The direction of the emf opposes the change. This third of Maxwell\u2019s equations is Faraday\u2019s law of induction, and includes Lenz\u2019s law.<\/li>\n<li id=\"import-auto-id1169738086650\">Magnetic fields are generated by moving charges or by changing electric fields. This fourth of Maxwell\u2019s equations encompasses Ampere\u2019s law and adds another source of magnetism\u2014changing electric fields.<\/li>\n<\/ol>\n<\/div>\n<p id=\"import-auto-id1169738150686\">Maxwell\u2019s equations encompass the major laws of electricity and magnetism. What is not so apparent is the symmetry that Maxwell introduced in his mathematical framework. Especially important is his addition of the hypothesis that changing electric fields create magnetic fields. This is exactly analogous (and symmetric) to Faraday\u2019s law of induction and had been suspected for some time, but fits beautifully into Maxwell\u2019s equations. <\/p>\n<p id=\"import-auto-id1169738114427\">Symmetry is apparent in nature in a wide range of situations. In contemporary research, symmetry plays a major part in the search for sub-atomic particles using massive multinational particle accelerators such as the new Large Hadron Collider at CERN.<\/p>\n<div data-type=\"note\" class=\"note\" data-has-label=\"true\" id=\"fs-id1169737712642\" data-label=\"\">\n<div data-type=\"title\" class=\"title\">Making Connections: Unification of Forces<\/div>\n<p id=\"import-auto-id1169736610500\">Maxwell\u2019s complete and symmetric theory showed that electric and magnetic forces are not separate, but different manifestations of the same thing\u2014the electromagnetic force. This classical unification of forces is one motivation for current attempts to unify the four basic forces in nature\u2014the gravitational, electrical, strong, and weak nuclear forces. <\/p>\n<\/div>\n<p id=\"import-auto-id1169738182595\">Since changing electric fields create relatively weak magnetic fields, they could not be easily detected at the time of Maxwell\u2019s hypothesis. Maxwell realized, however, that oscillating charges, like those in AC circuits, produce changing electric fields. He predicted that these changing fields would propagate from the source like waves generated on a lake by a jumping fish. <\/p>\n<p id=\"import-auto-id1169737926911\">The waves predicted by Maxwell would consist of oscillating electric and magnetic fields\u2014defined to be an electromagnetic wave (EM wave). Electromagnetic waves would be capable of exerting forces on charges great distances from their source, and they might thus be detectable. Maxwell calculated that electromagnetic waves would propagate at a speed given by the equation<\/p>\n<div data-type=\"equation\" class=\"equation\">[latex]c=\\frac{1}{\\sqrt{{\\mu }_{0}{\\epsilon }_{0}}}.[\/latex]<\/div>\n<p id=\"import-auto-id1169738085056\">When the values for [latex]{\\mu }_{0}[\/latex] and [latex]{\\epsilon }_{0}[\/latex] are entered into the equation for [latex]c[\/latex], we find that<\/p>\n<div data-type=\"equation\" class=\"equation\">[latex]c=\\frac{1}{\\sqrt{\\left(8\\text{.}\\text{85}\u00d7{\\text{10}}^{-\\text{12}}\\phantom{\\rule{0.25em}{0ex}}\\frac{{\\text{C}}^{2}}{\\text{N}\\cdot {\\text{m}}^{2}}\\right)\\left(4\\pi \u00d7{\\text{10}}^{-7}\\phantom{\\rule{0.25em}{0ex}}\\frac{\\text{T}\\cdot \\text{m}}{\\text{A}}\\right)}}=\\text{3}\\text{.}\\text{00}\u00d7{\\text{10}}^{8}\\phantom{\\rule{0.25em}{0ex}}\\text{m\/s},[\/latex]<\/div>\n<p id=\"import-auto-id1169738037323\">which is the speed of light. In fact, Maxwell concluded that light is an electromagnetic wave having such wavelengths that it can be detected by the eye. <\/p>\n<p id=\"import-auto-id1169738025388\">Other wavelengths should exist\u2014it remained to be seen if they did. If so, Maxwell\u2019s theory and remarkable predictions would be verified, the greatest triumph of physics since Newton. Experimental verification came within a few years, but not before Maxwell\u2019s death.