{"id":222,"date":"2019-04-09T01:20:46","date_gmt":"2019-04-09T05:20:46","guid":{"rendered":"https:\/\/pressbooks.bccampus.ca\/bcitphys8400\/?post_type=chapter&#038;p=222"},"modified":"2019-04-12T19:10:04","modified_gmt":"2019-04-12T23:10:04","slug":"4-2-orbital-magnetic-dipole-moment-of-the-electron","status":"publish","type":"chapter","link":"https:\/\/pressbooks.bccampus.ca\/bcitphys8400\/chapter\/4-2-orbital-magnetic-dipole-moment-of-the-electron\/","title":{"raw":"4.2 Orbital Magnetic Dipole Moment of the Electron","rendered":"4.2 Orbital Magnetic Dipole Moment of the Electron"},"content":{"raw":"<div data-type=\"abstract\" id=\"22668\" class=\"ui-has-child-title\"><header>\r\n<div class=\"textbox textbox--learning-objectives\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Learning Objectives<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nBy the end of this section, you will be able to:\r\n<ul>\r\n \t<li>Explain why the hydrogen atom has magnetic properties<\/li>\r\n \t<li>Explain why the energy levels of a hydrogen atom associated with orbital angular momentum are split by an external magnetic field<\/li>\r\n \t<li>Use quantum numbers to calculate the magnitude and direction of the orbital magnetic dipole moment of a hydrogen atom<\/li>\r\n<\/ul>\r\n<\/div>\r\n<\/div>\r\n<span style=\"font-size: 14pt\">In Bohr\u2019s model of the hydrogen atom, the electron moves in a circular orbit around the proton. The electron passes by a particular point on the loop in a certain time, so we can calculate a current<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-761-Frame\" style=\"font-size: 14pt\"><span class=\"MathJax_MathContainer\">I=Q\/t<\/span><\/span><span style=\"font-size: 14pt\">. An electron that orbits a proton in a hydrogen atom is therefore analogous to current flowing through a circular wire (<\/span>Figure 4.10<span style=\"font-size: 14pt\">). In the study of magnetism, we saw that a current-carrying wire produces magnetic fields. It is therefore reasonable to conclude that the hydrogen atom produces a magnetic field and interacts with other magnetic fields.<\/span>\r\n\r\n<\/header><\/div>\r\n<div class=\"os-figure\">\r\n<figure id=\"CNX_UPhysics_41_02_AtomicLoop\">\r\n\r\n[caption id=\"\" align=\"alignnone\" width=\"837\"]<img alt=\"Figure (a) shows a current carrying loop. The loop has current I circulating counterclockwise as viewed from above. A vector mu pointing upward is shown at the center of the loop. Figure (b) shows the hydrogen atom as an electron, represented as a small ball and labeled minus e, making a counterclockwise circular orbit, as viewed from above. A sphere, a vector mu pointing downward, and a vector L pointing upward are shown in the center of the orbit.\" data-media-type=\"image\/jpeg\" id=\"86901\" src=\"https:\/\/cnx.org\/resources\/331456eee02dc81fd687dd04689eed0a7e66073b\" width=\"837\" height=\"254\" \/> Figure 4.10 (a) Current flowing through a circular wire is analogous to (b) an electron that orbits a proton in a hydrogen atom.[\/caption]<\/figure>\r\n<\/div>\r\n<p id=\"fs-id1170902922212\">The<span>\u00a0<\/span><span data-type=\"term\" id=\"term338\">orbital magnetic dipole moment<\/span><span>\u00a0<\/span>is a measure of the strength of the magnetic field produced by the orbital angular momentum of an electron. From<span>\u00a0<\/span><a href=\"https:\/\/cnx.org\/contents\/af275420-6050-4707-995c-57b9cc13c358@11.1:31277ded-aa04-406e-a929-b692e0b8bf2b#fs-id1171360245659\" data-page=\"1\">Force and Torque on a Current Loop<\/a>, the magnitude of the orbital magnetic dipole moment for a current loop is<\/p>\r\n\r\n<div data-type=\"equation\" id=\"fs-id1170901575179\">\r\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-762-Frame\">\r\n<div class=\"textbox\">\r\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-762-Frame\"><span class=\"MathJax_MathContainer\"><span>\u03bc=IA,<\/span><\/span><\/div>\r\n<div class=\"os-equation-number\"><span class=\"os-number\">[4.13]<\/span><\/div>\r\n<\/div>\r\n<span style=\"font-size: 14pt\">where<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><em style=\"font-size: 14pt\" data-effect=\"italics\">I<\/em><span style=\"font-size: 14pt\">\u00a0<\/span><span style=\"font-size: 14pt\">is the current and<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><em style=\"font-size: 14pt\" data-effect=\"italics\">A<\/em><span style=\"font-size: 14pt\">\u00a0<\/span><span style=\"font-size: 14pt\">is the area of the loop. (For brevity, we refer to this as the magnetic moment.) The current<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><em style=\"font-size: 14pt\" data-effect=\"italics\">I<\/em><span style=\"font-size: 14pt\">\u00a0<\/span><span style=\"font-size: 14pt\">associated with an electron in orbit about a proton in a hydrogen atom is<\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"equation\" id=\"fs-id1170902672410\">\r\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-763-Frame\">\r\n<div class=\"textbox\">\r\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-763-Frame\"><span class=\"MathJax_MathContainer\"><span>I=eT,<\/span><\/span><\/div>\r\n<div class=\"os-equation-number\"><span class=\"os-number\">[4.14]<\/span><\/div>\r\n<\/div>\r\n<span style=\"font-size: 14pt\">where<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><em style=\"font-size: 14pt\" data-effect=\"italics\">e<\/em><span style=\"font-size: 14pt\">\u00a0<\/span><span style=\"font-size: 14pt\">is the magnitude of the electron charge and<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><em style=\"font-size: 14pt\" data-effect=\"italics\">T<\/em><span style=\"font-size: 14pt\">\u00a0<\/span><span style=\"font-size: 14pt\">is its orbital period. If we assume that the electron travels in a perfectly circular orbit, the orbital period is<\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"equation\" id=\"fs-id1170902891412\">\r\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-764-Frame\">\r\n<div class=\"textbox\">\r\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-764-Frame\"><span class=\"MathJax_MathContainer\"><span>T=2\u03c0rv,<\/span><\/span><\/div>\r\n<div class=\"os-equation-number\"><span class=\"os-number\">[4.15]<\/span><\/div>\r\n<\/div>\r\n<span style=\"font-size: 14pt\">where<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><em style=\"font-size: 14pt\" data-effect=\"italics\">r<\/em><span style=\"font-size: 14pt\">\u00a0<\/span><span style=\"font-size: 14pt\">is the radius of the orbit and<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><em style=\"font-size: 14pt\" data-effect=\"italics\">v<\/em><span style=\"font-size: 14pt\">\u00a0<\/span><span style=\"font-size: 14pt\">is the speed of the electron in its orbit. Given that the area of a circle is<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-765-Frame\" style=\"font-size: 14pt\"><span class=\"MathJax_MathContainer\">\u03c0r2<\/span><\/span><span style=\"font-size: 14pt\">, the absolute magnetic moment is<\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"equation\" id=\"fs-id1170902600684\">\r\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-766-Frame\">\r\n<div class=\"textbox\">\r\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-766-Frame\"><span class=\"MathJax_MathContainer\"><span>\u03bc=IA=e(2\u03c0rv)\u03c0r2=evr2.