Electromagnetism

Abstract

In this chapter, the way from experimental observations to Maxwell’s equations is laid out. Special attention is put at the behavior of the various fields at interfaces. Aberration and the Doppler effect is discussed in connection with the covariance of the Maxwell equations. Included is also a section on the relativistic Lagrangian mechanics of massless and of massive particles, the latter also under the influence of external forces, not always found in other textbooks. Another section deals with the classical Hamiltonian formalism for fields; this is also basic to the theory of condensed matter, but here is paving the grounds for quantum-mechanical field theories. Applications are presented for accelerated point charges, oscillating dipoles, brems and synchrotron radiation. There is a list of 41 problems.

Appendices

Problems

Problem 3.1

Reformulate $\mathbf {\nabla }({\mathbf {a}}\cdot {\mathbf {b}})$ and $\mathbf {\nabla }\times ({\mathbf {a}}\times {\mathbf {b}})$ such that the operator $\mathbf {\nabla }$ has only one vector to the right of it (on which it acts). Here the intermediate steps should be taken without components and the differential operator should treat both ${\mathbf {a}}_{\text {c}}$ and ${\mathbf {b}}_{\text {c}}$ as constant, so that the product rule reads ${\mathbf {\nabla }}({\mathbf {a}}\cdot {\mathbf {b}})={\mathbf {\nabla }}({\mathbf {a}}\cdot \,{\mathbf {b}}_{\text {c}})+ {\mathbf {\nabla }}({\mathbf {a}}_{\text {c}}\cdot \,{\mathbf {b}})$, or again $\mathbf {\nabla }\times ({\mathbf {a}}\times {\mathbf {b}})= {\mathbf {\nabla }}\times ({\mathbf {a}}\times {\mathbf {b}}_{\text {c}})+\cdots$. The equations ${\mathbf {a}}\times ({\mathbf {b}}\times {\mathbf {c}})= {\mathbf {b}}\;({\mathbf {c}}\cdot {\mathbf {a}})-{\mathbf {c}}\;({\mathbf {a}}\cdot {\mathbf {b}})= ({\mathbf {c}}\cdot {\mathbf {a}})\,{\mathbf {b}}-({\mathbf {a}}\cdot {\mathbf {b}})\,{\mathbf {c}}$ need not be proven. (4 P)

Problem 3.2

Using Cartesian components, determine ${\mathbf {\nabla }}\cdot {\mathbf {r}}$, ${\mathbf {\nabla }}\times {\mathbf {r}}$, and $({\mathbf {a}}\cdot {\mathbf {\nabla }})\,{\mathbf {r}}$. These results will be useful for the following problems. (3 P)

Problem 3.3

Consider an arbitrary (three-times differentiable) scalar function $\psi ({\mathbf {r}})$ and the three vector fields ${\mathbf {\nabla }}\psi$, ${\mathbf {r}}\times {\mathbf {\nabla }}\psi$, and ${\mathbf {\nabla }}\times ({\mathbf {r}}\times {\mathbf {\nabla }}\psi )$. Which of them are source-free and which curl-free? Determine the source and curl strengths as functions of $\psi$. What is their inversion behavior (parity) if $\psi (-{\mathbf {r}}) =\psi ({\mathbf {r}})$? (9 P)

Problem 3.4

Prove $\int _{(V)}(\mathrm {d}{\mathbf {f}}\cdot {\mathbf {a}})\,{\mathbf {b}}=\int _V\mathrm {d}V \;\{{\mathbf {b}}\; ({\mathbf {\nabla }}\cdot {\mathbf {a}})+({\mathbf {a}}\cdot {\mathbf {\nabla }})\,{\mathbf {b}}\}$ for arbitrary fields ${\mathbf {a}}({\mathbf {r}})$ and ${\mathbf {b}}({\mathbf {r}})$ and show that the volume integral of a source-free vector field ${\mathbf {a}}$ is always zero, if ${\mathbf {a}}$ vanishes on the surface (V). (4 P)

Problem 3.5

For which function $\psi (r)$ does the (spatial) central field ${\mathbf {a}} ({\mathbf {r}})=\psi (r)\,{\mathbf {r}}$ have sources only at the origin? Does it have curls? Investigate this also for a plane central field. Represent the solutions as gradient fields (gradients of scalar fields). (3 P)

Problem 3.6

Let $(\Delta +k^2)\,\psi ({\mathbf {r}})=0$. How can we prove that the three vector fields from Problem 3.3 satisfy the equation $(\Delta +k^2)\,{\mathbf {a}}({\mathbf {r}})={\mathbf {0}}$? Note the sources and curls of the vector fields. (4 P)