<\/p>\n<div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1169738224969\">\n<h1 data-type=\"title\">Hertz\u2019s Observations<\/h1>\n<p id=\"import-auto-id1169737952173\">The German physicist Heinrich Hertz (1857\u20131894) was the first to generate and detect certain types of electromagnetic waves in the laboratory. Starting in 1887, he performed a series of experiments that not only confirmed the existence of electromagnetic waves, but also verified that they travel at the speed of light. <\/p>\n<p id=\"import-auto-id1169738076649\">Hertz used an AC [latex]\\text{RLC}[\/latex] (resistor-inductor-capacitor) circuit that resonates at a known frequency [latex]{f}_{0}=\\frac{1}{2\\pi \\sqrt{\\text{LC}}}[\/latex] and connected it to a loop of wire as shown in <a href=\"#import-auto-id1169738181556\" class=\"autogenerated-content\">(Figure)<\/a>. High voltages induced across the gap in the loop produced sparks that were visible evidence of the current in the circuit and that helped generate electromagnetic waves. <\/p>\n<p id=\"import-auto-id1169738083742\">Across the laboratory, Hertz had another loop attached to another [latex]\\text{RLC}[\/latex] circuit, which could be tuned (as the dial on a radio) to the same resonant frequency as the first and could, thus, be made to receive electromagnetic waves. This loop also had a gap across which sparks were generated, giving solid evidence that electromagnetic waves had been received.<\/p>\n<div class=\"bc-figure figure\" id=\"import-auto-id1169738181556\">\n<div class=\"bc-figcaption figcaption\">The apparatus used by Hertz in 1887 to generate and detect electromagnetic waves. An [latex]\\text{RLC}[\/latex] circuit connected to the first loop caused sparks across a gap in the wire loop and generated electromagnetic waves. Sparks across a gap in the second loop located across the laboratory gave evidence that the waves had been received.\n               <\/div>\n<p><span data-type=\"media\" id=\"import-auto-id1169738249773\" data-alt=\"The circuit diagram shows a simple circuit containing an alternating voltage source, a resistor R, capacitor C and a transformer, which provides the impedance. The transformer is shown to consist of two coils separated by a core. In parallel with the transformer is connected a wire loop labeled as Loop one Transmitter with a small gap that creates sparks across the gap. The sparks create electromagnetic waves, which are transmitted through the air to a similar loop next to it labeled as Loop two Receiver. These waves induce sparks in Loop two, and are detected by the tuner shown as a rectangular box connected to it.\"><img src=\"\/resources\/bc820cfef32e1c2fdafe83dd3d7804063bbf0cb2\/Figure 25_01_02a.jpg#fixme#fixme\" data-media-type=\"image\/jpg\" alt=\"The circuit diagram shows a simple circuit containing an alternating voltage source, a resistor R, capacitor C and a transformer, which provides the impedance. The transformer is shown to consist of two coils separated by a core. In parallel with the transformer is connected a wire loop labeled as Loop one Transmitter with a small gap that creates sparks across the gap. The sparks create electromagnetic waves, which are transmitted through the air to a similar loop next to it labeled as Loop two Receiver. These waves induce sparks in Loop two, and are detected by the tuner shown as a rectangular box connected to it.\" width=\"350\"><\/span><\/p><\/div>\n<p id=\"import-auto-id1169738198969\">Hertz also studied the reflection, refraction, and interference patterns of the electromagnetic waves he generated, verifying their wave character. He was able to determine wavelength from the interference patterns, and knowing their frequency, he could calculate the propagation speed using the equation [latex]\\upsilon =\\mathrm{f\\lambda }[\/latex] (velocity\u2014or speed\u2014equals frequency times wavelength). Hertz was thus able to prove that electromagnetic waves travel at the speed of light. The SI unit for frequency, the hertz ([latex]1 Hz=\\text{1 cycle\/sec}[\/latex]), is named in his honor.<\/p>\n<\/div>\n<div class=\"section-summary\" data-depth=\"1\" id=\"fs-id1169737777222\">\n<h1 data-type=\"title\">Section Summary<\/h1>\n<ul id=\"fs-id1169738089740\">\n<li id=\"import-auto-id1169738234772\">Electromagnetic waves consist of oscillating electric and magnetic fields and propagate at the speed of light [latex]c[\/latex]. They were predicted by Maxwell, who also showed that\n<div data-type=\"equation\" class=\"equation\">[latex]c=\\frac{1}{\\sqrt{{\\mu }_{0}{\\epsilon }_{0}}},[\/latex]<\/div>\n<p id=\"import-auto-id1169737927591\">where [latex]{\\mu }_{0}[\/latex] is the permeability of free space and [latex]{\\epsilon }_{0}[\/latex] is the permittivity of free space.<\/p>\n<\/li>\n<li id=\"import-auto-id1169736621964\">Maxwell\u2019s prediction of electromagnetic waves resulted from his formulation of a complete and symmetric theory of electricity and magnetism, known as Maxwell\u2019s equations. <\/li>\n<li id=\"import-auto-id1169737924174\">These four equations are paraphrased in this text, rather than presented numerically, and encompass the major laws of electricity and magnetism. First is Gauss\u2019s law for electricity, second is Gauss\u2019s law for magnetism, third is Faraday\u2019s law of induction, including Lenz\u2019s law, and fourth is Ampere\u2019s law in a symmetric formulation that adds another source of magnetism\u2014changing electric fields.