# Chapter 12 Maxwell’s Equations; Magnetism of Matter

**Learning Objectives:**

In this chapter you will basically learn:

\(\bullet\) The basic magnetic structure is a magnetic dipole.

\(\bullet\) Calculate the magnetic flux \(\Phi\) through a surface by integrating the dot product of the magnetic field vector \(\vec B\) and the area vector over the surface.

\(\bullet\) That the net magnetic flux through a Gaussian surface (which is a closed surface) is zero.

\(\bullet\) That a changing electric flux induces a magnetic field.

\(\bullet\) That in the Ampere–Maxwell law, the contribution to the induced magnetic field by the changing electric flux can be attributed to a fictitious current (“displacement current”).

\(\bullet\) Apply the Ampere–Maxwell law to calculate the magnetic field of a real current and a displacement current.

\(\bullet\) In Earth’s magnetic field, identify that the field is approximately that of a dipole and also identify in which hemisphere the north geomagnetic pole is located.

\(\bullet\) Identify that a spin angular momentum \(\vec S\) (usually simply called spin) and a spin magnetic dipole moment \(\vec\mu\) are intrinsic properties of electrons (and also protons and neutrons).

\(\bullet\) Apply the relationship between the spin vector and the spin magnetic dipole moment vector

\(\bullet\) The \(\vec S\) and \(\vec\mu\) cannot be observed (measured); only their components on an axis of measurement (usually called the z axis) can be observed.

\(\bullet\) The observed components \(S_z\) and \(\mu_{s,z}\) are quantized.

\(\bullet\) The component \(S_z\) and the spin magnetic quantum number \(m_s\), specifying the allowed values of \(m_s\).

\(\bullet\) Determine the z components \(\mu_{s,z}\) of the spin magnetic dipole moment, both as a value and in terms of the Bohr magneton \(\mu_B\).

\(\bullet\) If an electron is in an external magnetic field, determine the orientation energy U of its spin magnetic dipole moment \(\vec\mu_s\).

\(\bullet\) An electron in an atom has an orbital angular momentum \(\vec L_{orb}\) and an orbital magnetic dipole moment \(\vec \mu_{orb}\).

\(\bullet\) Identity that \(\vec L_{orb}\) and \(\vec \mu_{orb}\) cannot be observed but their components \(L_{orb,z}\) and \(\mu_{orb,z}\) on a z (measurement) axis can be done.

\(\bullet\) The relationship between the component \(L_{orb,z}\) of the orbital angular momentum and the orbital magnetic quantum number \(m_l\), specifying the allowed values of \(m_l\).

\(\bullet\) The z components \(\mu_{orb,z}\) of the orbital magnetic dipole moment, both as a value and in terms of the Bohr magneton \(\mu_B\).

\(\bullet\) If an atom is in an external magnetic field, determine the orientation energy U of the orbital magnetic dipole moment \(\mu_{orb}\).

\(\bullet\) Distinguish diamagnetism, paramagnetism, and ferromagnetism.

\(\bullet\) The relationship between a sample’s magnetization M, its measured magnetic moment, and its volume.

\(\bullet\) Apply Curie’s law to relate a sample’s magnetization M to its temperature T, its Curie constant C, and the magnitude B of the external field.

\(\bullet\) The ferromagnetism is due to a quantum mechanical interaction called exchange coupling.

\(\bullet\) Explain why ferromagnetism disappears when the temperature exceeds the material’s Curie temperature.

\(\bullet\) For a ferromagnetic sample at a given temperature and in a given magnetic field, compare the energy associated with the dipole orientations and the thermal motion.

\(\bullet\) Describe and sketch a Rowland ring.

\(\bullet\) Identify the origin of lodestones.

## 12.1 Gauss’ Law for Magnetic Fields

James Clerk Maxwell (1831–1879) was one of the major contributors to physics in the nineteenth century (Figure 13.1.1). Although he died young, he made major contributions to the development of the kinetic theory of gases, to the understanding of color vision, and to the nature of Saturn’s rings. He is probably best known for having combined existing knowledge of the laws of electricity and of magnetism with insights of his own into a complete overarching electromagnetic theory, represented by Maxwell’s equations.

Maxwell’s Correction to the Laws of Electricity and Magnetism The four basic laws of electricity and magnetism had been discovered experimentally through the work of physicists such as Oersted, Coulomb, Gauss, and Faraday. Maxwell discovered logical inconsistencies in these earlier results and identified the incompleteness of Ampère’s law as their cause.

Recall that according to Ampère’s law, the integral of the magnetic field around a closed loop C is proportional to the current I passing through any surface whose boundary is loop C itself:

\[\begin{equation} \oint_C\vec{\mathbf{B}}\cdot d\vec{\mathbf{s}}=\mu_0I. \tag{12.1} \end{equation}\]

There are infinitely many surfaces that can be attached to any loop, and Ampère’s law stated in Equation 13.1.1 is independent of the choice of surface.