Problem 3.7

Determine the vector fields ${\mathbf {\nabla }}({\mathbf {p}}\cdot {\mathbf {r}}/r^3)$ and ${\mathbf {\nabla }}\times ({\mathbf {r}}\times {\mathbf {p}}/r^3)$ for constant ${\mathbf {p}}\,$ (dipole moment) when $r\ne 0$, and compare them. (5 P)

Problem 3.8

Derive the singular behavior of the two vector fields for $r=0$ from the volume integral of a sphere around the origin. Express the results in terms of the delta function. (8 P)

Problem 3.9

Prove the representation of the Fourier transform of $f(x)=g(x)\;h(x)$ as a convolution integral given on p. 22. (4 P)

Problem 3.10

For fixed $\alpha$, $\beta$, $\gamma$ (with $\alpha >0$, $\beta >0$, and $0<\gamma <\pi$), a rectilinear oblique coordinate system $x^1$, $x^2$ is given by the two equations $x^1=\alpha \;(x-y\cot \gamma )$ and $x^2=\beta \,y$. Which functions y(x) describe the coordinate lines $\{x^1,x^2\}$? At what angle do the coordinate lines cross? How do the basic vectors ${\mathbf {g}}_i=$ and ${{\mathbf {g}}}{\,}^i$ read as linear combinations of the Cartesian unit vectors? How do the fundamental tensors $g_{ik}$ and $g^{ik}$ read? (7 P)

Problem 3.11

For spherical coordinates ($r,\theta ,\varphi$), we have to introduce position-dependent unit vectors ${\mathbf {e}}_r$, ${\mathbf {e}}_\theta$, and ${\mathbf {e}}_\varphi$ in the direction of increasing coordinates. Decompose these three vectors in terms of ${\mathbf {e}}_x$, ${\mathbf {e}}_y$, and ${\mathbf {e}}_z$. Determine their partial derivatives with respect to $r,\theta ,\varphi$ and express them as multiples of the unit vectors ${\mathbf {e}}_r$, ${\mathbf {e}}_\theta$, ${\mathbf {e}}_\varphi$. (7 P)

Problem 3.12

Determine the covariant and contravariant base vectors $\{{\mathbf {g}}_i\}$ and $\{{\mathbf {g}}\,^i\}$ as multiples of the unit vectors ${\mathbf {e}}_i$ for spherical coordinates $x^1=r$, $x^2=\theta$, and $x^3=\varphi$. (2 P)

Problem 3.13

With the help the Maxwell construction, draw the force lines of two equally charged parallel lines with charges densities q / l and separated by a distance a. This uses the theorem that, for a source-free field, there is the same flux through any cross-section of a force tube. What changes with this construction for oppositely charged parallel lines, i.e., charge densities $\pm q/l$, separated by a distance a? Why is the construction more precise than the method of drawing trajectories orthogonal to the equipotential lines? (8 P)

Problem 3.14

Determine the equation $f(x,z)=0$ of the field line of an ideal dipole ${\mathbf {p}}=p {\mathbf {e}}_z$ which lies at the origin (${\mathbf {r}}\,'={\mathbf {0}}$). Note that, due to the cylindrical symmetry, we may set $y=0$. (4 P)

Problem 3.15

On the z-axis there are several point charges $q_i$ at the positions $z_i$. Determine their common potential $\Phi$ by Taylor series expansion up to order $(z_i/r)^3$. Examine the result for the potential when $r\gg a$ (write $\Phi$ as a multiple of $q_1=q$) for:

• a dipole ($q_1=-q_2$, $z_1=-z_2=\frac{1}{2}a$),

• a linear quadrupole ($q_1=-\frac{1}{2}q_2=q_3$, $z_1=-z_3=a$, $z_2=0$), and

• an octupole ($q_1=-\frac{1}{3}q_2=+\frac{1}{3}q_3=-q_4$, $z_1=3z_2= -3z_3=-z_4=\frac{3}{2}a$)?