<\/li>\n<\/ul>\n<\/div>\n<div class=\"problems-exercises\" data-depth=\"1\" id=\"fs-id1169736611212\" data-element-type=\"problems-exercises\">\n<h1 data-type=\"title\">Problems &amp; Exercises<\/h1>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1169738137813\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1169737712987\">\n<p id=\"import-auto-id1169737855668\">Verify that the correct value for the speed of light [latex]c[\/latex] is obtained when numerical values for the permeability and permittivity of free space ([latex]{\\mu }_{0}[\/latex] and [latex]{\\epsilon }_{0}[\/latex]) are entered into the equation [latex]c=\\frac{1}{\\sqrt{{\\mu }_{0}{\\epsilon }_{0}}}[\/latex].<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1169738015970\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1169738015971\">\n<p id=\"import-auto-id1169738114356\">Show that, when SI units for [latex]{\\mu }_{0}[\/latex] and [latex]{\\epsilon }_{0}[\/latex] are entered, the units given by the right-hand side of the equation in the problem above are m\/s. <\/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-id1169738209106\">\n<dt>electromagnetic waves<\/dt>\n<dd id=\"fs-id1169738076866\">radiation in the form of waves of electric and magnetic energy<\/dd>\n<\/dl>\n<dl class=\"definition\" id=\"import-auto-id1169738209108\">\n<dt>Maxwell\u2019s equations<\/dt>\n<dd id=\"fs-id1169738037473\">a set of four equations that comprise a complete, overarching theory of electromagnetism<\/dd>\n<\/dl>\n<dl class=\"definition\" id=\"import-auto-id1169737911378\">\n<dt><em data-effect=\"italics\">RLC<\/em> circuit<\/dt>\n<dd id=\"fs-id1169738135307\">an electric circuit that includes a resistor, capacitor and inductor<\/dd>\n<\/dl>\n<dl class=\"definition\" id=\"import-auto-id1169738202087\">\n<dt>hertz<\/dt>\n<dd id=\"fs-id1169738059862\">an SI unit denoting the frequency of an electromagnetic wave, in cycles per second<\/dd>\n<\/dl>\n<dl class=\"definition\" id=\"import-auto-id1169738202089\">\n<dt>speed of light<\/dt>\n<dd id=\"fs-id1169737967305\">in a vacuum, such as space, the speed of light is a constant 3 x 10<sup>8<\/sup> m\/s<\/dd>\n<\/dl>\n<dl class=\"definition\" id=\"import-auto-id1169738080261\">\n<dt>electromotive force (emf)<\/dt>\n<dd id=\"fs-id1169738010312\">energy produced per unit charge, drawn from a source that produces an electrical current<\/dd>\n<\/dl>\n<dl class=\"definition\" id=\"import-auto-id1169738208407\">\n<dt>electric field lines<\/dt>\n<dd id=\"fs-id1169737722530\">a pattern of imaginary lines that extend between an electric source and charged objects in the surrounding area, with arrows pointed away from positively charged objects and toward negatively charged objects. The more lines in the pattern, the stronger the electric field in that region<\/dd>\n<\/dl>\n<dl class=\"definition\" id=\"import-auto-id1169737855616\">\n<dt>magnetic field lines<\/dt>\n<dd id=\"fs-id1169738042878\">a pattern of continuous, imaginary lines that emerge from and enter into opposite magnetic poles. The density of the lines indicates the magnitude of the magnetic field<\/dd>\n<\/dl>\n<\/div>\n\n","rendered":"<div class=\"textbox learning-objectives\">\n<h3 itemprop=\"educationalUse\">Learning Objectives<\/h3>\n<ul>\n<li>Restate Maxwell\u2019s equations.<\/li>\n<\/ul>\n<\/div>\n<p id=\"import-auto-id1169736621559\">The Scotsman James Clerk Maxwell (1831\u20131879) is regarded as the greatest theoretical physicist of the 19th century. (See <a href=\"#import-auto-id1169738037078\" class=\"autogenerated-content\">(Figure)<\/a>.) Although he died young, Maxwell not only formulated a complete electromagnetic theory, represented by <span data-type=\"term\" id=\"import-auto-id1169738036336\">Maxwell\u2019s equations<\/span>, he also developed the kinetic theory of gases and made significant contributions to the understanding of color vision and the nature of Saturn\u2019s rings.<\/p>\n<div class=\"bc-figure figure\" id=\"import-auto-id1169738037078\">\n<div class=\"bc-figcaption figcaption\">James Clerk Maxwell, a 19th-century physicist, developed a theory that explained the relationship between electricity and magnetism and correctly predicted that visible light is caused by electromagnetic waves. (credit: G. J. Stodart)<\/div>\n<p><span data-type=\"media\" id=\"import-auto-id1169736583866\" data-alt=\"This black and white engraving shows physicist James Clerk Maxwell as a Victorian era gentleman dressed in bowtie, vest, and jacket, and sporting a full, graying beard and moustache.