Consider the set-up in Figure 13.1.2. A source of emf is abruptly connected across a parallel-plate capacitor so that a time-dependent current I develops in the wire. Suppose we apply Ampère’s law to loop C shown at a time before the capacitor is fully charged, so that I. Surface S_1 gives a nonzero value for the enclosed current I, whereas surface S_2 gives zero for the enclosed current because no current passes through it:

\[\oint_C\vec{\mathbf{B}}\cdot d\vec{\mathbf{s}}=\left\{\begin{array}{lc}\mu_0I&\mathrm{if~surface~}S_1~\mathrm{is~used}\\0&\mathrm{if~surface~}S_2~\mathrm{is~used}\end{array}\right..\]

Clearly, Ampère’s law in its usual form does not work here. This may not be surprising, because Ampère’s law as applied in earlier chapters required a steady current, whereas the current in this experiment is changing with time and is not steady at all.

(Figure 13.1.2) \[\begin{gather*}.\end{gather*}\]

Figure shows a wire connected to a plate of a parallel plate capacitor. A current I passes through it in the downward direction. The wire also passes through the flat surface of a cylinder at the top of the capacitor. This surface is labeled S1 and its circular boundary is labeled C. An arrow B is shown tangential to C. The sides of the cylinder taper downward and inward. This surface is labeled S2. Field lines labeled vector E are shown between two plates of the capacitor, pointing down. Figure 13.1.2 The currents through surface S_1 and surface S_2 are unequal, despite having the same boundary loop C. How can Ampère’s law be modified so that it works in all situations? Maxwell suggested including an additional contribution, called the displacement current I_}, to the real current I,

\[\begin{equation} \oint_C\vec{\mathbf{B}}\cdot d\vec{\mathbf{s}}=\mu_0(I+I_{\mathrm{d}}) \tag{12.2} \end{equation}\]

where the displacement current is defined to be

\[\begin{equation} I_{\mathrm{d}}=\epsilon_0\frac{d\Phi_{\mathrm{E}}}{dt}. \tag{12.1} \end{equation}\]

Here \(\epsilon_0\) is the permittivity of free space and \(\Phi_{\mathrm{E}}\) is the electric flux, defined as

\[\Phi_{\mathrm{E}}=\iint_{\mathrm{Surface}~S}\vec{\mathbf{E}}\cdot d\vec{\mathbf{A}}.\]

The displacement current is analogous to a real current in Ampère’s law, entering into Ampère’s law in the same way. It is produced, however, by a changing electric field. It accounts for a changing electric field producing a magnetic field, just as a real current does, but the displacement current can produce a magnetic field even where no real current is present. When this extra term is included, the modified Ampère’s law equation becomes

\[\begin{equation}\oint_C\vec{\mathbf{B}}\cdot d\vec{\mathbf{s}}=\mu_0I+\epsilon_0\mu_0\frac{d\Phi_{\mathrm{E}}}{dt} \tag{12.3} \end{equation}\]

and is independent of the surface S through which the current I is measured.

We can now examine this modified version of Ampère’s law to confirm that it holds independent of whether the surface S_1 or the surface S_2 in Figure 13.1.2 is chosen. The electric field \(\vec{\mathrm{E}}\) corresponding to the flux \(\Phi_{\mathrm{E}}\) in Equation 13.1.3 is between the capacitor plates. Therefore, the \(\vec{\mathbf{E}}\) field and the displacement current through the surface S_1 are both zero, and Equation 13.1.2 takes the form

\[\begin{equation} \oint_C\vec{\mathbf{B}}\cdot d\vec{\mathbf{s}}=\mu_0I. \tag{12.4} \end{equation}\]

We must now show that for surface S_2, through which no actual current flows, the displacement current leads to the same value \(\mu_0I\) for the right side of the Ampère’s law equation. For surface S_2, the equation becomes

\[\begin{equation}\oint_C\vec{\mathbf{B}}\cdot d\vec{\mathbf{s}}=\mu_0\frac{d}{dt}\left[\epsilon_0\iint_{\mathrm{Surface}~S_2}\vec{\mathbf{E}}\cdot d\vec{\mathbf{A}}\right]. \tag{12.5} \end{equation}\]

