Maxwell's Equations; Magnetism of Matter
This chapter shows off the breadth of physics, ranging from the basic science of electric and magnetic fields to the applied science of magnetic materials. First we close the loop on the last eleven chapters: nearly all of electromagnetism collapses into just four equations — Maxwell's equations. Along the way a beautiful symmetry surfaces — a changing electric field induces a magnetic field, the mirror image of Faraday's law — and with it the idea of the displacement current. Then we turn to the matter itself, asking why iron is magnetic and you are not, and uncover the answer in the spin and orbital motion of electrons, which sorts all materials into three families: diamagnetic, paramagnetic, and ferromagnetic.
- Gauss' law for magnetic fields \(\oint \vec{B}\cdot d\vec{A} = 0\): the net magnetic flux through any closed surface is zero, because magnetic monopoles do not exist — the simplest magnetic structure is a dipole.
- Induced magnetic fields: a changing electric flux induces a magnetic field, Maxwell's law of induction \(\oint \vec{B}\cdot d\vec{s} = \mu_0\varepsilon_0\dfrac{d\Phi_E}{dt}\), combined with Ampère's law into the Ampère–Maxwell law \(\oint \vec{B}\cdot d\vec{s} = \mu_0\varepsilon_0\dfrac{d\Phi_E}{dt} + \mu_0 i_{\text{enc}}\).
- The displacement current \(i_d = \varepsilon_0\dfrac{d\Phi_E}{dt}\), equal to the real current charging a capacitor, and the four Maxwell's equations that summarize all of electromagnetism.
- Magnetism and electrons: the spin moment \(\vec{\mu}_s = -\dfrac{e}{m}\vec{S}\) and orbital moment \(\vec{\mu}_{\text{orb}} = -\dfrac{e}{2m}\vec{L}_{\text{orb}}\), the Bohr magneton \(\mu_B = \dfrac{eh}{4\pi m} = 9.27\times10^{-24}\,\mathrm{J/T}\), and the orientation energy \(U = -\vec{\mu}\cdot\vec{B}_{\text{ext}}\).
- The three kinds of magnetic material — diamagnetism, paramagnetism with magnetization \(M = C\dfrac{B_{\text{ext}}}{T}\) (Curie's law), and ferromagnetism with exchange coupling, magnetic domains, and hysteresis.
What Is Physics?
This chapter ranges from the deepest principles to the most practical engineering. First we conclude our basic discussion of electric and magnetic fields, finding that most of the physics of the last eleven chapters can be summarized in only four equations, known as Maxwell's equations.
Second, we examine the science and engineering of magnetic materials. Many careers are devoted to understanding why some materials are magnetic and others are not, and to improving the magnets we already have. These researchers wonder why Earth has a magnetic field but you do not, and they find countless applications in cars, kitchens, offices, and hospitals. Magnetic materials also turn up in unexpected places — the inks of some tattoos contain magnetic particles, and some breakfast cereals are "iron fortified" with bits of iron you could collect with a magnet. Our first step is to revisit Gauss' law, but this time for magnetic fields.
Gauss' Law for Magnetic Fields
Sprinkle iron powder onto a sheet above a bar magnet and the grains line up to reveal the field. One end is a source, from which field lines diverge; the other is a sink, into which they converge. By convention the source is the north pole and the sink the south pole, and the two-poled magnet is a magnetic dipole. Try to isolate a single pole by breaking the magnet — and you fail. Each fragment, down to the atom, has both a north and a south pole. The simplest magnetic structure that can exist is the dipole; isolated magnetic monopoles do not exist, as far as we know.
The law holds for structures more complicated than a dipole, and even when the Gaussian surface does not enclose an entire magnet. A surface that seems to enclose only a north pole actually has a south pole associated with its boundary, where field lines enter — so the net flux is still zero. Every magnet is, at heart, a collection of dipoles.
Induced Magnetic Fields
In Chapter 30, Faraday's law told us that a changing magnetic flux induces an electric field, \(\oint \vec{E}\cdot d\vec{s} = -d\Phi_B/dt\). Because symmetry runs so deep in physics, we should ask the mirror question: can a changing electric flux induce a magnetic field? It can — and the governing equation, Maxwell's law of induction, is almost the twin of Faraday's law.
Picture a parallel-plate capacitor with circular plates of radius \(R\) being charged by a steady current. The field between the plates grows, the electric flux through a concentric loop changes, and a magnetic field is induced around that loop — both inside and outside the gap. Choosing a circular Amperian loop and exploiting the symmetry gives a tidy two-part result.
The left side of Maxwell's law is the same integral that appears in Ampère's law, \(\oint \vec{B}\cdot d\vec{s} = \mu_0 i_{\text{enc}}\). Since a magnetic field can be produced by a real current and by a changing electric flux, we merge the two into a single statement.