\"><img decoding=\"async\" src=\"\/resources\/ef86d1c63370fbe012f632b601c96463dd3ee059\/Figure 25_01_01a.jpg#fixme#fixme\" data-media-type=\"image\/jpg\" alt=\"This black and white engraving shows physicist James Clerk Maxwell as a Victorian era gentleman dressed in bowtie, vest, and jacket, and sporting a full, graying beard and moustache.\" width=\"175\" \/><\/span><\/p>\n<\/div>\n<p id=\"import-auto-id1169737968947\">Maxwell brought together all the work that had been done by brilliant physicists such as Oersted, Coulomb, Gauss, and Faraday, and added his own insights to develop the overarching theory of electromagnetism. Maxwell\u2019s equations are paraphrased here in words because their mathematical statement is beyond the level of this text. However, the equations illustrate how apparently simple mathematical statements can elegantly unite and express a multitude of concepts\u2014why mathematics is the language of science.<\/p>\n<div data-type=\"note\" class=\"note\" data-has-label=\"true\" id=\"fs-id1169738061393\" data-label=\"\">\n<div data-type=\"title\" class=\"title\">Maxwell\u2019s Equations<\/div>\n<ol id=\"fs-id1169737923940\" data-number-style=\"arabic\">\n<li id=\"import-auto-id1169736584314\"><span data-type=\"term\" id=\"import-auto-id1169736584124\">Electric field lines<\/span> originate on positive charges and terminate on negative charges. The electric field is defined as the force per unit charge on a test charge, and the strength of the force is related to the electric constant <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-de7b1782ae6a2ff50840b723fe5b9cfe_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#101;&#112;&#115;&#105;&#108;&#111;&#110;&#32;&#125;&#95;&#123;&#48;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"11\" width=\"14\" style=\"vertical-align: -3px;\" \/>, also known as the permittivity of free space. From Maxwell\u2019s first equation we obtain a special form of Coulomb\u2019s law known as Gauss\u2019s law for electricity. <\/li>\n<li id=\"import-auto-id1169738090192\"><span data-type=\"term\" id=\"import-auto-id1169737924198\">Magnetic field lines<\/span> are continuous, having no beginning or end. No magnetic monopoles are known to exist. The strength of the magnetic force is related to the magnetic constant <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-c0e771d3f7628916efa0341ca7783954_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#109;&#117;&#32;&#125;&#95;&#123;&#48;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"18\" style=\"vertical-align: -4px;\" \/>, also known as the permeability of free space. This second of Maxwell\u2019s equations is known as Gauss\u2019s law for magnetism. <\/li>\n<li id=\"import-auto-id1169738047114\">A changing magnetic field induces an electromotive force (emf) and, hence, an electric field. The direction of the emf opposes the change. This third of Maxwell\u2019s equations is Faraday\u2019s law of induction, and includes Lenz\u2019s law.<\/li>\n<li id=\"import-auto-id1169738086650\">Magnetic fields are generated by moving charges or by changing electric fields. This fourth of Maxwell\u2019s equations encompasses Ampere\u2019s law and adds another source of magnetism\u2014changing electric fields.<\/li>\n<\/ol>\n<\/div>\n<p id=\"import-auto-id1169738150686\">Maxwell\u2019s equations encompass the major laws of electricity and magnetism. What is not so apparent is the symmetry that Maxwell introduced in his mathematical framework. Especially important is his addition of the hypothesis that changing electric fields create magnetic fields. This is exactly analogous (and symmetric) to Faraday\u2019s law of induction and had been suspected for some time, but fits beautifully into Maxwell\u2019s equations. <\/p>\n<p id=\"import-auto-id1169738114427\">Symmetry is apparent in nature in a wide range of situations. In contemporary research, symmetry plays a major part in the search for sub-atomic particles using massive multinational particle accelerators such as the new Large Hadron Collider at CERN.