Gauss’s law for electric charge requires a closed surface and cannot ordinarily be applied to a surface like S_1 alone or S_2 alone. But the two surfaces S_1 and S_2 form a closed surface in Figure 13.1.2 and can be used in Gauss’s law. Because the electric field is zero on S_1, the flux contribution through S_1 is zero. This gives us

\[\begin{eqnarray} \oiint_{Surface~S_1+S_2}\vec{\mathbf{E}}\cdot d\vec{\mathbf{A}}&=&\iint_{\mathrm{Surface}~S_1}\vec{\mathbf{E}}\cdot d\vec{\mathbf{A}}+\iint_{\mathrm{Surface}~S_2}\vec{\mathbf{E}}\cdot d\vec{\mathbf{A}}\\&=&0+\iint_{\mathrm{Surface}~S_2}\vec{\mathbf{E}}\cdot d\vec{\mathbf{A}}\\&=&\iint_{\mathrm{Surface}~S_2}\vec{\mathbf{E}}\cdot d\vec{\mathbf{A}}. \tag{12.6} \end{eqnarray}\]

Therefore, we can replace the integral over S_2 in Equation 13.1.5 with the closed Gaussian surface S_1+S_2 and apply Gauss’s law to obtain

\[\begin{equation} \oint_{S_1}\vec{\mathbf{B}}\cdot d\vec{\mathbf{s}}=\mu_0\frac{dQ_{\mathrm{in}}}{dt}=\mu_0I. \tag{12.7} \end{equation}\]

Thus, the modified Ampère’s law equation is the same using surface S_2, where the right-hand side results from the displacement current, as it is for the surface S_1, where the contribution comes from the actual flow of electric charge.

EXAMPLE 13.1.1 Displacement current in a charging capacitor A parallel-plate capacitor with capacitance C whose plates have area A and separation distance d is connected to a resistor R and a battery of voltage V. The current starts to flow at t=0. (a) Find the displacement current between the capacitor plates at time t. (b) From the properties of the capacitor, find the corresponding real current I=, and compare the answer to the expected current in the wires of the corresponding RC circuit.

Strategy We can use the equations from the analysis of an RC circuit (Alternating-Current Circuits) plus Maxwell’s version of Ampère’s law.

Solution a. The voltage between the plates at time t is given by

\[V_C=\frac{1}{C}Q(t)=V_0\left(1-e^{-t/RC}\right).\]

Let the z-axis point from the positive plate to the negative plate. Then the z-component of the electric field between the plates as a function of time t is

\[E_z(t)=\frac{V_0}{d}\left(1-e^{-t/RC}\right).\]

Therefore, the z-component of the displacement current \(I_{\mathrm{d}}\) between the plates is

\[I_{\mathrm{d}}=\epsilon_0A\frac{\partial E_z(t)}{\partial t}=\epsilon_0A\frac{V_0}{d}\times\frac{1}{RC}e^{-t/RC}=\frac{V_0}{R}e^{-t/RC},\]

where we have used \(C=\frac{\epsilon_0{A}}{d}\) for the capacitance.

- From the expression for V_C, the charge on the capacitor is

\[Q(t)=CV_C=CV_0\left(1-e^{-t/RC}\right).\]

The current into the capacitor after the circuit is closed, is therefore

\[I=\frac{dQ}{dt}=\frac{V_0}{R}e^{-t/RC}.\]

This current is the same as \(I_{\mathrm{d}}\) found in (a).

Maxwell’s Equations With the correction for the displacement current, Maxwell’s equations take the form

\[\begin{equation}\oint\vec{\mathbf{E}}\cdot d\vec{\mathbf{A}}=\frac{Q_{\mathrm{in}}}{\epsilon_0}~~~~~~~~(\mathrm{Gauss{\text '}s~law}) \tag{12.8} \end{equation}\]

\[\begin{equation} \oint\vec{\mathbf{B}}\cdot d\vec{\mathbf{A}}=0~~~~~~~~(\mathrm{Gauss{\text '}s~law~for~magnetism}) \tag{12.9} \end{equation}\]

\[\begin{equation} \oint\vec{\mathbf{E}}\cdot d\vec{\mathbf{s}}=\frac{d\Phi_{\mathrm{m}}}{dt}~~~~~~~~(\mathrm{Faraday{\text '}s~law}) \tag{12.10} \end{equation}\]

\[\begin{equation} \oint\vec{\mathbf{B}}\cdot d\vec{\mathbf{s}}=\mu_0I+\epsilon_0\mu_0\frac{d\Phi_E}{dt}~~~~~~~~(\mathrm{Ampère{\text -}Maxwell~law}) \tag{12.11} \end{equation}\]

Once the fields have been calculated using these four equations, the Lorentz force equation

\[\begin{equation} \vec{\mathbf{F}}=q\vec{\mathbf{E}}+q\vec{\mathbf{v}}\times\vec{\mathbf{B}} \tag{12.12} \end{equation}\]

gives the force that the fields exert on a particle with charge q moving with velocity \(\vec{\mathbf{v}}\). The Lorentz force equation combines the force of the electric field and of the magnetic field on the moving charge. The magnetic and electric forces have been examined in earlier modules. These four Maxwell’s equations are, respectively,

MAXWELL’S EQUATIONS 1. Gauss’s law

The electric flux through any closed surface is equal to the electric charge \(Q_{\mathrm{in}}\) enclosed by the surface. Gauss’s law [Equation 13.1.7] describes the relation between an electric charge and the electric field it produces. This is often pictured in terms of electric field lines originating from positive charges and terminating on negative charges, and indicating the direction of the electric field at each point in space.

