Chapter 37, Radiation I Video Solutions, Electromagnetic Fields and Waves: Including Electric Circuits

Problem 1

The retarded vector potential near a long, straight wire carrying a time-dependent current
A long, straight wire of length $C$ carries a current that increases linearly with time: $I=K t$.
Show that, at a distance $\rho$ from the wire such that $4 \rho^{2} \ll C^{2}$, and away from the ends,
$$
A=\frac{\mu_{0} I}{2 \pi} \ln \frac{C}{\rho}
$$
Refer to the first example in Sec. 18.4. We have disregarded a term that is independent of both the time and the coordinates and that does not, therefore, affect either $\boldsymbol{E}$ or $\boldsymbol{B}$. This particular $A$ therefore has the same form as if the current were constant.

Eduard Sanchez

Numerade Educator

05:02

Problem 2

The propagation speed of $V$ and $\boldsymbol{A}$ in the field of an electric dipole Find the propagation speed of the scalar and vector potentials $V$ and $\boldsymbol{A}$ in the field of an electric dipole, on the assumption that $s^{3} \ll r^{3}$ and that $s^{3} \ll x^{3} .$
Observe that the addition of two waves of $V$ traveling at the speed $c$ gives a wave whose phase speed is larger than $c$. This is an interference effect. Observe also that the speed of propagation $V$, close to the dipole, is a function of the wavelength and hence of the frequency. The dispersion originates, not in the properties of the medium, but rather in the geometry, as in rectangular metallic waveguides and in dielectric waveguides.

Nick Johnson

Numerade Educator

00:41

Problem 3

In the field of a magnetic dipole, $V=0, \boldsymbol{A} \neq 0$. Do there exist radiation fields where the inverse is true?

Averell Hause

Carnegie Mellon University

01:33

Problem 4

We have shown that
$$
\boldsymbol{B}=\frac{\mu_{0}}{4 \pi} \int_{v^{\prime}} \frac{\boldsymbol{\nabla}^{\prime} \times \boldsymbol{J}}{r} d v^{\prime}
$$
if retardation is negligible. The volume $v^{\prime}$ encloses all the currents.
Show that the term on the right is equal to $\boldsymbol{\nabla} \times \boldsymbol{A}$ for a finite current distribution. Refer to the identities on the inside of the front cover.

James Kiss

Numerade Educator

04:46

Problem 5

Show that, for any finite charge distribution, the integrals for $\boldsymbol{E}$ given in Sec. $37.5$ are equal:
$$
\frac{1}{4 \pi \epsilon_{0}} \int_{v^{\prime}} \frac{\rho \hat{\boldsymbol{r}}}{r^{2}} d v^{\prime}=-\frac{1}{4 \pi \epsilon_{0}} \int_{v^{\prime}} \frac{\boldsymbol{\nabla}^{\prime} \rho}{r} d v^{\prime}
$$

Suhas Katkar

Numerade Educator

00:37

Problem 6

Show that, for any finite current distribution, the integrals for $\boldsymbol{B}$ given in Sec. $37.5$ are equal:
$$
\frac{\mu_{0}}{4 \pi} \int_{v^{\prime}} \frac{\boldsymbol{J} \times \hat{\boldsymbol{r}}}{r^{2}} d v^{\prime}=\frac{\mu_{0}}{4 \pi} \int_{v^{\prime}} \frac{\boldsymbol{\nabla}^{\prime} \times \boldsymbol{J}}{r} d v^{\prime}
$$

Frank Lin

Numerade Educator

02:46

Problem 7

Show that
$$
\int_{v^{\prime}} \boldsymbol{\nabla}^{\prime} \times \frac{\boldsymbol{J}}{r} d v^{\prime}=0
$$

Bobby Barnes

University of North Texas

05:29

Problem 8

The electric field of the electric dipole, calculated from the third integral for $\boldsymbol{E}$
Calculate $\boldsymbol{E}$ in the field of the electric dipole, starting from the integral of Sec. 37.6. To simplify the calculation, disregard terms in $s^{2} / r^{2}$ and $s^{3} / r^{3}$ as well as the higher-order terms in $s / \chi$. This will leave you only the radiation term.

Doruk Isik

Numerade Educator

04:40

Problem 9

Suppose you have a disk of radius $R$ and thickness $2 s \ll R$. It carries a charge density $Q^{\prime}=K(R-\rho)(s-z)^{2}$. We use $Q^{\prime}$ for the charge density in order to use $\rho$ for the radial coodinate.
You are required to find the value of $\boldsymbol{B}$ at the center when the disk rotates as a solid at the angular velocity $\omega$. Of course, $B$ is normal to the plane of the disk. See Prob. 18-4.
Calculate $\boldsymbol{B}$ from Eq. 37-60 and then from Eq. 37-63.

Supratim Pal

Numerade Educator