<\/span><\/span><\/div>\r\n<div class=\"os-equation-number\"><span class=\"os-number\">[4.16]<\/span><\/div>\r\n<\/div>\r\n<span style=\"font-size: 14pt\">It is helpful to express the magnetic momentum<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-767-Frame\" style=\"font-size: 14pt\"><span class=\"MathJax_MathContainer\">\u03bc<\/span><\/span><span style=\"font-size: 14pt\">\u00a0<\/span><span style=\"font-size: 14pt\">in terms of the orbital angular momentum<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-768-Frame\" style=\"font-size: 14pt\"><span class=\"MathJax_MathContainer\">(L\u2192=r\u2192\u00d7p\u2192).<\/span><\/span><span style=\"font-size: 14pt\">\u00a0<\/span><span style=\"font-size: 14pt\">Because the electron orbits in a circle, the position vector<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-769-Frame\" style=\"font-size: 14pt\"><span class=\"MathJax_MathContainer\">r\u2192<\/span><\/span><span style=\"font-size: 14pt\">\u00a0<\/span><span style=\"font-size: 14pt\">and the momentum vector<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-770-Frame\" style=\"font-size: 14pt\"><span class=\"MathJax_MathContainer\">p\u2192<\/span><\/span><span style=\"font-size: 14pt\">\u00a0<\/span><span style=\"font-size: 14pt\">form a right angle. Thus, the magnitude of the orbital angular momentum is<\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"equation\" id=\"fs-id1170901533720\">\r\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-771-Frame\">\r\n<div class=\"textbox\">\r\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-771-Frame\"><span class=\"MathJax_MathContainer\"><span>L=|L\u2192|=|r\u2192\u00d7p\u2192|=rpsin\u03b8=rp=rmv.<\/span><\/span><\/div>\r\n<div class=\"os-equation-number\"><span class=\"os-number\">[4.17]<\/span><\/div>\r\n<\/div>\r\n<span style=\"font-size: 14pt\">Combining these two equations, we have<\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"equation\" id=\"fs-id1170901597936\">\r\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-772-Frame\">\r\n<div class=\"textbox\">\r\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-772-Frame\"><span class=\"MathJax_MathContainer\"><span>\u03bc=(e2me)L.<\/span><\/span><\/div>\r\n<div class=\"os-equation-number\"><span class=\"os-number\">[4.18]<\/span><\/div>\r\n<\/div>\r\n<span style=\"font-size: 14pt\">In full vector form, this expression is written as<\/span>\r\n<div class=\"textbox\">\r\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-773-Frame\"><span class=\"MathJax_MathContainer\"><span>\u03bc\u2192=\u2212(e2me)L\u2192.<\/span><\/span><\/div>\r\n<div class=\"os-equation-number\"><span class=\"os-number\">[4.19]<\/span><\/div>\r\n<\/div>\r\n<span style=\"font-size: 14pt\">The negative sign appears because the electron has a negative charge. Notice that the direction of the magnetic moment of the electron is antiparallel to the orbital angular momentum, as shown in<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><span style=\"font-size: 14pt\">Figure 4.10(b). In the Bohr model of the atom, the relationship between<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-774-Frame\" style=\"font-size: 14pt\"><span class=\"MathJax_MathContainer\">\u03bc\u2192<\/span><\/span><span style=\"font-size: 14pt\">\u00a0<\/span><span style=\"font-size: 14pt\">and<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-775-Frame\" style=\"font-size: 14pt\"><span class=\"MathJax_MathContainer\">L\u2192<\/span><\/span><span style=\"font-size: 14pt\">\u00a0<\/span><span style=\"font-size: 14pt\">in<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><span style=\"font-size: 14pt\">Equation 4.19<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><span style=\"font-size: 14pt\">is independent of the radius of the orbit.<\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<p id=\"fs-id1170902756483\">The magnetic moment<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-776-Frame\"><span class=\"MathJax_MathContainer\"><span>\u03bc<\/span><\/span><\/span><span>\u00a0<\/span>can also be expressed in terms of the orbital angular quantum number<span>\u00a0<\/span><em data-effect=\"italics\">l<\/em>. Combining<span>\u00a0<\/span>Equation 4.18<span>\u00a0<\/span>and<span>\u00a0<\/span>Equation 4.15, the magnitude of the magnetic moment is<\/p>\r\n\r\n<div data-type=\"equation\" id=\"fs-id1170902750454\">\r\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-777-Frame\">\r\n<div class=\"textbox\">\r\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-777-Frame\"><span class=\"MathJax_MathContainer\"><span>\u03bc=(e2me)L=(e2me)l(l+1)\u210f=\u03bcBl(l+1).<\/span><\/span><\/div>\r\n<div class=\"os-equation-number\"><span class=\"os-number\">[4.20]<\/span><\/div>\r\n<\/div>\r\n<span style=\"font-size: 14pt\">The<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><em style=\"font-size: 14pt\" data-effect=\"italics\">z<\/em><span style=\"font-size: 14pt\">-component of the magnetic moment is<\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"equation\" id=\"fs-id1170902849388\">\r\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-778-Frame\">\r\n<div class=\"textbox\">\r\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-778-Frame\"><span class=\"MathJax_MathContainer\"><span>\u03bcz=\u2212(e2me)Lz=\u2212(e2me)m\u210f=\u2212\u03bcBm.<\/span><\/span><\/div>\r\n<div class=\"os-equation-number\"><span class=\"os-number\">[4.21]<\/span><\/div>\r\n<\/div>\r\n<span style=\"font-size: 14pt\">The quantity<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-779-Frame\" style=\"font-size: 14pt\"><span class=\"MathJax_MathContainer\">\u03bcB<\/span><\/span><span style=\"font-size: 14pt\">\u00a0<\/span><span style=\"font-size: 14pt\">is a fundamental unit of magnetism called the<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><span data-type=\"term\" id=\"term339\" style=\"font-size: 14pt\">Bohr magneton<\/span><span style=\"font-size: 14pt\">, which has the value<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-780-Frame\" style=\"font-size: 14pt\"><span class=\"MathJax_MathContainer\">9.3\u00d710\u221224joule\/tesla<\/span><\/span><span style=\"font-size: 14pt\">\u00a0<\/span><span style=\"font-size: 14pt\">(J\/T) or<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-781-Frame\" style=\"font-size: 14pt\"><span class=\"MathJax_MathContainer\">5.8\u00d710\u22125eV\/T.<\/span><\/span><span style=\"font-size: 14pt\">\u00a0<\/span><span style=\"font-size: 14pt\">Quantization of the magnetic moment is the result of quantization of the orbital angular momentum.<\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<p id=\"fs-id1170901624819\">As we will see in the next section, the total magnetic dipole moment of the hydrogen atom is due to both the orbital motion of the electron and its intrinsic spin. For now, we ignore the effect of electron spin.<\/p>\r\n\r\n<div data-type=\"example\" id=\"fs-id1170902848374\" class=\"ui-has-child-title\"><header><\/header><section>\r\n<div class=\"textbox shaded\"><header>\r\n<h3 class=\"os-title\"><span class=\"os-title-label\">EXAMPLE<span>\u00a04<\/span><\/span><span class=\"os-number\">.3<\/span><span class=\"os-divider\"><\/span><\/h3>\r\n<\/header><section>\r\n<p id=\"fs-id1170901635325\"><span data-type=\"title\"><strong>Orbital Magnetic Dipole Moment<\/strong><\/span><\/p>\r\nWhat is the magnitude of the orbital dipole magnetic moment<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-782-Frame\"><span class=\"MathJax_MathContainer\"><span>\u03bc<\/span><\/span><\/span><span>\u00a0<\/span>of an electron in the hydrogen atom in the (a)<span>\u00a0<\/span><em data-effect=\"italics\">s<\/em><span>\u00a0<\/span>state, (b)<span>\u00a0<\/span><em data-effect=\"italics\">p<\/em>state, and (c)<span>\u00a0<\/span><em data-effect=\"italics\">d<\/em><span>\u00a0<\/span>state? (Assume that the spin of the electron is zero.)