Chapter 12, Electric Fields X Video Solutions, Electromagnetic Fields and Waves: Including Electric Circuits

Problem 1

A grounded, infinite, circular cylindrical conductor of radius $a$ lies in a previously uniform electric field, with its axis perpendicular to $\boldsymbol{E}_{0}$, as in Fig. 12-3. Show that $V=-E_{6}\left(1-a^{2} / \rho^{2}\right) \rho \cos \phi$.

Nathan Silvano

Numerade Educator

04:23

Problem 2

Show that $\int_{v} \cdot \boldsymbol{\nabla}^{\prime} \times(\boldsymbol{E} / r) d v^{\prime}=0$ for any finite charge distribution. Use identity 19 from the front endpaper.

Suzanne W.

Numerade Educator

04:33

Problem 3

Show that the two integrals for $\boldsymbol{E}$,
$$
\boldsymbol{E}=\frac{1}{4 \pi \epsilon_{0}} \int_{v} \cdot \frac{\rho}{r^{2}} \hat{\boldsymbol{r}} d v^{\prime} \quad \text { and } \quad \boldsymbol{E}=-\frac{1}{4 \pi \epsilon_{0}} \int_{v^{*}} \boldsymbol{\nabla}^{\prime} \frac{\rho}{r} d v^{\prime}
$$
are equal, at least if $v^{\prime}$ is finite. Use identities 15 and 18 from the front endpaper.

Eduard Sanchez

Numerade Educator

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Problem 4

We have shown that, for static fields,
$$
\boldsymbol{E}=\frac{1}{4 \pi \epsilon_{0}} \int_{v} \cdot \frac{\rho}{r^{2}} \hat{\boldsymbol{r}} d v^{\prime}=-\frac{1}{4 \pi \epsilon_{0}} \int_{v^{\prime}} \frac{\boldsymbol{\nabla} \rho}{r} d v^{\prime}
$$
Show that, as a consequence,
$$
\int_{v^{\prime}} \boldsymbol{\nabla}^{\prime} \frac{\rho}{r} d v^{\prime}=0
$$

Victor Salazar

Numerade Educator

01:03

Problem 5

Dirac proposed at one time that a positron could be considered as a hole in an infinite sea of negative electrons. Assume that positrons and electrons have finite dimensions.
Deduce the field of a positron from Coulomb's law on these assumptions. You can find this field without having to integrate!

Carlos Henrique De Lima

Numerade Educator

01:16

Problem 6

Figure $12-4$ shows the density as a function of the radius for a given sphere of charge. This corresponds roughly to the charge distribution inside an atomic nucleus.
(a) Use Gauss's law to find $E$ at a distance $r>\beta$ from the center of the sphere.
(b) Show that the expression for $E$ given in Eq. 12-52 leads to the same result. Both calculations remain valid when $\beta-\alpha$ tends to zero.

Prem Bijarniya

Numerade Educator