<\/p>\n<div data-type=\"note\" class=\"note\" data-has-label=\"true\" id=\"fs-id1169737712642\" data-label=\"\">\n<div data-type=\"title\" class=\"title\">Making Connections: Unification of Forces<\/div>\n<p id=\"import-auto-id1169736610500\">Maxwell\u2019s complete and symmetric theory showed that electric and magnetic forces are not separate, but different manifestations of the same thing\u2014the electromagnetic force. This classical unification of forces is one motivation for current attempts to unify the four basic forces in nature\u2014the gravitational, electrical, strong, and weak nuclear forces. <\/p>\n<\/div>\n<p id=\"import-auto-id1169738182595\">Since changing electric fields create relatively weak magnetic fields, they could not be easily detected at the time of Maxwell\u2019s hypothesis. Maxwell realized, however, that oscillating charges, like those in AC circuits, produce changing electric fields. He predicted that these changing fields would propagate from the source like waves generated on a lake by a jumping fish. <\/p>\n<p id=\"import-auto-id1169737926911\">The waves predicted by Maxwell would consist of oscillating electric and magnetic fields\u2014defined to be an electromagnetic wave (EM wave). Electromagnetic waves would be capable of exerting forces on charges great distances from their source, and they might thus be detectable. Maxwell calculated that electromagnetic waves would propagate at a speed given by the equation<\/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-1ca308b39bb7461b0528d142bc617e03_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#99;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#92;&#115;&#113;&#114;&#116;&#123;&#123;&#92;&#109;&#117;&#32;&#125;&#95;&#123;&#48;&#125;&#123;&#92;&#101;&#112;&#115;&#105;&#108;&#111;&#110;&#32;&#125;&#95;&#123;&#48;&#125;&#125;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"27\" width=\"78\" style=\"vertical-align: -11px;\" \/><\/div>\n<p id=\"import-auto-id1169738085056\">When the values for <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-c0e771d3f7628916efa0341ca7783954_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#109;&#117;&#32;&#125;&#95;&#123;&#48;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"18\" style=\"vertical-align: -4px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-de7b1782ae6a2ff50840b723fe5b9cfe_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#101;&#112;&#115;&#105;&#108;&#111;&#110;&#32;&#125;&#95;&#123;&#48;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"11\" width=\"14\" style=\"vertical-align: -3px;\" \/> are entered into the equation for <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-41a04eeea923a1a0c28094a8a4680525_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#99;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"8\" style=\"vertical-align: 0px;\" \/>, we find 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-2b261f5cabe6a3457f557e276aea62b2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#99;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#92;&#115;&#113;&#114;&#116;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#56;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#56;&#53;&#125;&times;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#125;&#125;&#94;&#123;&#45;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#50;&#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;&#102;&#114;&#97;&#99;&#123;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#67;&#125;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#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;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#52;&#92;&#112;&#105;&#32;&times;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#125;&#125;&#94;&#123;&#45;&#55;&#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;&#102;&#114;&#97;&#99;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#84;&#125;&#92;&#99;&#100;&#111;&#116;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#109;&#125;&#125;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#65;&#125;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#125;&#61;&#92;&#116;&#101;&#120;&#116;&#123;&#51;&#125;&#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;&#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;&#109;&#47;&#115;&#125;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"45\" width=\"372\" style=\"vertical-align: -29px;\" \/><\/div>\n<p id=\"import-auto-id1169738037323\">which is the speed of light. In fact, Maxwell concluded that light is an electromagnetic wave having such wavelengths that it can be detected by the eye. <\/p>\n<p id=\"import-auto-id1169738025388\">Other wavelengths should exist\u2014it remained to be seen if they did. If so, Maxwell\u2019s theory and remarkable predictions would be verified, the greatest triumph of physics since Newton. Experimental verification came within a few years, but not before Maxwell\u2019s death.