\r\n\r\n<span data-type=\"title\"><strong>Strategy<\/strong><\/span>\r\n\r\nThe magnetic momentum of the electron is related to its orbital angular momentum<span>\u00a0<\/span><em data-effect=\"italics\">L<\/em>. For the hydrogen atom, this quantity is related to the orbital angular quantum number<span>\u00a0<\/span><em data-effect=\"italics\">l<\/em>. The states are given in spectroscopic notation, which relates a letter (<em data-effect=\"italics\">s<\/em>,<span>\u00a0<\/span><em data-effect=\"italics\">p<\/em>,<span>\u00a0<\/span><em data-effect=\"italics\">d<\/em>, etc.) to a quantum number.\r\n\r\n<span data-type=\"title\"><strong>Solution<\/strong><\/span>\r\n\r\nThe magnitude of the magnetic moment is given in<span>\u00a0<\/span>Equation 4.20:\r\n<div class=\"unnumbered\" data-label=\"\" data-type=\"equation\" id=\"fs-id1170901769839\">\r\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-783-Frame\"><span class=\"MathJax_MathContainer\"><span>\u03bc=(e2me)L=(e2me)l(l+1)\u210f=\u03bcBl(l+1).<\/span><\/span><\/div>\r\n<\/div>\r\n<ol id=\"fs-id1170903053484\" type=\"a\">\r\n \t<li>For the<span>\u00a0<\/span><em data-effect=\"italics\">s<\/em><span>\u00a0<\/span>state,<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-784-Frame\"><span class=\"MathJax_MathContainer\"><span>l=0<\/span><\/span><\/span><span>\u00a0<\/span>so we have<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-785-Frame\"><span class=\"MathJax_MathContainer\"><span>\u03bc=0<\/span><\/span><\/span><span>\u00a0<\/span>and<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-786-Frame\"><span class=\"MathJax_MathContainer\"><span>\u03bcz=0.<\/span><\/span><\/span><\/li>\r\n \t<li>For the<span>\u00a0<\/span><em data-effect=\"italics\">p<\/em><span>\u00a0<\/span>state,<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-787-Frame\"><span class=\"MathJax_MathContainer\"><span>l=1<\/span><\/span><\/span><span>\u00a0<\/span>and we have<span data-type=\"newline\">\r\n<\/span>\r\n<div class=\"unnumbered\" data-label=\"\" data-type=\"equation\" id=\"fs-id1170903023586\">\r\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-788-Frame\"><span class=\"MathJax_MathContainer\"><span>\u03bc=\u03bcB1(1+1)=2\u03bcB\u03bcz=\u2212\u03bcBm,wherem=(\u22121,0,1),so\u03bcz=\u03bcB,0,\u2212\u03bcB.<\/span><\/span><\/div>\r\n<\/div><\/li>\r\n \t<li>For the<span>\u00a0<\/span><em data-effect=\"italics\">d<\/em><span>\u00a0<\/span>state,<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-789-Frame\"><span class=\"MathJax_MathContainer\"><span>l=2<\/span><\/span><\/span><span>\u00a0<\/span>and we obtain<span data-type=\"newline\">\r\n<\/span>\r\n<div class=\"unnumbered\" data-label=\"\" data-type=\"equation\" id=\"fs-id1170902786762\">\r\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-790-Frame\">\r\n\r\n<span class=\"MathJax_MathContainer\"><span>\u03bc=\u03bcB2(2+1)=6\u03bcB\u03bcz=\u2212\u03bcBm,wherem=(\u22122,\u22121,0,1,2),so\u03bcz=2\u03bcB,\u03bcB,0,\u2212\u03bcB,\u22122\u03bcB.<\/span><\/span>\r\n\r\n<span data-type=\"title\" style=\"text-indent: 1em;font-size: 1rem\"><strong>Significance<\/strong><\/span>\r\n\r\n<span style=\"text-indent: 1em;font-size: 1rem\">In the<\/span><span style=\"text-indent: 1em;font-size: 1rem\">\u00a0<\/span><em style=\"text-indent: 1em;font-size: 1rem\" data-effect=\"italics\">s<\/em><span style=\"text-indent: 1em;font-size: 1rem\">\u00a0<\/span><span style=\"text-indent: 1em;font-size: 1rem\">state, there is no orbital angular momentum and therefore no magnetic moment. This does not mean that the electron is at rest, just that the overall motion of the electron does not produce a magnetic field. In the<\/span><span style=\"text-indent: 1em;font-size: 1rem\">\u00a0<\/span><em style=\"text-indent: 1em;font-size: 1rem\" data-effect=\"italics\">p<\/em><span style=\"text-indent: 1em;font-size: 1rem\">\u00a0<\/span><span style=\"text-indent: 1em;font-size: 1rem\">state, the electron has a magnetic moment with three possible values for the<\/span><span style=\"text-indent: 1em;font-size: 1rem\">\u00a0<\/span><em style=\"text-indent: 1em;font-size: 1rem\" data-effect=\"italics\">z<\/em><span style=\"text-indent: 1em;font-size: 1rem\">-component of this magnetic moment; this means that magnetic moment can point in three different polar directions\u2014each antiparallel to the orbital angular momentum vector. In the<\/span><span style=\"text-indent: 1em;font-size: 1rem\">\u00a0<\/span><em style=\"text-indent: 1em;font-size: 1rem\" data-effect=\"italics\">d<\/em><span style=\"text-indent: 1em;font-size: 1rem\">\u00a0<\/span><span style=\"text-indent: 1em;font-size: 1rem\">state, the electron has a magnetic moment with five possible values for the<\/span><span style=\"text-indent: 1em;font-size: 1rem\">\u00a0<\/span><em style=\"text-indent: 1em;font-size: 1rem\" data-effect=\"italics\">z<\/em><span style=\"text-indent: 1em;font-size: 1rem\">-component of this magnetic moment. In this case, the magnetic moment can point in five different polar directions.<\/span>\r\n\r\n<\/div>\r\n<\/div><\/li>\r\n<\/ol>\r\n<\/section><\/div>\r\n<span style=\"font-size: 14pt\">A hydrogen atom has a magnetic field, so we expect the hydrogen atom to interact with an external magnetic field\u2014such as the push and pull between two bar magnets. From<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><a href=\"https:\/\/cnx.org\/contents\/af275420-6050-4707-995c-57b9cc13c358@11.1:31277ded-aa04-406e-a929-b692e0b8bf2b#fs-id1171360288680\" data-page=\"1\" style=\"font-size: 14pt\">Force and Torque on a Current Loop<\/a><span style=\"font-size: 14pt\">, we know that when a current loop interacts with an external magnetic field<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-791-Frame\" style=\"font-size: 14pt\"><span class=\"MathJax_MathContainer\">B\u2192<\/span><\/span><span style=\"font-size: 14pt\">, it experiences a torque given by<\/span>\r\n\r\n<\/section><\/div>\r\n<div data-type=\"equation\" id=\"fs-id1170902787960\">\r\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-792-Frame\">\r\n<div class=\"textbox\">\r\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-792-Frame\"><span class=\"MathJax_MathContainer\"><span>\u03c4\u2192=I(A\u2192\u00d7B\u2192)=\u03bc\u2192\u00d7B\u2192,<\/span><\/span><\/div>\r\n<div class=\"os-equation-number\"><span class=\"os-number\">[4.22]<\/span><\/div>\r\n<\/div>\r\n<span style=\"font-size: 14pt\">where<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><em style=\"font-size: 14pt\" data-effect=\"italics\">I<\/em><span style=\"font-size: 14pt\">\u00a0<\/span><span style=\"font-size: 14pt\">is the current,<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-793-Frame\" style=\"font-size: 14pt\"><span class=\"MathJax_MathContainer\">A\u2192<\/span><\/span><span style=\"font-size: 14pt\">\u00a0<\/span><span style=\"font-size: 14pt\">is the area of the loop,<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-794-Frame\" style=\"font-size: 14pt\"><span class=\"MathJax_MathContainer\">\u03bc\u2192<\/span><\/span><span style=\"font-size: 14pt\">\u00a0<\/span><span style=\"font-size: 14pt\">is the magnetic moment, and<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-795-Frame\" style=\"font-size: 14pt\"><span class=\"MathJax_MathContainer\">B\u2192<\/span><\/span><span style=\"font-size: 14pt\">\u00a0<\/span><span style=\"font-size: 14pt\">is the external magnetic field. This torque acts to rotate the magnetic moment vector of the hydrogen atom to align with the external magnetic field. Because mechanical work is done by the external magnetic field on the hydrogen atom, we can talk about energy transformations in the atom. The potential energy of the hydrogen atom associated with this magnetic interaction is given by<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><span style=\"font-size: 14pt\">Equation 4.23:<\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"equation\" id=\"fs-id1170902875239\">\r\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-796-Frame\">\r\n<div class=\"textbox\">\r\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-796-Frame\"><span class=\"MathJax_MathContainer\"><span>U=\u2212\u03bc\u2192\u00b7B\u2192.