Show that the field of a finite dipole may be written approximately as a superposition of a dipole field and an octupole field. How strong is the octupole field compared with the field of a pure quadrupole? Justify with the examples above that an ideal $2^n$-pole can be viewed as a superposition of two $2^{n-1}$-poles. (8 P)

Problem 3.16

Determine the potential and field strength of a hollow sphere with outer and inner radii R and $\eta R$ and a charge Q distributed evenly over its volume. Here $0\le \eta \le 1$, so a solid sphere has $\eta =0$ and a surface charge $\eta =1$. Sketch the results $\Phi (r)$ and E(r) in the limiting cases $\eta =0$ and $\eta =1$. In these limiting cases, how much field energy is in the space with $r\le R$? How much is in the external space? (7 P)

Problem 3.17

Express the potential of a metal ring of radius R and charge Q in terms of the complete elliptic integral of the first kind K(m) (with $0\le m\le 1$) of p. 202. Here it will be convenient to replace the spherical coordinate $\varphi$ by $\pi -2x$. Determine the potential and the field strength on the axis of the ring. (6 P)

Problem 3.18

Determine the potential and field strength on the axis of a thin metal disc of radius R and charge Q for constant charge density. What is the jump in the field strength at the disc? (3 P)

Problem 3.19

What is obtained for the potential on the axis if the disc has a constant dipole density ${\mathbf {p}}_A$? What is the jump in the potential? (4 P)

Problem 3.20

On a straight line at distance a from the origin, let there be a point charge $q>0$, and at distance $a'$ on the same side of the origin, a charge $-q'<0$. For suitable $q'(q,a,a')$, the potential vanishes on the surface of a sphere about the origin. What is its radius? Use this to determine the charge density $\rho _A$ on a grounded metal sphere of radius R induced by a point charge q at distance a from the center of the sphere. What changes for an ungrounded metal sphere? (6 P)

Problem 3.21

How does the Maxwell stress tensor read for a homogeneous field of strength $\mathbf {E}=E\,{\mathbf {e}}_z$ in vacuum? How strong is the force on a volume element $\mathrm {d}x \;\mathrm {d}y\;\mathrm {d}z$? Using the stress tensor, determine the force on an area A if its normal is ${\mathbf {n}}={\mathbf {e}}_x\,\sin \theta +{\mathbf {e}}_z\,\cos \theta$.

Hint: Decompose the force into components along ${\mathbf {n}}$, ${\mathbf {t}}={\mathbf {e}}_x\, \cos \theta -{\mathbf {e}}_z\,\sin \theta$, and ${\mathbf {b}}={\mathbf {t}}\times {\mathbf {n}}$. Draw the vectors $\mathbf {E}$, ${\mathbf {n}}$, ${\mathbf {t}}$, and ${\mathbf {F}}$ for $\theta =30^0$. Interpret the result for opposite sides of a cube. (7 P)

Problem 3.22

How does the stress tensor change at the x, y-plane if it carries the charge density $\rho _A$ and is placed in an external (homogeneous) field in the z direction? Can the force on an enclosing layer be related to the mean value of the field strength above and below the plane? Determine the Cartesian components of the Maxwell stress tensor on the plane midway between two equal charges q (each at distance a from this plane)? What force is thus exerted from one of the sides on the plane?

Hint: Express the strength of the field in cylindrical coordinates. (7 P)

Problem 3.23

Determine the electric field around a metal sphere in a homogeneous electric field. Superpose the field of a dipole ${\mathbf {p}}$ on a suitable homogeneous field $\mathbf {E}_0$ in such a way that the tangential component of the total field vanishes on the surface of the sphere of radius r around the dipole. How large is the normal component (in particular in the direction of $\mathbf {E}_0$, opposite and perpendicular to it)? (4 P)

Problem 3.24

Determine the current density and resistance for half a metal ring with circular cross-section (area $\pi a^2$), whose axis forms a semi-circle of radius A (conductivity $\sigma$), if there is a voltage U between the faces.¹ Note the special case $a\ll A$. (5 P)

Problem 3.25

In an otherwise homogeneous conductor, there is a spherical void of radius $r_0$ containing air. Determine the current density ${\mathbf {j}}$ if it is equal to ${\mathbf {j}} _0$ for large r. (3 P)

Problem 3.26

Equal currents I flow through two equal coaxial circles (radii R) a distance a apart. For which ratio a / R is the magnetic field strength at the center of the setup as homogeneous as possible? What does that mean? Where would we have to place a further pair of loop currents with radius $\frac{1}{2}R$ in order to amplify the homogeneous field? Can the homogeneity be improved by a suitable choice of current strengths in two pairs of loops? (8 P)

Problem 3.27

A closed iron ring with permeability $\mu$ and dimensions a and A, as in Problem 3.24, is wrapped around N times with a thin wire. How large is the induction flux $\Phi =\int \mathrm {d}{\mathbf {f}}\cdot {\mathbf {B}}$ in the ring? How large is the relative error , if we assume a constant magnetic field ${\mathbf {H}}$ equal to the value $\overline{\Phi }$ at the center of the cross-section? Determine $\overline{\Phi }$ and for $N=600$, $\mu =500\mu _0$, $A=20~{\text {cm}}$, $\pi a^2=10~{\text {cm}}^2$, and $I=1$ A. The iron ring may have a narrow discontinuity (air gap) of width d. It can be so narrow that no field lines escape from the slit. How does the induction flux depend on the width d if we use a constant magnetic field ${\mathbf {H}}$ in the cross-section? (7 P)