<\/p>\n<div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1169738224969\">\n<h1 data-type=\"title\">Hertz\u2019s Observations<\/h1>\n<p id=\"import-auto-id1169737952173\">The German physicist Heinrich Hertz (1857\u20131894) was the first to generate and detect certain types of electromagnetic waves in the laboratory. Starting in 1887, he performed a series of experiments that not only confirmed the existence of electromagnetic waves, but also verified that they travel at the speed of light. <\/p>\n<p id=\"import-auto-id1169738076649\">Hertz used an AC <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-43cda88f72fc3f02910d22a1bb8fc1a1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#82;&#76;&#67;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"36\" style=\"vertical-align: 0px;\" \/> (resistor-inductor-capacitor) circuit that resonates at a known frequency <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-af945c964403218114b307ecf9f6504e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#102;&#125;&#95;&#123;&#48;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#92;&#112;&#105;&#32;&#92;&#115;&#113;&#114;&#116;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#76;&#67;&#125;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"27\" width=\"88\" style=\"vertical-align: -11px;\" \/> and connected it to a loop of wire as shown in <a href=\"#import-auto-id1169738181556\" class=\"autogenerated-content\">(Figure)<\/a>. High voltages induced across the gap in the loop produced sparks that were visible evidence of the current in the circuit and that helped generate electromagnetic waves. <\/p>\n<p id=\"import-auto-id1169738083742\">Across the laboratory, Hertz had another loop attached to another <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-43cda88f72fc3f02910d22a1bb8fc1a1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#82;&#76;&#67;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"36\" style=\"vertical-align: 0px;\" \/> circuit, which could be tuned (as the dial on a radio) to the same resonant frequency as the first and could, thus, be made to receive electromagnetic waves. This loop also had a gap across which sparks were generated, giving solid evidence that electromagnetic waves had been received.<\/p>\n<div class=\"bc-figure figure\" id=\"import-auto-id1169738181556\">\n<div class=\"bc-figcaption figcaption\">The apparatus used by Hertz in 1887 to generate and detect electromagnetic waves. An <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-43cda88f72fc3f02910d22a1bb8fc1a1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#82;&#76;&#67;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"36\" style=\"vertical-align: 0px;\" \/> circuit connected to the first loop caused sparks across a gap in the wire loop and generated electromagnetic waves. Sparks across a gap in the second loop located across the laboratory gave evidence that the waves had been received.\n               <\/div>\n<p><span data-type=\"media\" id=\"import-auto-id1169738249773\" data-alt=\"The circuit diagram shows a simple circuit containing an alternating voltage source, a resistor R, capacitor C and a transformer, which provides the impedance. The transformer is shown to consist of two coils separated by a core. In parallel with the transformer is connected a wire loop labeled as Loop one Transmitter with a small gap that creates sparks across the gap. The sparks create electromagnetic waves, which are transmitted through the air to a similar loop next to it labeled as Loop two Receiver. These waves induce sparks in Loop two, and are detected by the tuner shown as a rectangular box connected to it.\"><img decoding=\"async\" src=\"\/resources\/bc820cfef32e1c2fdafe83dd3d7804063bbf0cb2\/Figure 25_01_02a.jpg#fixme#fixme\" data-media-type=\"image\/jpg\" alt=\"The circuit diagram shows a simple circuit containing an alternating voltage source, a resistor R, capacitor C and a transformer, which provides the impedance. The transformer is shown to consist of two coils separated by a core. In parallel with the transformer is connected a wire loop labeled as Loop one Transmitter with a small gap that creates sparks across the gap. The sparks create electromagnetic waves, which are transmitted through the air to a similar loop next to it labeled as Loop two Receiver. These waves induce sparks in Loop two, and are detected by the tuner shown as a rectangular box connected to it.