<\/span><\/span><\/div>\r\n<div class=\"os-equation-number\"><span class=\"os-number\">[4.23]<\/span><\/div>\r\n<\/div>\r\n<span style=\"font-size: 14pt\">If the magnetic moment is antiparallel to the external magnetic field, the potential energy is large, but if the magnetic moment is parallel to the field, the potential energy is small. Work done on the hydrogen atom to rotate the atom\u2019s magnetic moment vector in the direction of the external magnetic field is therefore associated with a drop in potential energy. The energy of the system is conserved, however, because a drop in potential energy produces radiation (the emission of a photon). These energy transitions are quantized because the magnetic moment can point in only certain directions.<\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<p id=\"fs-id1170903053910\">If the external magnetic field points in the positive<span>\u00a0<\/span><em data-effect=\"italics\">z<\/em>-direction, the potential energy associated with the orbital magnetic dipole moment is<\/p>\r\n\r\n<div class=\"textbox\">\r\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-797-Frame\"><span class=\"MathJax_MathContainer\"><span>U(\u03b8)=\u2212\u03bcBcos\u03b8=\u2212\u03bczB=\u2212(\u2212\u03bcBm)B=m\u03bcBB,<\/span><\/span><\/div>\r\n<div class=\"os-equation-number\"><span class=\"os-number\">[4.24]<\/span><\/div>\r\n<\/div>\r\n<p id=\"fs-id1170902736440\">where<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-798-Frame\"><span class=\"MathJax_MathContainer\"><span>\u03bcB<\/span><\/span><\/span><span>\u00a0<\/span>is the Bohr magneton and<span>\u00a0<\/span><em data-effect=\"italics\">m<\/em><span>\u00a0<\/span>is the angular momentum projection quantum number (or<span>\u00a0<\/span><span data-type=\"term\" id=\"term340\">magnetic orbital quantum number<\/span>), which has the values<\/p>\r\n\r\n<div data-type=\"equation\" id=\"fs-id1170901767676\">\r\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-799-Frame\">\r\n<div class=\"textbox\">\r\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-799-Frame\"><span class=\"MathJax_MathContainer\"><span>m=\u2212l,\u2212l+1,...,0,...,l\u22121,l.<\/span><\/span><\/div>\r\n<div class=\"os-equation-number\"><span class=\"os-number\">[4.25]<\/span><\/div>\r\n<\/div>\r\n<span style=\"font-size: 14pt\">For example, in the<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-800-Frame\" style=\"font-size: 14pt\"><span class=\"MathJax_MathContainer\">l=1<\/span><\/span><span style=\"font-size: 14pt\">\u00a0<\/span><span style=\"font-size: 14pt\">electron state, the total energy of the electron is split into three distinct energy levels corresponding to<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-801-Frame\" style=\"font-size: 14pt\"><span class=\"MathJax_MathContainer\">U=\u2212\u03bcBB,0,\u03bcBB.<\/span><\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<p id=\"fs-id1170902741598\">The splitting of energy levels by an external magnetic field is called the<span>\u00a0<\/span><span data-type=\"term\" id=\"term341\">Zeeman effect<\/span>. Ignoring the effects of electron spin, transitions from the<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-803-Frame\"><span class=\"MathJax_MathContainer\"><span>l=1<\/span><\/span><\/span><span>\u00a0<\/span>state to a common lower energy state produce three closely spaced spectral lines (Figure 4.11, left column). Likewise, transitions from the<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-804-Frame\"><span class=\"MathJax_MathContainer\"><span>l=2<\/span><\/span><\/span><span>\u00a0<\/span>state produce five closely spaced spectral lines (right column). The separation of these lines is proportional to the strength of the external magnetic field. This effect has many applications. For example, the splitting of lines in the hydrogen spectrum of the Sun is used to determine the strength of the Sun\u2019s magnetic field. Many such magnetic field measurements can be used to make a map of the magnetic activity at the Sun\u2019s surface called a<span>\u00a0<\/span><span data-type=\"term\" id=\"term342\">magnetogram<\/span><span>\u00a0<\/span>(Figure 4.12).<\/p>\r\n\r\n<div class=\"os-figure\">\r\n<figure id=\"CNX_UPhysics_41_02_Zeeman\">\r\n\r\n[caption id=\"\" align=\"alignnone\" width=\"490\"]<img alt=\"The figure shows the effect of magnetic field, B sub ext, on two different spectral lines, corresponding to the l=1 to l=0 transition on the left and the l=2 to l=0 transition on the right. The spectra are shown for no external field, for a non zero external field and for a large external field. With no external field, both transitions appear as single lines. In the second case, when magnetic field is applied, the spectral lines split into several lines; the line on the left splits into three lines. The line on the right splits into five. In the third case, the magnetic field is large. The left line is again split into three lines and the right into five, but the split lines are farther apart than they are when the external magnetic field is not as strong.\" data-media-type=\"image\/jpeg\" id=\"5110\" src=\"https:\/\/cnx.org\/resources\/d10c950d884c87b3c395dd6bb97b6c587800022a\" width=\"490\" height=\"402\" \/> Figure 4.11 The Zeeman effect refers to the splitting of spectral lines by an external magnetic field. In the left column, the energy splitting occurs due to transitions from the state (n=2,l=1) to a lower energy state; and in the right column, energy splitting occurs due to transitions from the state (n=2,l=2) to a lower-energy state. The separation of these lines is proportional to the strength of the external magnetic field.[\/caption]\r\n\r\n&nbsp;<\/figure>\r\n<\/div>\r\n<div class=\"os-figure\">\r\n<figure id=\"CNX_UPhysics_41_02_Magnetogrm\">\r\n\r\n[caption id=\"\" align=\"alignnone\" width=\"487\"]<img alt=\"A magnetogram of the sun, which appears as a gray disc against a black background, with white and black spots scattered on it. Most of the spots are concentrated in the center right part of the image.\" data-media-type=\"image\/jpeg\" id=\"66788\" src=\"https:\/\/cnx.org\/resources\/109d8548d50af1f2b6c7ceefb740978ef6093452\" width=\"487\" height=\"487\" \/> Figure 4.12 A magnetogram of the Sun. The bright and dark spots show significant magnetic activity at the surface of the Sun. (credit: NASA, SDO)[\/caption]<\/figure>\r\n<div>\r\n\r\n&nbsp;\r\n<div class=\"textbox\"><em>Download for free at http:\/\/cnx.org\/contents\/af275420-6050-4707-995c-57b9cc13c358@11.1<\/em><\/div>\r\n<\/div>\r\n<\/div>","rendered":"<div data-type=\"abstract\" id=\"22668\" class=\"ui-has-child-title\">\n<header>\n<div class=\"textbox textbox--learning-objectives\"><\/div>\n<\/header>\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Learning Objectives<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>By the end of this section, you will be able to:<\/p>\n<ul>\n<li>Explain why the hydrogen atom has magnetic properties<\/li>\n<li>Explain why the energy levels of a hydrogen atom associated with orbital angular momentum are split by an external magnetic field<\/li>\n<li>Use quantum numbers to calculate the magnitude and direction of the orbital magnetic dipole moment of a hydrogen atom<\/li>\n<\/ul>\n<\/div>\n<\/div>\n<p><span style=\"font-size: 14pt\">In Bohr\u2019s model of the hydrogen atom, the electron moves in a circular orbit around the proton. The electron passes by a particular point on the loop in a certain time, so we can calculate a current<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-761-Frame\" style=\"font-size: 14pt\"><span class=\"MathJax_MathContainer\">I=Q\/t<\/span><\/span><span style=\"font-size: 14pt\">. An electron that orbits a proton in a hydrogen atom is therefore analogous to current flowing through a circular wire (<\/span>Figure 4.10<span style=\"font-size: 14pt\">). In the study of magnetism, we saw that a current-carrying wire produces magnetic fields. It is therefore reasonable to conclude that the hydrogen atom produces a magnetic field and interacts with other magnetic fields.