- Gauss’s law for magnetism

The magnetic field flux through any closed surface is zero [Equation 13.1.8]. This is equivalent to the statement that magnetic field lines are continuous, having no beginning or end. Any magnetic field line entering the region enclosed by the surface must also leave it. No magnetic monopoles, where magnetic field lines would terminate, are known to exist (see Magnetic Fields and Lines).

- Faraday’s law

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’s equations, Equation 13.1.9, is Faraday’s law of induction and includes Lenz’s law. The electric field from a changing magnetic field has field lines that form closed loops, without any beginning or end.

- Ampère-Maxwell law

Magnetic fields are generated by moving charges or by changing electric fields. This fourth of Maxwell’s equations, Equation 13.1.10, encompasses Ampère’s law and adds another source of magnetic fields, namely changing electric fields.

Maxwell’s equations and the Lorentz force law together encompass all the laws of electricity and magnetism. The symmetry that Maxwell introduced into his mathematical framework may not be immediately apparent. Faraday’s law describes how changing magnetic fields produce electric fields. The displacement current introduced by Maxwell results instead from a changing electric field and accounts for a changing electric field producing a magnetic field. The equations for the effects of both changing electric fields and changing magnetic fields differ in form only where the absence of magnetic monopoles leads to missing terms. This symmetry between the effects of changing magnetic and electric fields is essential in explaining the nature of electromagnetic waves.

Later application of Einstein’s theory of relativity to Maxwell’s complete and symmetric theory showed that electric and magnetic forces are not separate but are different manifestations of the same thing—the electromagnetic force. The electromagnetic force and weak nuclear force are similarly unified as the electroweak force. This unification of forces has been one motivation for attempts to unify all of the four basic forces in nature—the gravitational, electrical, strong, and weak nuclear forces.

The Mechanism of Electromagnetic Wave Propagation To see how the symmetry introduced by Maxwell accounts for the existence of combined electric and magnetic waves that propagate through space, imagine a time-varying magnetic field \(\vec{\mathbf{B}}_0(t)\) produced by the high-frequency alternating current seen in Figure 13.1.3. We represent \(\vec{\mathbf{B}}_0(t)\) in the diagram by one of its field lines. From Faraday’s law, the changing magnetic field through a surface induces a time-varying electric field \(\vec{\mathbf{E}}_0(t)\) at the boundary of that surface. The displacement current source for the electric field, like the Faraday’s law source for the magnetic field, produces only closed loops of field lines, because of the mathematical symmetry involved in the equations for the induced electric and induced magnetic fields. A field line representation of \(\vec{\mathbf{E}}_0(t)\) is shown. In turn, the changing electric field \(\vec{\mathbf{E}}_0(t)\) creates a magnetic field \(\vec{\mathbf{B}}_1(t)\) according to the modified Ampère’s law. This changing field induces _1(t), which induces \(\vec{\mathbf{B}}_2(t)\), and so on. We then have a self-continuing process that leads to the creation of time-varying electric and magnetic fields in regions farther and farther away from O. This process may be visualized as the propagation of an electromagnetic wave through space.

(Figure 13.1.3) \[\begin{gather*}.\end{gather*}\]

Figure shows a 3 dimensional diagram. A wire carrying an AC current is along the z axis. A circle labeled B0 goes around the wire. It lies in the xy plane. Another circle, labeled E0 goes through B0. E0 lies in the xz plane. Circle B1 goes through E0 and E1 goes through B1, and so on forming what looks like a chain. Circles B0, B1 and B2 are in the xy plane, with their centres along the x axis. These are interspersed with circles E0, E1 and E2 in the xz plane, whose centers lie on the y axis. Figure 13.1.3 How changing \(\vec{\mathbf{E}}\) and \(\vec{\mathbf{B}}\) fields propagate through space. In the next section, we show in more precise mathematical terms how Maxwell’s equations lead to the prediction of electromagnetic waves that can travel through space without a material medium, implying a speed of electromagnetic waves equal to the speed of light.