Displacement Current
Look at the first term on the right of the Ampère–Maxwell law: the product \(\varepsilon_0\,(d\Phi_E/dt)\) carries the dimensions of a current. Maxwell treated it as a fictitious current, the displacement current \(i_d\). ("Displacement" is a poor name — nothing is displaced — but we are stuck with it.)
This picture pays off: to find the induced field, just treat the gap as an imaginary wire of radius \(R\) carrying the current \(i_d\), and apply the right-hand rule and Chapter 29's solenoid/wire results directly.
Maxwell's Equations
The Ampère–Maxwell law completes the set of four fundamental equations of electromagnetism — Maxwell's equations. These four explain a remarkable range of phenomena, from why a compass points north to why a car starts when you turn the key, and they underpin electric motors, radio and television, radar, and microwave ovens. Nearly every equation since Chapter 21 can be derived from them, as can the optics of the chapters to come.
\(\oint \vec{E}\cdot d\vec{A} = \dfrac{q_{\text{enc}}}{\varepsilon_0}\) — relates net electric flux to net enclosed charge.
\(\oint \vec{B}\cdot d\vec{A} = 0\) — net magnetic flux is zero; no monopoles.
\(\oint \vec{E}\cdot d\vec{s} = -\dfrac{d\Phi_B}{dt}\) — a changing magnetic flux induces \(\vec{E}\).
\(\oint \vec{B}\cdot d\vec{s} = \mu_0\varepsilon_0\dfrac{d\Phi_E}{dt} + \mu_0 i_{\text{enc}}\) — current and changing \(\Phi_E\) induce \(\vec{B}\).
Magnets and the Magnetism of Earth
The first known magnets were lodestones, naturally magnetized rocks the ancients used in compasses. The largest magnet we deal with daily is Earth itself: near the surface its field is well approximated by that of a giant bar magnet — a magnetic dipole — straddling the center of the planet, with magnetic dipole moment of magnitude about \(8.0\times10^{22}\,\mathrm{J/T}\) tilted \(11.5^\circ\) from the rotation axis.
A subtle point of nomenclature: because field lines emerge from a dipole's north pole, and Earth's lines emerge in the Southern Hemisphere, the pole we call the "north magnetic pole" is really the south pole of Earth's dipole. The field's direction at any spot is given by two angles: the declination (the horizontal angle east or west of geographic north) and the inclination or "dip" (the angle above or below horizontal).
Local readings can differ markedly from the idealized dipole; between 1580 and 1820 the compass direction in London swung by \(35^\circ\). The frozen magnetism of magma on either side of the Mid-Atlantic Ridge records that Earth's field has reversed its polarity roughly every million years — for reasons still not understood.
Magnetism and Electrons
Magnetic materials are magnetic because of their electrons. Beyond moving as a current, an electron carries two intrinsic sources of magnetism, each a magnetic dipole moment: one tied to its spin, one to its orbital motion in an atom. Both rest on quantum physics, which we only outline here.
Magnetic Materials
Each electron's spin and orbital moments combine vectorially; those add across the atom, and atom by atom across a sample. Whether the grand total produces a field — and how the sample responds to an external field — sorts all matter into three families.
Present in all materials but very weak. An external field induces opposing dipole moments; they vanish when the field is removed.
Atoms carry permanent moments, randomly oriented. A field partially aligns them with itself; alignment is lost when the field is removed.
Iron, nickel, and kin: moments lock into aligned regions, producing strong magnetism that partly persists after the field is removed.
Diamagnetism
In a diamagnetic atom the electron orbits cancel in pairs, leaving no net moment until a field is applied. Turn on a field and, by Faraday's and Lenz's laws, the changing flux speeds up the orbit going one way and slows the one going the other — leaving the atom with a small net moment opposite the applied field. Diamagnetism is the magnetic analogue of induced polarization.
The effect is feeble but real. In a strong enough field gradient even a living frog — diamagnetic, like all animals — can be levitated, every atom gently repelled upward until the magnetic force balances gravity. There is no discomfort, because every atom feels the same force.
Paramagnetism
In a paramagnetic material the spin and orbital moments do not cancel, so each atom has a permanent moment \(\vec{\mu}\). Left alone these point every which way and average to nothing. An external field nudges them into partial alignment with the field — the opposite of diamagnetism — giving the sample a net moment that is pulled toward stronger field. We quantify the degree of alignment with the magnetization.
Ferromagnetism
This is the strong, permanent magnetism of everyday life. In iron, cobalt, nickel, and certain alloys, a purely quantum effect called exchange coupling locks neighboring electron spins into alignment, overpowering thermal agitation. Heat the material past its Curie temperature (\(1043\,\mathrm{K}\) for iron) and the coupling fails — the material becomes merely paramagnetic.