<\/span><\/p>\n<div class=\"os-figure\">\n<figure id=\"CNX_UPhysics_41_02_AtomicLoop\">\n<figure style=\"width: 837px\" class=\"wp-caption alignnone\"><img loading=\"lazy\" decoding=\"async\" alt=\"Figure (a) shows a current carrying loop. The loop has current I circulating counterclockwise as viewed from above. A vector mu pointing upward is shown at the center of the loop. Figure (b) shows the hydrogen atom as an electron, represented as a small ball and labeled minus e, making a counterclockwise circular orbit, as viewed from above. A sphere, a vector mu pointing downward, and a vector L pointing upward are shown in the center of the orbit.\" data-media-type=\"image\/jpeg\" id=\"86901\" src=\"https:\/\/cnx.org\/resources\/331456eee02dc81fd687dd04689eed0a7e66073b\" width=\"837\" height=\"254\" \/><figcaption class=\"wp-caption-text\">Figure 4.10 (a) Current flowing through a circular wire is analogous to (b) an electron that orbits a proton in a hydrogen atom.<\/figcaption><\/figure>\n<\/figure>\n<\/div>\n<p id=\"fs-id1170902922212\">The<span>\u00a0<\/span><span data-type=\"term\" id=\"term338\">orbital magnetic dipole moment<\/span><span>\u00a0<\/span>is a measure of the strength of the magnetic field produced by the orbital angular momentum of an electron. From<span>\u00a0<\/span><a href=\"https:\/\/cnx.org\/contents\/af275420-6050-4707-995c-57b9cc13c358@11.1:31277ded-aa04-406e-a929-b692e0b8bf2b#fs-id1171360245659\" data-page=\"1\">Force and Torque on a Current Loop<\/a>, the magnitude of the orbital magnetic dipole moment for a current loop is<\/p>\n<div data-type=\"equation\" id=\"fs-id1170901575179\">\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-762-Frame\">\n<div class=\"textbox\">\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-762-Frame\"><span class=\"MathJax_MathContainer\"><span>\u03bc=IA,<\/span><\/span><\/div>\n<div class=\"os-equation-number\"><span class=\"os-number\">[4.13]<\/span><\/div>\n<\/div>\n<p><span style=\"font-size: 14pt\">where<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><em style=\"font-size: 14pt\" data-effect=\"italics\">I<\/em><span style=\"font-size: 14pt\">\u00a0<\/span><span style=\"font-size: 14pt\">is the current and<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><em style=\"font-size: 14pt\" data-effect=\"italics\">A<\/em><span style=\"font-size: 14pt\">\u00a0<\/span><span style=\"font-size: 14pt\">is the area of the loop. (For brevity, we refer to this as the magnetic moment.) The current<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><em style=\"font-size: 14pt\" data-effect=\"italics\">I<\/em><span style=\"font-size: 14pt\">\u00a0<\/span><span style=\"font-size: 14pt\">associated with an electron in orbit about a proton in a hydrogen atom is<\/span><\/p>\n<\/div>\n<\/div>\n<div data-type=\"equation\" id=\"fs-id1170902672410\">\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-763-Frame\">\n<div class=\"textbox\">\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-763-Frame\"><span class=\"MathJax_MathContainer\"><span>I=eT,<\/span><\/span><\/div>\n<div class=\"os-equation-number\"><span class=\"os-number\">[4.14]<\/span><\/div>\n<\/div>\n<p><span style=\"font-size: 14pt\">where<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><em style=\"font-size: 14pt\" data-effect=\"italics\">e<\/em><span style=\"font-size: 14pt\">\u00a0<\/span><span style=\"font-size: 14pt\">is the magnitude of the electron charge and<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><em style=\"font-size: 14pt\" data-effect=\"italics\">T<\/em><span style=\"font-size: 14pt\">\u00a0<\/span><span style=\"font-size: 14pt\">is its orbital period. If we assume that the electron travels in a perfectly circular orbit, the orbital period is<\/span><\/p>\n<\/div>\n<\/div>\n<div data-type=\"equation\" id=\"fs-id1170902891412\">\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-764-Frame\">\n<div class=\"textbox\">\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-764-Frame\"><span class=\"MathJax_MathContainer\"><span>T=2\u03c0rv,<\/span><\/span><\/div>\n<div class=\"os-equation-number\"><span class=\"os-number\">[4.15]<\/span><\/div>\n<\/div>\n<p><span style=\"font-size: 14pt\">where<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><em style=\"font-size: 14pt\" data-effect=\"italics\">r<\/em><span style=\"font-size: 14pt\">\u00a0<\/span><span style=\"font-size: 14pt\">is the radius of the orbit and<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><em style=\"font-size: 14pt\" data-effect=\"italics\">v<\/em><span style=\"font-size: 14pt\">\u00a0<\/span><span style=\"font-size: 14pt\">is the speed of the electron in its orbit. Given that the area of a circle is<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-765-Frame\" style=\"font-size: 14pt\"><span class=\"MathJax_MathContainer\">\u03c0r2<\/span><\/span><span style=\"font-size: 14pt\">, the absolute magnetic moment is<\/span><\/p>\n<\/div>\n<\/div>\n<div data-type=\"equation\" id=\"fs-id1170902600684\">\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-766-Frame\">\n<div class=\"textbox\">\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-766-Frame\"><span class=\"MathJax_MathContainer\"><span>\u03bc=IA=e(2\u03c0rv)\u03c0r2=evr2.<\/span><\/span><\/div>\n<div class=\"os-equation-number\"><span class=\"os-number\">[4.16]<\/span><\/div>\n<\/div>\n<p><span style=\"font-size: 14pt\">It is helpful to express the magnetic momentum<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-767-Frame\" style=\"font-size: 14pt\"><span class=\"MathJax_MathContainer\">\u03bc<\/span><\/span><span style=\"font-size: 14pt\">\u00a0<\/span><span style=\"font-size: 14pt\">in terms of the orbital angular momentum<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-768-Frame\" style=\"font-size: 14pt\"><span class=\"MathJax_MathContainer\">(L\u2192=r\u2192\u00d7p\u2192).<\/span><\/span><span style=\"font-size: 14pt\">\u00a0<\/span><span style=\"font-size: 14pt\">Because the electron orbits in a circle, the position vector<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-769-Frame\" style=\"font-size: 14pt\"><span class=\"MathJax_MathContainer\">r\u2192<\/span><\/span><span style=\"font-size: 14pt\">\u00a0<\/span><span style=\"font-size: 14pt\">and the momentum vector<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-770-Frame\" style=\"font-size: 14pt\"><span class=\"MathJax_MathContainer\">p\u2192<\/span><\/span><span style=\"font-size: 14pt\">\u00a0<\/span><span style=\"font-size: 14pt\">form a right angle. Thus, the magnitude of the orbital angular momentum is<\/span><\/p>\n<\/div>\n<\/div>\n<div data-type=\"equation\" id=\"fs-id1170901533720\">\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-771-Frame\">\n<div class=\"textbox\">\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-771-Frame\"><span class=\"MathJax_MathContainer\"><span>L=|L\u2192|=|r\u2192\u00d7p\u2192|=rpsin\u03b8=rp=rmv.