Prior to Maxwell’s work, experiments had already indicated that light was a wave phenomenon, although the nature of the waves was yet unknown. In 1801, Thomas Young (1773–1829) showed that when a light beam was separated by two narrow slits and then recombined, a pattern made up of bright and dark fringes was formed on a screen. Young explained this behavior by assuming that light was composed of waves that added constructively at some points and destructively at others. Subsequently, Jean Foucault (1819–1868), with measurements of the speed of light in various media, and Augustin Fresnel (1788–1827), with detailed experiments involving interference and diffraction of light, provided further conclusive evidence that light was a wave. So, light was known to be a wave, and Maxwell had predicted the existence of electromagnetic waves that traveled at the speed of light. The conclusion seemed inescapable: Light must be a form of electromagnetic radiation. But Maxwell’s theory showed that other wavelengths and frequencies than those of light were possible for electromagnetic waves. He showed that electromagnetic radiation with the same fundamental properties as visible light should exist at any frequency. It remained for others to test, and confirm, this prediction.

CHECK YOUR UNDERSTANDING 13.1 When the emf across a capacitor is turned on and the capacitor is allowed to charge, when does the magnetic field induced by the displacement current have the greatest magnitude?

Hertz’s Observations The German physicist Heinrich Hertz (1857–1894) 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.

Hertz used an alternating-current RLC (resistor-inductor-capacitor) circuit that resonates at a known frequency \(f_0=\frac{1}{2\pi\sqrt{LC}}\) and connected it to a loop of wire, as shown in Figure 13.1.4. High voltages induced across the gap in the loop produced sparks that were visible evidence of the current in the circuit and helped generate electromagnetic waves.

Across the laboratory, Hertz placed another loop attached to another RLC 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.

(Figure 13.1.4) \[\begin{gather*}.\end{gather*}\]

Figure shows a circuit on the left with R, L and C connected in series to an AC voltage source. This resonates at f subscript 0 equal to 1 upon 2 pi root LC. The inductor in this circuit forms the primary coil of a transformer. The secondary coil is connected to a loop labeled loop 1 transmitter. Within this loop are the words spark gap. Some distance to the right of this is another loop labeled loop 2 receiver. Within this loop are the words induced sparks. This is connected to a box labeled tuner. Figure 13.1.4 The apparatus used by Hertz in 1887 to generate and detect electromagnetic waves. Hertz also studied the reflection, refraction, and interference patterns of the electromagnetic waves he generated, confirming their wave character. He was able to determine the wavelengths from the interference patterns, and knowing their frequencies, he could calculate the propagation speed using the equation \(v=f\lambda\) where v is the speed of a wave, f is its frequency, and \(\lambda\) is its wavelength. Hertz was thus able to prove that electromagnetic waves travel at the speed of light. The SI unit for frequency, the hertz \((1~\mathrm{Hz}=1~\mathrm{cycle/second})\), is named in his honour.

## 12.2 Maxwell’s Equations; Magnetism of Matter:

## 12.3 Gauss’ Law for Magnetic Fields:

## 12.4 Induced Magnetic Fields:

## 12.5 Displacement Current:

## 12.6 Maxwell’s Equations:

## 12.7 Magnets:

## 12.8 Magnetism and Electrons:

## 12.9 Diamagnetism:

## 12.10 Paramagnetism:

## 12.11 Ferromagnetism:

## Solved Problems Maxwells Equations

**Section 12-1 Gauss’ Law for Magnetic Fields**

[1] The magnetic flux through each of five faces of a die (singular of “dice”) is given by \(\Phi_B=\pm NWb\), where N (1 to 5) is the number of spots on the face. The flux is positive (outward) for N even and negative (inward) for N odd.What is the flux through the sixth face of the die?

[8] Nonuniform electric flux. Figure 32-30 shows a circular region of radius R 3.00 cm in which an electric flux is directed out of the plane of the page. The flux encircled by a concentric circle of radius r is given by \(\Phi_{E,enc}\) = (0.600 V(m/s)(r/R)t, where r \(\le\) R and t is in seconds. What is the magnitude of the induced magnetic field at radial distances (a) 2.00 cm and (b) 5.00 cm?

[11] Suppose that a parallel-plate capacitor has circular plates with radius R ! 30 mm and a plate separation of 5.0 mm. Suppose also that a sinusoidal potential difference with a maximum value of 150 V and a frequency of 60 Hz is applied across the plates; that is, \[V = (150 V) \sin[2\pi(60 Hz)t]\].

- Find \(B_{max}(R)\), the maximum value of the induced magnetic field that occurs at r = R. (b) Plot \(B_{max}(r)\) for 0 \(\lt\) r \(\lt\) 10 cm.

[15] Prove that the displacement current in a parallel-plate capacitor of capacitance C can be written as \(i_d\) = C(dV/dt), where V is the potential difference between the plates.