Measured in a Rowland ring (a toroidal sample wound with a coil), the total field is the sum of the coil's field and the iron's contribution.
Within each domain the dipoles are essentially perfectly aligned, but the domains point every way, so a fresh crystal shows little net field. An applied field grows the aligned domains and rotates others toward it. Because the domain walls do not retrace their steps, the \(B_M\)–\(B_0\) curve forms a hysteresis loop: the iron stays magnetized even after the field is removed. This "memory" is permanent magnetism — and the basis of magnetic data storage.
Putting It to Work
Problem. A parallel-plate capacitor with circular plates of radius \(R\) is being charged so that \(dE/dt = 1.50\times10^{12}\,\mathrm{V/m\cdot s}\). Find the induced magnetic field at \(r = R/5 = 11.0\,\mathrm{mm}\) (inside the gap).
Solution. Inside the gap the Ampère–Maxwell law (no real current) gives \(B = \tfrac{1}{2}\mu_0\varepsilon_0 r\,(dE/dt)\).
About \(0.09\,\mu\mathrm{T}\) — so small it can scarcely be measured with simple apparatus, in sharp contrast to easily measured induced electric fields. Note that inside the gap \(B\) grows linearly with \(r\), peaking at the plate edge \(r = R\).
Problem. A circular parallel-plate capacitor of plate radius \(R\) is charged by a current \(i\). Between the plates, what is \(\oint \vec{B}\cdot d\vec{s}\) around a loop of radius \(r = R/5\), in terms of \(\mu_0\) and \(i\)?
Solution. With no real current in the gap, \(\oint \vec{B}\cdot d\vec{s} = \mu_0 i_{d,\text{enc}}\). Taking \(i_d\) uniform over the plate area, the loop encircles the fraction \(\pi r^{2}/\pi R^{2}\).
Since the displacement current equals the real current, \(i_d = i\), the answer is simply \(\mu_0 i/25\) — and the field there is one-fifth of its maximum value, as expected from the linear growth of \(B\) with \(r\).
Problem. At a New Hampshire site the horizontal component of Earth's field is \(B_h = 16\,\mu\mathrm{T}\) and the inclination (dip) is \(73^\circ\). Find the magnitude of the total field.
Solution. The horizontal component is the projection of the field onto the horizontal: \(B_h = B\cos\phi_i\).
The steep \(73^\circ\) dip — typical of mid-northern latitudes — means most of the field points into the ground, so the full field is over three times the horizontal piece a simple compass detects.
Problem. A paramagnetic gas at \(T = 300\,\mathrm{K}\) sits in a field \(B = 1.5\,\mathrm{T}\); each atom has \(\mu = 1.0\,\mu_B\). Compare the mean translational kinetic energy \(K\) with the energy difference \(\Delta U_B\) between aligned and anti-aligned orientations.
Solution. Use \(K = \tfrac{3}{2}kT\) and \(\Delta U_B = 2\mu B\).
Here \(K\) is about 230 times \(\Delta U_B\). Collisions easily knock aligned dipoles loose, so the net magnetization is only a fleeting partial alignment — exactly why paramagnetism is weak and Curie's law has temperature in the denominator.
Problem. A pure-iron needle (density \(7900\,\mathrm{kg/m^3}\)) measures \(3.0\,\mathrm{cm}\times1.0\,\mathrm{mm}\times0.50\,\mathrm{mm}\). Each iron atom has \(\mu_{\text{Fe}} = 2.1\times10^{-23}\,\mathrm{J/T}\), and \(10\%\) of the atoms are aligned. Find the needle's dipole moment (molar mass \(M = 0.0558\,\mathrm{kg/mol}\)).
Solution. Volume \(= 1.5\times10^{-8}\,\mathrm{m^3}\), so mass \(m = (7900)(1.5\times10^{-8}) = 1.185\times10^{-4}\,\mathrm{kg}\). The atom count is \(N = mN_A/M\).
Only one atom in ten contributes, yet a trillion-trillion aligned atoms still add up to a measurable moment — the reason a humble iron needle makes a working compass.
Chapter Summary
\(\oint \vec{B}\cdot d\vec{A} = 0\): no net magnetic flux, no monopoles; simplest structure is a dipole.
\(\oint \vec{B}\cdot d\vec{s} = \mu_0\varepsilon_0\dfrac{d\Phi_E}{dt}\): changing \(\Phi_E\) induces \(\vec{B}\).
\(\oint \vec{B}\cdot d\vec{s} = \mu_0\varepsilon_0\dfrac{d\Phi_E}{dt} + \mu_0 i_{\text{enc}}\); displacement current \(i_d = \varepsilon_0\dfrac{d\Phi_E}{dt} = i\).