Problem 3.28

The mutual inductance of two coaxial circular rings of radii R and $R'$ a distance a apart is determined as $L=\mu _0\sqrt{RR'}\;\{2\,(\text{ K }-\text{ E })\!-\!k^2\text{ K }\}/k$, with the parameter $k^2=4RR'/\{a^2+(R\!+\!R')^2\}$, involving the complete elliptic functions of the first kind, viz., K$(k^2)$ as in Problem 3.17, and the second kind, viz., E$(k^2)=\int _0^{\pi /2}\sqrt{1\!-\!k^2\sin ^2z}\;\mathrm {d}z$. What is obtained to leading order for L at very large distances ($R\ll a$, $R'\ll a$)?

Hint: Expand K and E in powers of k. (3 P)

Problem 3.29

In the limit of small distances ($R\approx R'\gg a$), we use the Landen transformation F$(x|k^2)=2/(1\!+\!k)\;\text{ F }(x'|k'^2)$ for the incomplete elliptic integral of the first kind F$(x|k^2)=\int _0^x \mathrm {d}z/\sqrt{1\!-\!k^2\sin ^2\!z}$, with $x'=\frac{1}{2}\{x\!+\!\arcsin (k\sin x)\!\}$ and $k'^2=4k/(1\!+\!k)^2$.

With $\sin (2z_1-z)=k\sin z$, we have $\cos (2z_1-z)\,(2\,\mathrm {d}z_1-\mathrm {d}z) =k\cos z\,\mathrm {d}z$, hence also $\mathrm {d}z\,\{k\cos z+\cos \,(2z_1-z)\}=2\,\mathrm {d}z_1\,\cos (2z_1-z)=2\,\mathrm {d}z_1 \, (1-k^2\sin ^2z)^{1/2}$. The square of the curly brackets is equal to $k^2\cos ^2z+2k\cos z\cos \,(2z_1-z)+1-k^2\sin ^2z$, or again $1+k^2+2k\,\{\cos z\cos \,(2z_1-z)-\sin z\sin \,(2z_1-z)\}$. The curly bracket may be reformulated as $\cos 2z_1=1-2\sin ^2z_1$. Then

$$ \mathrm {d}z/(1-k^2\sin ^2z)^{1/2}=2\mathrm {d}z_1/\{(1+k)^2-4k\sin ^2z_1\}^{1/2}\;, $$

which is important for the proof of Landen’s transformation.

Prove that K$(1-\varepsilon )\approx \ln (4/\sqrt{\varepsilon })$. What follows for the inductance $L(R,R',a)$? (5 P)

Problem 3.30

Derive from this the self-inductance of a thin ring of wire with circular cross-section (abbreviation as in Problem 3.24), which is composed of the mutual inductances $L=(\pi a^2)^{-2}\int \mathrm {d}f_1\,df_2\,L_{12}$ of its filaments. Here $\int _0^{\pi }\ln (A+B\cos \varphi )\;\mathrm {d}\varphi =\pi \,\ln \{\frac{1}{2}(A+\sqrt{A^2-B^2})\}$ for $A\ge |B|$. (For ferromagnetic materials, there is an additional term, not required here.) (5 P)

Problem 3.31

For a current strength I, determine the vector potential of a circular ring of radius $R_0$ at an arbitrary point ${\mathbf {r}}$. The circular ring suggests using cylindrical coordinates ($R,\varphi ,z$) with ${\mathbf {r}}=R{\mathbf {e}}_R+z{\mathbf {e}}_z$. (6 P)

Problem 3.32

A very long hollow cylinder with inner radius $R_\text {i}$, outer radius $R_\text {a}$, and permeability $\mu$ is brought into a homogeneous magnetic field ${\mathbf {H}}_0$ perpendicular to its axis. Determine ${\mathbf {B}}$ and ${\mathbf {H}}$ for all ${\mathbf {r}}$. How large is the field $H_0$ compared to its value on the axis for $\mu \gg \mu _0$? (9 P)

Problem 3.33

Perpendicular to the circuit shown in Fig. 3.36, made of a thin wire with resistance $R=R_1+R_2$, a homogeneous magnetic field changes by equal amounts in equal time intervals. What voltage does the voltmeter show, and in particular, if the circuit forms a circle and the voltmeter sits at the center of the circle and is connected with straight wires? (5 P)