\" width=\"350\" \/><\/span><\/p>\n<\/div>\n<p id=\"import-auto-id1169738198969\">Hertz also studied the reflection, refraction, and interference patterns of the electromagnetic waves he generated, verifying their wave character. He was able to determine wavelength from the interference patterns, and knowing their frequency, he could calculate the propagation speed using the equation <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-7b03076a44733da73c6780226339b345_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#117;&#112;&#115;&#105;&#108;&#111;&#110;&#32;&#61;&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#102;&#92;&#108;&#97;&#109;&#98;&#100;&#97;&#32;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"51\" style=\"vertical-align: -1px;\" \/> (velocity\u2014or speed\u2014equals frequency times wavelength). Hertz was thus able to prove that electromagnetic waves travel at the speed of light. The SI unit for frequency, the hertz (<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-128207af873470ce6429bc2873c61d46_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#32;&#72;&#122;&#61;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#32;&#99;&#121;&#99;&#108;&#101;&#47;&#115;&#101;&#99;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"141\" style=\"vertical-align: -4px;\" \/>), is named in his honor.<\/p>\n<\/div>\n<div class=\"section-summary\" data-depth=\"1\" id=\"fs-id1169737777222\">\n<h1 data-type=\"title\">Section Summary<\/h1>\n<ul id=\"fs-id1169738089740\">\n<li id=\"import-auto-id1169738234772\">Electromagnetic waves consist of oscillating electric and magnetic fields and propagate at the speed of light <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-41a04eeea923a1a0c28094a8a4680525_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#99;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"8\" style=\"vertical-align: 0px;\" \/>. They were predicted by Maxwell, who also showed that\n<div data-type=\"equation\" class=\"equation\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-f961148830b3ce47fa4d5adfbe416bf0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#99;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#92;&#115;&#113;&#114;&#116;&#123;&#123;&#92;&#109;&#117;&#32;&#125;&#95;&#123;&#48;&#125;&#123;&#92;&#101;&#112;&#115;&#105;&#108;&#111;&#110;&#32;&#125;&#95;&#123;&#48;&#125;&#125;&#125;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"27\" width=\"78\" style=\"vertical-align: -11px;\" \/><\/div>\n<p id=\"import-auto-id1169737927591\">where <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-c0e771d3f7628916efa0341ca7783954_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#109;&#117;&#32;&#125;&#95;&#123;&#48;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"18\" style=\"vertical-align: -4px;\" \/> is the permeability of free space and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-de7b1782ae6a2ff50840b723fe5b9cfe_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#101;&#112;&#115;&#105;&#108;&#111;&#110;&#32;&#125;&#95;&#123;&#48;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"11\" width=\"14\" style=\"vertical-align: -3px;\" \/> is the permittivity of free space.<\/p>\n<\/li>\n<li id=\"import-auto-id1169736621964\">Maxwell\u2019s prediction of electromagnetic waves resulted from his formulation of a complete and symmetric theory of electricity and magnetism, known as Maxwell\u2019s equations. <\/li>\n<li id=\"import-auto-id1169737924174\">These four equations are paraphrased in this text, rather than presented numerically, and encompass the major laws of electricity and magnetism. First is Gauss\u2019s law for electricity, second is Gauss\u2019s law for magnetism, third is Faraday\u2019s law of induction, including Lenz\u2019s law, and fourth is Ampere\u2019s law in a symmetric formulation that adds another source of magnetism\u2014changing electric fields.<\/li>\n<\/ul>\n<\/div>\n<div class=\"problems-exercises\" data-depth=\"1\" id=\"fs-id1169736611212\" data-element-type=\"problems-exercises\">\n<h1 data-type=\"title\">Problems &amp; Exercises<\/h1>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1169738137813\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1169737712987\">\n<p id=\"import-auto-id1169737855668\">Verify that the correct value for the speed of light <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-41a04eeea923a1a0c28094a8a4680525_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#99;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"8\" style=\"vertical-align: 0px;\" \/> is obtained when numerical values for the permeability and permittivity of free space (<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-c0e771d3f7628916efa0341ca7783954_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#109;&#117;&#32;&#125;&#95;&#123;&#48;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"18\" style=\"vertical-align: -4px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-de7b1782ae6a2ff50840b723fe5b9cfe_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#101;&#112;&#115;&#105;&#108;&#111;&#110;&#32;&#125;&#95;&#123;&#48;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"11\" width=\"14\" style=\"vertical-align: -3px;\" \/>) are entered into the equation <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-9fc1a1d7ac143e76a4e42ffd1b80ce25_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#99;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#92;&#115;&#113;&#114;&#116;&#123;&#123;&#92;&#109;&#117;&#32;&#125;&#95;&#123;&#48;&#125;&#123;&#92;&#101;&#112;&#115;&#105;&#108;&#111;&#110;&#32;&#125;&#95;&#123;&#48;&#125;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"27\" width=\"73\" style=\"vertical-align: -11px;\" \/>.