Four laws — two Gauss', Faraday, Ampère–Maxwell — summarize all of electromagnetism and optics.
A tilted dipole (\(11.5^\circ\), \(8.0\times10^{22}\,\mathrm{J/T}\)); described by declination and inclination; reverses over ages.
\(\vec{\mu}_s = -\dfrac{e}{m}\vec{S}\), \(\vec{\mu}_{\text{orb}} = -\dfrac{e}{2m}\vec{L}_{\text{orb}}\); \(\mu_B = \dfrac{eh}{4\pi m}\); \(U = -\vec{\mu}\cdot\vec{B}_{\text{ext}}\).
Diamagnetic: induced moment opposite \(\vec{B}_{\text{ext}}\), repelled. Paramagnetic: aligns with field, \(M = C\dfrac{B_{\text{ext}}}{T}\).
Exchange coupling, domains, and hysteresis; field \(B = B_0 + B_M\); permanent above-zero magnetism below the Curie temperature.
Problems
Take \(\mu_0 = 4\pi\times10^{-7}\,\mathrm{T\cdot m/A}\), \(\varepsilon_0 = 8.85\times10^{-12}\,\mathrm{C^2/N\cdot m^2}\), \(\mu_B = 9.27\times10^{-24}\,\mathrm{J/T}\), and \(k = 1.38\times10^{-23}\,\mathrm{J/K}\). For Gauss'-law problems, remember the net flux through any closed surface is zero. For induced-field problems, decide first whether you are inside (\(r \le R\)) or outside (\(r \ge R\)) the gap.
- The magnetic flux through five faces of a die is \(\Phi_B = \pm N\,\mathrm{Wb}\), where \(N\) (1 to 5) is the number of spots, positive (outward) for even \(N\) and negative for odd. What is the flux through the sixth face?
- A closed surface has a flat top face (radius 2.0 cm) through which a perpendicular 0.30 T field is directed outward, and a flat bottom face through which 0.70 mWb is directed outward. Find the magnitude and direction of the flux through the curved part.
- The induced magnetic field at radial distance 6.0 mm from the axis of a circular parallel-plate capacitor is \(2.0\times10^{-7}\,\mathrm{T}\). The plate radius is 3.0 mm. At what rate is the electric field between the plates changing?
- A parallel-plate capacitor with circular plates of radius \(R = 30\,\mathrm{mm}\) and separation 5.0 mm has \(V = (150\,\mathrm{V})\sin[2\pi(60\,\mathrm{Hz})t]\) applied. Find the maximum induced magnetic field, which occurs at \(r = R\).
- A parallel-plate capacitor with circular plates of radius 40 mm is discharged by a 6.0 A current. At what radius (a) inside and (b) outside the gap is the induced magnetic field 75% of its maximum, and (c) what is that maximum?
- At what rate must the potential difference across a \(2.0\,\mu\mathrm{F}\) capacitor change to produce a displacement current of 1.5 A?
- Prove that the displacement current in a parallel-plate capacitor of capacitance \(C\) can be written \(i_d = C\,(dV/dt)\).
- In New Hampshire (1912) the average horizontal component of Earth's field was \(16\,\mu\mathrm{T}\) and the dip was \(73^\circ\). Find the magnitude of Earth's field there.
- What is the energy difference between parallel and antiparallel alignment of the \(z\)-component of an electron's spin magnetic moment with a 0.25 T field directed along \(z\)?
- An electron in an atom has orbital quantum number \(m_\ell = 3\). In a 35 mT field along \(z\), find (a) \(L_{\text{orb},z}\), (b) \(\mu_{\text{orb},z}\), and (c) the orientation energy \(U_{\text{orb}}\).
- A cylindrical magnet rod of length 5.00 cm and diameter 1.00 cm has uniform magnetization \(5.30\times10^{3}\,\mathrm{A/m}\). Find its magnetic dipole moment.
- A 0.50 T field is applied to a paramagnetic gas whose atoms have moment \(1.0\times10^{-23}\,\mathrm{J/T}\). At what temperature does the mean translational kinetic energy equal the energy to reverse such a dipole end-for-end?
- The saturation magnetization of nickel is \(4.70\times10^{5}\,\mathrm{A/m}\). Compute the magnetic dipole moment of a single nickel atom (density \(8.90\,\mathrm{g/cm^3}\), molar mass \(58.71\,\mathrm{g/mol}\)).
- A Rowland ring of ferromagnetic material has inner radius 5.0 cm, outer radius 6.0 cm, and 400 turns. (a) What current gives a toroidal field \(B_0 = 0.20\,\mathrm{mT}\)? (b) If \(B_M = 800\,B_0\), find the total field \(B\) inside.