Fig. 3.36

Between two points of a circuit, a voltmeter is connected with thin (loss-free) wires (resistance $R_0$), such that the area A spanned by the circuit is divided in the ratio $A_1{:}A_2$

Problem 3.34

An insulating cuboid $(0\le x\le L_x$, $0\le y\le L_y$, $0\le z \le L_z)$ of homogeneous material with scalar permittivity and permeability is enclosed by ideally conducting walls. Investigate the following ansatz for the vector potential:

$$\begin{aligned} A_x= & {} a_x\cos (\omega t)\cos (k_xx+\varphi _{xx})\cos (k_yy+\varphi _{xy}) \cos \,(k_zz+\varphi _{xz})\;,\\ A_y= & {} a_y\cos (\omega t)\cos (k_xx+\varphi _{yx})\cos (k_yy+\varphi _{yy}) \cos \,(k_zz+\varphi _{yz})\;,\\ A_z= & {} a_z\cos (\omega t)\cos (k_xx+\varphi _{zx})\cos (k_yy+\varphi _{zy}) \cos \,(k_zz+\varphi _{zz})\;, \end{aligned}$$

with the radiation gauge. Can we restrict ourselves here to $0\le \varphi _{ik}<\pi$? What is the relation between $\omega$ and ${\mathbf {k}}$ if all the Maxwell equations are valid? What requirements follow from the boundary conditions ${\mathbf {n}}\times \mathbf {E}={\mathbf {0}}$ and ${\mathbf {n}}\cdot {\mathbf {B}}=0$? (7 P)

Problem 3.35

What requirement does the gauge condition ${\mathbf {\nabla }}\cdot {\mathbf {A}}=0$ lead to for the ansatz above? What do we then obtain for the three fields ${\mathbf {A}}$, $\mathbf {E}$, and ${\mathbf {B}}$? (5 P)

Problem 3.36

What do we obtain if ${\mathbf {k}}$ is parallel to one of the edges of the cuboid? What is the general ansatz for ${\mathbf {A}}$ in Problem 3.34? (3 P)

Problem 3.37

Express the energy density $w(t,{\mathbf {r}})$ of an electromagnetic wave in terms of its vector potential in the radiation gauge, i.e., with $\Phi =0$ and ${\mathbf {\nabla }}\cdot {\mathbf {A}}=0$. How can Parseval’s equation help to re-express the total energy of the wave (integrated over the whole space) as an integral of the square of the absolute value of ${\mathbf {A}}(t,{\mathbf {k}})$ and $\partial {\mathbf {A}}/\partial t$ as weight factors? What is the unknown expression? (5 P)

Problem 3.38

How does the electric field amplitude of the reflected and transmitted waves depend on the incoming amplitude in the limiting cases $\theta =0^0$ and $90^0$ (expressed in terms of the refractive index n)? To what extent are the parallel and perpendicular components to be distinguished for perpendicular incidence ($\theta =0^0$)? (4 P)

Problem 3.39

How large is the energy flux $\dot{W}$, averaged over time, for an electromagnetic wave with wave vector ${\mathbf {k}}$ passing through an area A perpendicular to ${\mathbf {k}}$? What do we obtain for the reflected and the transmitted waves in the limiting cases investigated above? (4 P)

Problem 3.40

Does the energy conservation law hold true for an electromagnetic wave, incident with the wave vector ${\mathbf {k}}$ on the interface between two homogeneous insulators (with an arbitrary angle of incidence)? Investigate this question for arbitrary scalar material constants $\varepsilon$ and $\mu$, i.e., also with $\mu \ne \mu _0$. (5 P)

Problem 3.41

For a homogeneous conductor (with scalar $\sigma$, $\varepsilon$, and $\mu$), derive the relation between $\overline{w}$, ${\mathbf {E}}^*\cdot {\mathbf {D}}$, and ${{\mathbf {H}}}^*\cdot {\mathbf {B}}$ from the Maxwell equations, if only one wave vector is given. How is the time average of the Poynting vector connected to the averaged energy density $\overline{w}$?

Hint: Use the approximation $\alpha ^2\approx \beta ^2\approx \sigma /(2\varepsilon \omega )\gg 1$.    (7 P)

List of Symbols

We stick closely to the recommendations of the International Union of Pure and Applied Physics (IUPAP) and the Deutsches Institut für Normung (DIN). These are listed in Symbole, Einheiten und Nomenklatur in der Physik (Physik-Verlag, Weinheim 1980) and are marked here with an asterisk. However, one and the same symbol may represent different quantities in different branches of physics. Therefore, we have to divide the list of symbols into different parts (Table 3.2).

Table 3.2 Symbols used in electromagnetism