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1169738015970\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1169738015971\">\n<p id=\"import-auto-id1169738114356\">Show that, when SI units for <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-c0e771d3f7628916efa0341ca7783954_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#109;&#117;&#32;&#125;&#95;&#123;&#48;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"18\" style=\"vertical-align: -4px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-de7b1782ae6a2ff50840b723fe5b9cfe_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#101;&#112;&#115;&#105;&#108;&#111;&#110;&#32;&#125;&#95;&#123;&#48;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"11\" width=\"14\" style=\"vertical-align: -3px;\" \/> are entered, the units given by the right-hand side of the equation in the problem above are m\/s. <\/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-id1169738209106\">\n<dt>electromagnetic waves<\/dt>\n<dd id=\"fs-id1169738076866\">radiation in the form of waves of electric and magnetic energy<\/dd>\n<\/dl>\n<dl class=\"definition\" id=\"import-auto-id1169738209108\">\n<dt>Maxwell\u2019s equations<\/dt>\n<dd id=\"fs-id1169738037473\">a set of four equations that comprise a complete, overarching theory of electromagnetism<\/dd>\n<\/dl>\n<dl class=\"definition\" id=\"import-auto-id1169737911378\">\n<dt><em data-effect=\"italics\">RLC<\/em> circuit<\/dt>\n<dd id=\"fs-id1169738135307\">an electric circuit that includes a resistor, capacitor and inductor<\/dd>\n<\/dl>\n<dl class=\"definition\" id=\"import-auto-id1169738202087\">\n<dt>hertz<\/dt>\n<dd id=\"fs-id1169738059862\">an SI unit denoting the frequency of an electromagnetic wave, in cycles per second<\/dd>\n<\/dl>\n<dl class=\"definition\" id=\"import-auto-id1169738202089\">\n<dt>speed of light<\/dt>\n<dd id=\"fs-id1169737967305\">in a vacuum, such as space, the speed of light is a constant 3 x 10<sup>8<\/sup> m\/s<\/dd>\n<\/dl>\n<dl class=\"definition\" id=\"import-auto-id1169738080261\">\n<dt>electromotive force (emf)<\/dt>\n<dd id=\"fs-id1169738010312\">energy produced per unit charge, drawn from a source that produces an electrical current<\/dd>\n<\/dl>\n<dl class=\"definition\" id=\"import-auto-id1169738208407\">\n<dt>electric field lines<\/dt>\n<dd id=\"fs-id1169737722530\">a pattern of imaginary lines that extend between an electric source and charged objects in the surrounding area, with arrows pointed away from positively charged objects and toward negatively charged objects. The more lines in the pattern, the stronger the electric field in that region<\/dd>\n<\/dl>\n<dl class=\"definition\" id=\"import-auto-id1169737855616\">\n<dt>magnetic field lines<\/dt>\n<dd id=\"fs-id1169738042878\">a pattern of continuous, imaginary lines that emerge from and enter into opposite magnetic poles. The density of the lines indicates the magnitude of the magnetic field<\/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-1377","chapter","type-chapter","status-publish","hentry","license-all-rights-reserved"],"part":1374,"_links":{"self":[{"href":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-json\/pressbooks\/v2\/chapters\/1377","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\/1377\/revisions"}],"predecessor-version":[{"id":1378,"href":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-json\/pressbooks\/v2\/chapters\/1377\/revisions\/1378"}],"part":[{"href":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-json\/pressbooks\/v2\/parts\/1374"}],"metadata":[{"href":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-json\/pressbooks\/v2\/chapters\/1377\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-json\/wp\/v2\/media?parent=1377"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-json\/pressbooks\/v2\/chapter-type?post=1377"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-json\/wp\/v2\/contributor?post=1377"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-json\/wp\/v2\/license?post=1377"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}