[23] In Fig. 32-32, a parallel-plate capacitor has square plates of edge length L = 1.0 m. A current of 2.0 A charges the capacitor, producing a uniform electric field \(\vec E\) between the plates, with perpendicular to the plates. (a) What is the displacement current \(i_d\) through the region between the plates? (b) What is dE/dt in this region? (c) What is the displacement current encircled by the square dashed path of edge length d = 0.50 m? (d) What is the value of \(\oint \vec B.d\vec s\) around this square dashed path?

[29] In Fig. 32-36, a capacitor with circular plates of radius R = 18.0 cm is connected to a source of emf \(\varepsilon = \varepsilon_m \sin\omega t\), where \(\varepsilon_m\) = 220 V and v = 130 rad/s. The maximum value of the displacement current is \(i_d = 7.60 \mu A\). Neglect fringing of the electric field at the edges of the plates. (a) What is the maximum value of the current i in the circuit? (b) What is the maximum value of \(d\Phi_E/dt\), where \(\Phi_E\) is the electric flux through the region between the plates? (c) What is the separation d between the plates? (d) Find the maximum value of the magnitude of \(\vec B\) between the plates at a distance r = 11.0 cm from the center.

[31] In New Hampshire the average horizontal component of Earth’s magnetic field in 1912 was 16 \(\mu T\), and the average inclination or “dip” was \(73^\circ\). What was the corresponding magnitude of Earth’s magnetic field?

[33] If an electron in an atom has an orbital angular momentum with m = 0, what are the components (a) \(L_{orb,z}\) and (b) $_{orb,z}? If the atom is in an external magnetic field that has magnitude 35 mT and is directed along the z axis, what are (c) the energy \(U_{orb}\) associated with \(\vec\mu_{orb}\) and (d) the energy \(U_{spin}\) associated with \(\vec\mu_s\)? If, instead, the electron has m=-3, what are (e) \(L_{orb,z}\), (f) \(\mu_{orb,z}\), (g) \(U_{orb}\), and (h) \(U_{spin}\) ?

[35] What is the measured component of the orbital magnetic dipole moment of an electron with \(m_{\ell}\)=1 (a) and (b) \(m_{\ell}\)=-2?

[38] Assume that an electron of mass m and charge magnitude e moves in a circular orbit of radius r about a nucleus. A uniform magnetic field \(\vec B\) is then established perpendicular to the plane of the orbit. Assuming also that the radius of the orbit does not change and that the change in the speed of the electron due to field \(\vec B\) is small, find an expression for the change in the orbital magnetic dipole moment of the electron due to the field.

[40] A sample of the paramagnetic salt to which the magnetization curve of Fig. 32-14 applies is held at room temperature (300 K). At what applied magnetic field will the degree of magnetic saturation of the sample be (a) 50% and (b) 90%? (c) Are these fields attainable in the laboratory?

[42] A 0.50 T magnetic field is applied to a paramagnetic gas whose atoms have an intrinsic magnetic dipole moment of \(1.0 \times 10^{-23}\) J/T. At what temperature will the mean kinetic energy of translation of the atoms equal the energy required to reverse such a dipole end for end in this magnetic field?

[43] An electron with kinetic energy \(K_e\) travels in a circular path that is perpendicular to a uniform magnetic field, which is in the positive direction of a z axis. The electron’s motion is subject only to the force due to the field. (a) Show that the magnetic dipole moment of the electron due to its orbital motion has magnitude m ! Ke/B and that it is in the direction opposite that of .What are the

- magnitude and (c) direction of the magnetic dipole moment of a positive ion with kinetic energy Ki under the same circumstances?
- An ionized gas consists of \(5.3 \times 10^{21}\) electrons/\(m^3\) and the same number density of ions.Take the average electron kinetic energy to be \(6.2 \times 10^{-20}\) J and the average ion kinetic energy to be \(7.6 \times 10^{-21}\) J. Calculate the magnetization of the gas when it is in a magnetic field of 1.2 T.

[45] Consider a solid containing N atoms per unit volume, each atom having a magnetic dipole moment \(\vec\mu\). Suppose the direction of \(\vec\mu\) can be only parallel or antiparallel to an externally applied magnetic field \(\vec B\) (this will be the case if is due to the spin of a single electron). According to statistical mechanics, the probability of an atom being in a state with energy U is proportional to \(e^{-U/kT}\), where T is the temperature and k is Boltzmann’s constant. Thus, because energy U is \(-\mu\cdot\vec B\), the fraction of atoms whose dipole moment is parallel to \(\vec B\) is proportional to \(e^{\mu B/kT}\) and the fraction of atoms whose dipole moment is antiparallel to \(\vec B\) is proportional to \(e^{-\mu B/kT}\). (a) Show that the magnitude of the magnetization of this solid is \(M = N\mu \tanh(\mu B/kT)\). Here tanh is the hyperbolic tangent function: \(tanh(x) = (e^x - e^{-x})/(e^x + e^{-x})\). (b) Show that the result given in (a) reduces to \(M = N\mu^2 B/kT\) for \(\mu B \ll kT\). (c) Show that the result of (a) reduces to \(M = N\mu\) for \(\mu B \gg kT\). (d) Show that both (b) and (c) agree qualitatively with Fig. 32-14.