<\/span><\/span><\/div>\n<div class=\"os-equation-number\"><span class=\"os-number\">[4.17]<\/span><\/div>\n<\/div>\n<p><span style=\"font-size: 14pt\">Combining these two equations, we have<\/span><\/p>\n<\/div>\n<\/div>\n<div data-type=\"equation\" id=\"fs-id1170901597936\">\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-772-Frame\">\n<div class=\"textbox\">\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-772-Frame\"><span class=\"MathJax_MathContainer\"><span>\u03bc=(e2me)L.<\/span><\/span><\/div>\n<div class=\"os-equation-number\"><span class=\"os-number\">[4.18]<\/span><\/div>\n<\/div>\n<p><span style=\"font-size: 14pt\">In full vector form, this expression is written as<\/span><\/p>\n<div class=\"textbox\">\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-773-Frame\"><span class=\"MathJax_MathContainer\"><span>\u03bc\u2192=\u2212(e2me)L\u2192.<\/span><\/span><\/div>\n<div class=\"os-equation-number\"><span class=\"os-number\">[4.19]<\/span><\/div>\n<\/div>\n<p><span style=\"font-size: 14pt\">The negative sign appears because the electron has a negative charge. Notice that the direction of the magnetic moment of the electron is antiparallel to the orbital angular momentum, as shown in<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><span style=\"font-size: 14pt\">Figure 4.10(b). In the Bohr model of the atom, the relationship between<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-774-Frame\" style=\"font-size: 14pt\"><span class=\"MathJax_MathContainer\">\u03bc\u2192<\/span><\/span><span style=\"font-size: 14pt\">\u00a0<\/span><span style=\"font-size: 14pt\">and<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-775-Frame\" style=\"font-size: 14pt\"><span class=\"MathJax_MathContainer\">L\u2192<\/span><\/span><span style=\"font-size: 14pt\">\u00a0<\/span><span style=\"font-size: 14pt\">in<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><span style=\"font-size: 14pt\">Equation 4.19<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><span style=\"font-size: 14pt\">is independent of the radius of the orbit.<\/span><\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1170902756483\">The magnetic moment<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-776-Frame\"><span class=\"MathJax_MathContainer\"><span>\u03bc<\/span><\/span><\/span><span>\u00a0<\/span>can also be expressed in terms of the orbital angular quantum number<span>\u00a0<\/span><em data-effect=\"italics\">l<\/em>. Combining<span>\u00a0<\/span>Equation 4.18<span>\u00a0<\/span>and<span>\u00a0<\/span>Equation 4.15, the magnitude of the magnetic moment is<\/p>\n<div data-type=\"equation\" id=\"fs-id1170902750454\">\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-777-Frame\">\n<div class=\"textbox\">\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-777-Frame\"><span class=\"MathJax_MathContainer\"><span>\u03bc=(e2me)L=(e2me)l(l+1)\u210f=\u03bcBl(l+1).<\/span><\/span><\/div>\n<div class=\"os-equation-number\"><span class=\"os-number\">[4.20]<\/span><\/div>\n<\/div>\n<p><span style=\"font-size: 14pt\">The<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><em style=\"font-size: 14pt\" data-effect=\"italics\">z<\/em><span style=\"font-size: 14pt\">-component of the magnetic moment is<\/span><\/p>\n<\/div>\n<\/div>\n<div data-type=\"equation\" id=\"fs-id1170902849388\">\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-778-Frame\">\n<div class=\"textbox\">\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-778-Frame\"><span class=\"MathJax_MathContainer\"><span>\u03bcz=\u2212(e2me)Lz=\u2212(e2me)m\u210f=\u2212\u03bcBm.<\/span><\/span><\/div>\n<div class=\"os-equation-number\"><span class=\"os-number\">[4.21]<\/span><\/div>\n<\/div>\n<p><span style=\"font-size: 14pt\">The quantity<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-779-Frame\" style=\"font-size: 14pt\"><span class=\"MathJax_MathContainer\">\u03bcB<\/span><\/span><span style=\"font-size: 14pt\">\u00a0<\/span><span style=\"font-size: 14pt\">is a fundamental unit of magnetism called the<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><span data-type=\"term\" id=\"term339\" style=\"font-size: 14pt\">Bohr magneton<\/span><span style=\"font-size: 14pt\">, which has the value<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-780-Frame\" style=\"font-size: 14pt\"><span class=\"MathJax_MathContainer\">9.3\u00d710\u221224joule\/tesla<\/span><\/span><span style=\"font-size: 14pt\">\u00a0<\/span><span style=\"font-size: 14pt\">(J\/T) or<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-781-Frame\" style=\"font-size: 14pt\"><span class=\"MathJax_MathContainer\">5.8\u00d710\u22125eV\/T.<\/span><\/span><span style=\"font-size: 14pt\">\u00a0<\/span><span style=\"font-size: 14pt\">Quantization of the magnetic moment is the result of quantization of the orbital angular momentum.<\/span><\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1170901624819\">As we will see in the next section, the total magnetic dipole moment of the hydrogen atom is due to both the orbital motion of the electron and its intrinsic spin. For now, we ignore the effect of electron spin.<\/p>\n<div data-type=\"example\" id=\"fs-id1170902848374\" class=\"ui-has-child-title\">\n<header><\/header>\n<section>\n<div class=\"textbox shaded\">\n<header>\n<h3 class=\"os-title\"><span class=\"os-title-label\">EXAMPLE<span>\u00a04<\/span><\/span><span class=\"os-number\">.3<\/span><span class=\"os-divider\"><\/span><\/h3>\n<\/header>\n<section>\n<p id=\"fs-id1170901635325\"><span data-type=\"title\"><strong>Orbital Magnetic Dipole Moment<\/strong><\/span><\/p>\n<p>What is the magnitude of the orbital dipole magnetic moment<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-782-Frame\"><span class=\"MathJax_MathContainer\"><span>\u03bc<\/span><\/span><\/span><span>\u00a0<\/span>of an electron in the hydrogen atom in the (a)<span>\u00a0<\/span><em data-effect=\"italics\">s<\/em><span>\u00a0<\/span>state, (b)<span>\u00a0<\/span><em data-effect=\"italics\">p<\/em>state, and (c)<span>\u00a0<\/span><em data-effect=\"italics\">d<\/em><span>\u00a0<\/span>state? (Assume that the spin of the electron is zero.)<\/p>\n<p><span data-type=\"title\"><strong>Strategy<\/strong><\/span><\/p>\n<p>The magnetic momentum of the electron is related to its orbital angular momentum<span>\u00a0<\/span><em data-effect=\"italics\">L<\/em>. For the hydrogen atom, this quantity is related to the orbital angular quantum number<span>\u00a0<\/span><em data-effect=\"italics\">l<\/em>. The states are given in spectroscopic notation, which relates a letter (<em data-effect=\"italics\">s<\/em>,<span>\u00a0<\/span><em data-effect=\"italics\">p<\/em>,<span>\u00a0<\/span><em data-effect=\"italics\">d<\/em>, etc.) to a quantum number.<\/p>\n<p><span data-type=\"title\"><strong>Solution<\/strong><\/span><\/p>\n<p>The magnitude of the magnetic moment is given in<span>\u00a0<\/span>Equation 4.20:<\/p>\n<div class=\"unnumbered\" data-label=\"\" data-type=\"equation\" id=\"fs-id1170901769839\">\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-783-Frame\"><span class=\"MathJax_MathContainer\"><span>\u03bc=(e2me)L=(e2me)l(l+1)\u210f=\u03bcBl(l+1).<\/span><\/span><\/div>\n<\/div>\n<ol id=\"fs-id1170903053484\" type=\"a\">\n<li>For the<span>\u00a0<\/span><em data-effect=\"italics\">s<\/em><span>\u00a0<\/span>state,<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-784-Frame\"><span class=\"MathJax_MathContainer\"><span>l=0<\/span><\/span><\/span><span>\u00a0<\/span>so we have<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-785-Frame\"><span class=\"MathJax_MathContainer\"><span>\u03bc=0<\/span><\/span><\/span><span>\u00a0<\/span>and<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-786-Frame\"><span class=\"MathJax_MathContainer\"><span>\u03bcz=0.<\/span><\/span><\/span><\/li>\n<li>For the<span>\u00a0<\/span><em data-effect=\"italics\">p<\/em><span>\u00a0<\/span>state,<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-787-Frame\"><span class=\"MathJax_MathContainer\"><span>l=1<\/span><\/span><\/span><span>\u00a0<\/span>and we have<span data-type=\"newline\"><br \/>\n<\/span><\/p>\n<div class=\"unnumbered\" data-label=\"\" data-type=\"equation\" id=\"fs-id1170903023586\">\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-788-Frame\"><span class=\"MathJax_MathContainer\"><span>\u03bc=\u03bcB1(1+1)=2\u03bcB\u03bcz=\u2212\u03bcBm,wherem=(\u22121,0,1),so\u03bcz=\u03bcB,0,\u2212\u03bcB.<\/span><\/span><\/div>\n<\/div>\n<\/li>\n<li>For the<span>\u00a0<\/span><em data-effect=\"italics\">d<\/em><span>\u00a0<\/span>state,<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-789-Frame\"><span class=\"MathJax_MathContainer\"><span>l=2<\/span><\/span><\/span><span>\u00a0<\/span>and we obtain<span data-type=\"newline\"><br \/>\n<\/span><\/p>\n<div class=\"unnumbered\" data-label=\"\" data-type=\"equation\" id=\"fs-id1170902786762\">\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-790-Frame\">\n<p><span class=\"MathJax_MathContainer\"><span>\u03bc=\u03bcB2(2+1)=6\u03bcB\u03bcz=\u2212\u03bcBm,wherem=(\u22122,\u22121,0,1,2),so\u03bcz=2\u03bcB,\u03bcB,0,\u2212\u03bcB,\u22122\u03bcB.<\/span><\/span><\/p>\n<p><span data-type=\"title\" style=\"text-indent: 1em;font-size: 1rem\"><strong>Significance<\/strong><\/span><\/p>\n<p><span style=\"text-indent: 1em;font-size: 1rem\">In the<\/span><span style=\"text-indent: 1em;font-size: 1rem\">\u00a0<\/span><em style=\"text-indent: 1em;font-size: 1rem\" data-effect=\"italics\">s<\/em><span style=\"text-indent: 1em;font-size: 1rem\">\u00a0<\/span><span style=\"text-indent: 1em;font-size: 1rem\">state, there is no orbital angular momentum and therefore no magnetic moment. This does not mean that the electron is at rest, just that the overall motion of the electron does not produce a magnetic field. In the<\/span><span style=\"text-indent: 1em;font-size: 1rem\">\u00a0<\/span><em style=\"text-indent: 1em;font-size: 1rem\" data-effect=\"italics\">p<\/em><span style=\"text-indent: 1em;font-size: 1rem\">\u00a0<\/span><span style=\"text-indent: 1em;font-size: 1rem\">state, the electron has a magnetic moment with three possible values for the<\/span><span style=\"text-indent: 1em;font-size: 1rem\">\u00a0<\/span><em style=\"text-indent: 1em;font-size: 1rem\" data-effect=\"italics\">z<\/em><span style=\"text-indent: 1em;font-size: 1rem\">-component of this magnetic moment; this means that magnetic moment can point in three different polar directions\u2014each antiparallel to the orbital angular momentum vector. In the<\/span><span style=\"text-indent: 1em;font-size: 1rem\">\u00a0<\/span><em style=\"text-indent: 1em;font-size: 1rem\" data-effect=\"italics\">d<\/em><span style=\"text-indent: 1em;font-size: 1rem\">\u00a0<\/span><span style=\"text-indent: 1em;font-size: 1rem\">state, the electron has a magnetic moment with five possible values for the<\/span><span style=\"text-indent: 1em;font-size: 1rem\">\u00a0<\/span><em style=\"text-indent: 1em;font-size: 1rem\" data-effect=\"italics\">z<\/em><span style=\"text-indent: 1em;font-size: 1rem\">-component of this magnetic moment. In this case, the magnetic moment can point in five different polar directions.<\/span><\/p>\n<\/div>\n<\/div>\n<\/li>\n<\/ol>\n<\/section>\n<\/div>\n<p><span style=\"font-size: 14pt\">A hydrogen atom has a magnetic field, so we expect the hydrogen atom to interact with an external magnetic field\u2014such as the push and pull between two bar magnets. From<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><a href=\"https:\/\/cnx.org\/contents\/af275420-6050-4707-995c-57b9cc13c358@11.1:31277ded-aa04-406e-a929-b692e0b8bf2b#fs-id1171360288680\" data-page=\"1\" style=\"font-size: 14pt\">Force and Torque on a Current Loop<\/a><span style=\"font-size: 14pt\">, we know that when a current loop interacts with an external magnetic field<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-791-Frame\" style=\"font-size: 14pt\"><span class=\"MathJax_MathContainer\">B\u2192<\/span><\/span><span style=\"font-size: 14pt\">, it experiences a torque given by<\/span><\/p>\n<\/section>\n<\/div>\n<div data-type=\"equation\" id=\"fs-id1170902787960\">\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-792-Frame\">\n<div class=\"textbox\">\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-792-Frame\"><span class=\"MathJax_MathContainer\"><span>\u03c4\u2192=I(A\u2192\u00d7B\u2192)=\u03bc\u2192\u00d7B\u2192,<\/span><\/span><\/div>\n<div class=\"os-equation-number\"><span class=\"os-number\">[4.22]<\/span><\/div>\n<\/div>\n<p><span style=\"font-size: 14pt\">where<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><em style=\"font-size: 14pt\" data-effect=\"italics\">I<\/em><span style=\"font-size: 14pt\">\u00a0<\/span><span style=\"font-size: 14pt\">is the current,<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-793-Frame\" style=\"font-size: 14pt\"><span class=\"MathJax_MathContainer\">A\u2192<\/span><\/span><span style=\"font-size: 14pt\">\u00a0<\/span><span style=\"font-size: 14pt\">is the area of the loop,<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-794-Frame\" style=\"font-size: 14pt\"><span class=\"MathJax_MathContainer\">\u03bc\u2192<\/span><\/span><span style=\"font-size: 14pt\">\u00a0<\/span><span style=\"font-size: 14pt\">is the magnetic moment, and<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-795-Frame\" style=\"font-size: 14pt\"><span class=\"MathJax_MathContainer\">B\u2192<\/span><\/span><span style=\"font-size: 14pt\">\u00a0<\/span><span style=\"font-size: 14pt\">is the external magnetic field. This torque acts to rotate the magnetic moment vector of the hydrogen atom to align with the external magnetic field. Because mechanical work is done by the external magnetic field on the hydrogen atom, we can talk about energy transformations in the atom. The potential energy of the hydrogen atom associated with this magnetic interaction is given by<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><span style=\"font-size: 14pt\">Equation 4.23:<\/span><\/p>\n<\/div>\n<\/div>\n<div data-type=\"equation\" id=\"fs-id1170902875239\">\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-796-Frame\">\n<div class=\"textbox\">\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-796-Frame\"><span class=\"MathJax_MathContainer\"><span>U=\u2212\u03bc\u2192\u00b7B\u2192.<\/span><\/span><\/div>\n<div class=\"os-equation-number\"><span class=\"os-number\">[4.23]<\/span><\/div>\n<\/div>\n<p><span style=\"font-size: 14pt\">If the magnetic moment is antiparallel to the external magnetic field, the potential energy is large, but if the magnetic moment is parallel to the field, the potential energy is small. Work done on the hydrogen atom to rotate the atom\u2019s magnetic moment vector in the direction of the external magnetic field is therefore associated with a drop in potential energy. The energy of the system is conserved, however, because a drop in potential energy produces radiation (the emission of a photon). These energy transitions are quantized because the magnetic moment can point in only certain directions.