[49] The exchange coupling mentioned in Section 12-8 as being responsible for ferromagnetism is not the mutual magnetic interaction between two elementary magnetic dipoles. To show this, calculate (a) the magnitude of the magnetic field a distance of 10 nm away, along the dipole axis, from an atom with magnetic dipole moment \(1.5 \times 10^{-23}\) J/T (cobalt), and (b) the minimum energy required to turn a second identical dipole end for end in this field. (c) By comparing the latter with the mean translational kinetic energy of 0.040 eV, what can you conclude?

[51] The saturation magnetization \(M_{max}\) of the ferromagnetic metal nickel is \(4.70 \times 10^5\) A/m. Calculate the magnetic dipole moment of a single nickel atom. (The density of nickel is 8.90 \(g/cm^3\), and its molar mass is 58.71 g/mol.)

## Problems Maxwells Equations

**Section 12-1 Gauss’ Law for Magnetic Fields**

- [4] Two wires, parallel to a z axis and a distance 4r apart, carry equal currents i in opposite directions, as shown in Fig. 32-28. A circular cylinder of radius r and length L has its axis on the z axis, midway between the wires. Use Gauss’ law for magnetism to derive an expression for the net outward magnetic flux through the half of the cylindrical surface above the x axis. (Hint: Find the flux through the portion of the xz plane that lies within the cylinder.)

**Section 12-2 Induced Magnetic Fields**

[6] A capacitor with square plates of edge length L is being discharged by a current of 0.75 A. Figure 32-29 is a head-on view of one of the plates from inside the capacitor. A dashed rectangular path is shown. If L = 12 cm, W = 4.0 cm, and H = 2.0 cm, what is the value of \(\oint \vec B.d\vec s\) around the dashed path?

[10] Nonuniform electric field. In Fig. 32-30, an electric field is directed out of the page within a circular region of radius R = 3.00 cm.The field magnitude is E = (0.500 V/m.s)(1 - r/R)t, where t is in seconds and r is the radial distance (r \(\le\) R).What is the magnitude of the induced magnetic field at radial distances (a) 2.00 cm and (b) 5.00 cm?

[12] A parallel-plate capacitor with circular plates of radius 40 mm is being discharged by a current of 6.0 A. At what radius

- inside and (b) outside the capacitor gap is the magnitude of the induced magnetic field equal to 75% of its maximum value? (c)What is that maximum value?

**Section 12-3 Displacement Current**

[14] A parallel-plate capacitor with circular plates of radius R is being charged. Show that the magnitude of the current density of the displacement current is \(J_d = \varepsilon_0\)(dE/dt) for r \(\le\) R.

[18] The circuit in Fig. 32-31 consists of switch S, a 12.0 V ideal battery, a 20.0 \(M\Omega\) resistor, and an air-filled capacitor. The capacitor has parallel circular plates of radius 5.00 cm, separated by 3.00 mm. At time t = 0, switch S is closed to begin charging the capacitor. The electric field between the plates is uniform. At t = 250 \(\mu s\), what is the magnitude of the magnetic field within the capacitor, at radial distance 3.00 cm?

[28] Figure 32-35a shows the current i that is produced in a wire of resistivity \(1.62 \times 10^{-8} \Omega.m\). The magnitude of the current versus time t is shown in Fig. 32-35b. The vertical axis scale is set by \(i_s\) = 10.0 A, and the horizontal axis scale is set by \(t_s\) = 50.0 ms. Point P is at radial distance 9.00 mm from the wire’s center. Determine the magnitude of the magnetic field \(\vec B_i\) at point P due to the actual current i in the wire at (a) t = 20 ms, (b) t = 40 ms, and (c) t = 60 ms. Next, assume that the electric field driving the current is confined to the wire. Then determine the magnitude of the magnetic field \(\vec B_{id}\) at point P due to the displacement current \(i_d\) in the wire at (d) t = 20 ms, (e) t = 40 ms, and (f ) t = 60 ms. At point P at t = 20 s, what is the direction (into or out of the page) of (g) \(\vec B_i\) and (h) \(\vec B_{id}\)?

**Section 12-4 Magnets**

- [30] Assume the average value of the vertical component of Earth’s magnetic field is 43 \(\mu T\) (downward) for all of Arizona, which has an area of \(2.95 \times 10^5 km^2\).What then are the (a) magnitude and (bdirection (inward or outward) of the net magnetic flux through the rest of Earth’s surface (the entire surface excluding Arizona)?

**Section 12-5 Magnetism and Electrons**

[32] Figure 32-37a is a one-axis graph along which two of the allowed energy values (levels) of an atom are plotted. When the atom is placed in a magnetic field of 0.500 T, the graph changes to that of Fig. 32-37b because of the energy associated with \(\vec\mu_{orb}\cdot\vec B\). (We neglect \(\vec\mu_s\).) Level \(E_1\) is unchanged, but level \(E_2\) splits into a (closely spaced) triplet of levels. What are the allowed values of associated with (a) energy level \(E_1\) and (b) energy level \(E_2\)? (c) In joules, what amount of energy is represented by the spacing between the triplet levels?

[34] What is the energy difference between parallel and antiparallel alignment of the z component of an electron’s spin magnetic dipole moment with an external magnetic field of magnitude 0.25 T, directed parallel to the z axis?

**Section 12-6 Diamagnetism**

- [37] Figure 32-38 shows a loop model (loop L) for a diamagnetic material. (a) Sketch the magnetic field lines within and about the material due to the bar magnet. What is the direction of (b) the loop’s net magnetic dipole moment \(\vec\mu_s\), (c) the conventional current i in the loop (clockwise or counterclockwise in the figure), and (d) the magnetic force on the loop?

**Section 12-7 Paramagnetism**

[39] A sample of the paramagnetic salt to which the magnetization curve of Fig. 32-14 applies is to be tested to see whether it obeys Curie’s law.The sample is placed in a uniform 0.50 T magnetic field that remains constant throughout the experiment. The magnetization M is then measured at temperatures ranging from 10 to 300 K. Will it be found that Curie’s law is valid under these conditions?

[41] A magnet in the form of a cylindrical rod has a length of 5.00 cm and a diameter of 1.00 cm. It has a uniform magnetization of 5.30 & 103 A/m.What is its magnetic dipole moment?

[44] Figure 32-39 gives the magnetization curve for a paramagnetic material. The vertical axis scale is set by a ! 0.15, and the horizontal axis scale is set by b ! 0.2 T/K. Let msam be the measured net magnetic moment of a sample of the material and mmax be the maximum possible net magnetic moment of that sample. According to Curie’s law, what would be the ratio msam/mmax were the sample placed in a uniform magnetic field of magnitude 0.800 T, at a temperature of 2.00 K?

**Section 12-8 Ferromagnetism**

[46] You place a magnetic compass on a horizontal surface, allow the needle to settle, and then give the compass a gentle wiggle to cause the needle to oscillate about its equilibrium position. The oscillation frequency is 0.312 Hz. Earth’s magnetic field at the location of the compass has a horizontal component of 18.0 \(\mu T\). The needle has a magnetic moment of 0.680 mJ/T. What is the needle’s rotational inertia about its (vertical) axis of rotation?

[47] The magnitude of the magnetic dipole moment of Earth is \(8.0 \times 10^{22}\) J/T. (a) If the origin of this magnetism were a magnetized iron sphere at the center of Earth, what would be its radius? (b) What fraction of the volume of Earth would such a sphere occupy? Assume complete alignment of the dipoles.The density of Earth’s inner core is 14 \(g/cm^3\). The magnetic dipole moment of an iron atom is \(2.1 \times 10^{-23}\) J/T. (Note: Earth’s inner core is in fact thought to be in both liquid and solid forms and partly iron, but a permanent magnet as the source of Earth’s magnetism has been ruled out by several considerations. For one, the temperature is certainly above the Curie point.)

[48] The magnitude of the dipole moment associated with an atom of iron in an iron bar is \(2.1 \times 10^{-23}\) J/T. Assume that all the atoms in the bar, which is 5.0 cm long and has a cross-sectional area of 1.0 \(cm^2\), have their dipole moments aligned. (a) What is the dipole moment of the bar? (b) What torque must be exerted to hold this magnet perpendicular to an external field of magnitude 1.5 T? (The density of iron is 7.9 g/\(cm^3\).)

[50] A magnetic rod with length 6.00 cm, radius 3.00 mm, and (uniform) magnetization \(2.70 \times 10^3\) A/m can turn about its center like a compass needle. It is placed in a uniform magnetic field \(\vec B\) of magnitude 35.0 mT, such that the directions of its dipole moment and make an angle of \(\68.0^\circ\). (a) What is the magnitude of the torque on the rod due to \(\vec B\)? (b) What is the change in the orientation energy of the rod if the angle changes to \(34^\circ\)?