<\/span><\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1170903053910\">If the external magnetic field points in the positive<span>\u00a0<\/span><em data-effect=\"italics\">z<\/em>-direction, the potential energy associated with the orbital magnetic dipole moment is<\/p>\n<div class=\"textbox\">\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-797-Frame\"><span class=\"MathJax_MathContainer\"><span>U(\u03b8)=\u2212\u03bcBcos\u03b8=\u2212\u03bczB=\u2212(\u2212\u03bcBm)B=m\u03bcBB,<\/span><\/span><\/div>\n<div class=\"os-equation-number\"><span class=\"os-number\">[4.24]<\/span><\/div>\n<\/div>\n<p id=\"fs-id1170902736440\">where<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-798-Frame\"><span class=\"MathJax_MathContainer\"><span>\u03bcB<\/span><\/span><\/span><span>\u00a0<\/span>is the Bohr magneton and<span>\u00a0<\/span><em data-effect=\"italics\">m<\/em><span>\u00a0<\/span>is the angular momentum projection quantum number (or<span>\u00a0<\/span><span data-type=\"term\" id=\"term340\">magnetic orbital quantum number<\/span>), which has the values<\/p>\n<div data-type=\"equation\" id=\"fs-id1170901767676\">\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-799-Frame\">\n<div class=\"textbox\">\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-799-Frame\"><span class=\"MathJax_MathContainer\"><span>m=\u2212l,\u2212l+1,&#8230;,0,&#8230;,l\u22121,l.<\/span><\/span><\/div>\n<div class=\"os-equation-number\"><span class=\"os-number\">[4.25]<\/span><\/div>\n<\/div>\n<p><span style=\"font-size: 14pt\">For example, in the<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-800-Frame\" style=\"font-size: 14pt\"><span class=\"MathJax_MathContainer\">l=1<\/span><\/span><span style=\"font-size: 14pt\">\u00a0<\/span><span style=\"font-size: 14pt\">electron state, the total energy of the electron is split into three distinct energy levels corresponding to<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-801-Frame\" style=\"font-size: 14pt\"><span class=\"MathJax_MathContainer\">U=\u2212\u03bcBB,0,\u03bcBB.<\/span><\/span><\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1170902741598\">The splitting of energy levels by an external magnetic field is called the<span>\u00a0<\/span><span data-type=\"term\" id=\"term341\">Zeeman effect<\/span>. Ignoring the effects of electron spin, transitions from the<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-803-Frame\"><span class=\"MathJax_MathContainer\"><span>l=1<\/span><\/span><\/span><span>\u00a0<\/span>state to a common lower energy state produce three closely spaced spectral lines (Figure 4.11, left column). Likewise, transitions from the<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-804-Frame\"><span class=\"MathJax_MathContainer\"><span>l=2<\/span><\/span><\/span><span>\u00a0<\/span>state produce five closely spaced spectral lines (right column). The separation of these lines is proportional to the strength of the external magnetic field. This effect has many applications. For example, the splitting of lines in the hydrogen spectrum of the Sun is used to determine the strength of the Sun\u2019s magnetic field. Many such magnetic field measurements can be used to make a map of the magnetic activity at the Sun\u2019s surface called a<span>\u00a0<\/span><span data-type=\"term\" id=\"term342\">magnetogram<\/span><span>\u00a0<\/span>(Figure 4.12).<\/p>\n<div class=\"os-figure\">\n<figure id=\"CNX_UPhysics_41_02_Zeeman\">\n<figure style=\"width: 490px\" class=\"wp-caption alignnone\"><img loading=\"lazy\" decoding=\"async\" alt=\"The figure shows the effect of magnetic field, B sub ext, on two different spectral lines, corresponding to the l=1 to l=0 transition on the left and the l=2 to l=0 transition on the right. The spectra are shown for no external field, for a non zero external field and for a large external field. With no external field, both transitions appear as single lines. In the second case, when magnetic field is applied, the spectral lines split into several lines; the line on the left splits into three lines. The line on the right splits into five. In the third case, the magnetic field is large. The left line is again split into three lines and the right into five, but the split lines are farther apart than they are when the external magnetic field is not as strong.\" data-media-type=\"image\/jpeg\" id=\"5110\" src=\"https:\/\/cnx.org\/resources\/d10c950d884c87b3c395dd6bb97b6c587800022a\" width=\"490\" height=\"402\" \/><figcaption class=\"wp-caption-text\">Figure 4.11 The Zeeman effect refers to the splitting of spectral lines by an external magnetic field. In the left column, the energy splitting occurs due to transitions from the state (n=2,l=1) to a lower energy state; and in the right column, energy splitting occurs due to transitions from the state (n=2,l=2) to a lower-energy state. The separation of these lines is proportional to the strength of the external magnetic field.<\/figcaption><\/figure>\n<p>&nbsp;<\/figure>\n<\/div>\n<div class=\"os-figure\">\n<figure id=\"CNX_UPhysics_41_02_Magnetogrm\">\n<figure style=\"width: 487px\" class=\"wp-caption alignnone\"><img loading=\"lazy\" decoding=\"async\" alt=\"A magnetogram of the sun, which appears as a gray disc against a black background, with white and black spots scattered on it. Most of the spots are concentrated in the center right part of the image.\" data-media-type=\"image\/jpeg\" id=\"66788\" src=\"https:\/\/cnx.org\/resources\/109d8548d50af1f2b6c7ceefb740978ef6093452\" width=\"487\" height=\"487\" \/><figcaption class=\"wp-caption-text\">Figure 4.12 A magnetogram of the Sun. The bright and dark spots show significant magnetic activity at the surface of the Sun. (credit: NASA, SDO)<\/figcaption><\/figure>\n<\/figure>\n<div>\n<p>&nbsp;<\/p>\n<div class=\"textbox\"><em>Download for free at http:\/\/cnx.org\/contents\/af275420-6050-4707-995c-57b9cc13c358@11.1<\/em><\/div>\n<\/div>\n<\/div>\n","protected":false},"author":615,"menu_order":3,"template":"","meta":{"pb_show_title":"on","pb_short_title":"Atomic Structure","pb_subtitle":"4.2 Orbital Magnetic Dipole Moment of the Electron","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-222","chapter","type-chapter","status-publish","hentry"],"part":213,"_links":{"self":[{"href":"https:\/\/pressbooks.bccampus.ca\/bcitphys8400\/wp-json\/pressbooks\/v2\/chapters\/222","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/pressbooks.bccampus.ca\/bcitphys8400\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/pressbooks.bccampus.ca\/bcitphys8400\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/bcitphys8400\/wp-json\/wp\/v2\/users\/615"}],"version-history":[{"count":5,"href":"https:\/\/pressbooks.bccampus.ca\/bcitphys8400\/wp-json\/pressbooks\/v2\/chapters\/222\/revisions"}],"predecessor-version":[{"id":450,"href":"https:\/\/pressbooks.bccampus.ca\/bcitphys8400\/wp-json\/pressbooks\/v2\/chapters\/222\/revisions\/450"}],"part":[{"href":"https:\/\/pressbooks.bccampus.ca\/bcitphys8400\/wp-json\/pressbooks\/v2\/parts\/213"}],"metadata":[{"href":"https:\/\/pressbooks.bccampus.ca\/bcitphys8400\/wp-json\/pressbooks\/v2\/chapters\/222\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/pressbooks.bccampus.ca\/bcitphys8400\/wp-json\/wp\/v2\/media?parent=222"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/bcitphys8400\/wp-json\/pressbooks\/v2\/chapter-type?post=222"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/bcitphys8400\/wp-json\/wp\/v2\/contributor?post=222"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/bcitphys8400\/wp-json\/wp\/v2\/license?post=222"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}