Chapter 4, Electric Fields Ii Video Solutions, Electromagnetic Fields and Waves: Including Electric Circuits

Problem 1

The object of this problem is to illustrate the enormous magnitude of the electric charge densities in matter.
We take the example of the conduction electrons in copper. A copper atom contains 29 electrons, one of which is a conduction electron. Copper has an atomic weight of 64 and a density of $8.9 \times 10^{3}$ kilograms/meter $^{3}$. Suppose that you have two copper spheres, each one having a volume of I centimeter $^{3}$. The spheres are depleted of their conduction electrons and separated by a distance of 100 millimeters. Calculate the force of repulsion. Show that this force is equal to about $0.5 \%$ of the force of attraction between the sun and the earth. See the table of physical constants on the page facing the back cover.

Dading Chen

Numerade Educator

03:22

Problem 2

Copper has an atomic weight of 64 and a density of $8.9 \times 10^{3}$ kilograms/ meter $^{3}$.
(a) Calculate the number of atoms per cubic meter and the approximate diameter of an atom.
(b) Calculate the charge $\lambda$ carried by the conduction electrons in 1 meter of copper wire 1 millimeter in diameter. There is one conduction electron per atom.
(c) Calculate the drift velocity of the conduction electrons in meters per hour when the wire carries a current of 1 ampere.

Dading Chen

Numerade Educator

01:43

Problem 3

Refraction of lines of $\boldsymbol{E}$ at the interface between media of different conductivities
We shall see in Sec. $10.2 .3$ that the tangential component of $\boldsymbol{E}$ is continuous at the interface between two media.
Show that, at the boundary between two media of conductivities $\sigma_{1}$ and $\sigma_{2}$, a line of $\boldsymbol{E}$, or a line of $\boldsymbol{J}$, is "refracted" in such a way that $\tan \theta_{1} / \sigma_{1}=\tan \theta_{2} / \sigma_{2}$, where $\theta_{1}$ and $\theta_{2}$ are the angles formed by a line of $\boldsymbol{E}$ with the normal to the interface.

Ajay Singhal

Numerade Educator

01:48

Problem 4

A current of density $J$ flows in the direction normal to the interface between two media of conductivities $\sigma_{\mathrm{col}}$ and $\sigma_{\mathrm{co} 2}$. The current flows from medium 1 to medium 2 .
Show that the surface charge density $\sigma_{\mathrm{ch}}$ is $\epsilon_{r} \epsilon_{0} J\left(1 / \sigma_{\mathrm{co2}}-1 / \sigma_{\mathrm{co1}}\right)$. Assume that $\epsilon_{r}$ has the same value on both sides. If the current is not normal to the interface, then the above $J$ is the normal component of the current density.

Varsha Aggarwal

Numerade Educator

01:38

Problem 5

In a nonhomogeneus medium, the conductivity $\sigma$ is a function of the coordinates.
Show that, under static conditions, or when $\boldsymbol{E}=-\boldsymbol{\nabla} V$, and if $\sigma$ is nowhere equal to zero,
$$
\boldsymbol{\nabla}^{2} V+\boldsymbol{\nabla} V \cdot \boldsymbol{\nabla} \tau=0
$$
where $\tau=\ln \sigma$.

Suhas Katkar

Numerade Educator

01:01

Problem 6

One can locate resistivity anomalies in the ground as in Fig. 4-5. The current $I$ flowing between electrodes $C_{1}$ and $C_{2}$ establishes an electric field in the ground, and one measures the voltage $V$ between a pair of electrodes $P_{1}$ and $P_{2}$ maintained at a fixed spacing $b$. With $b \ll a, V / b$ is equal to $E$ at the position $x$. Anomalies in ground conductivity show up in the curve of $E$ as a function of $x$. Show that if the substrate conductivity is uniform and equal to $\sigma$, then
$$
\frac{V}{b}=-\frac{2 a x I}{\pi \sigma\left(x^{2}-a^{2}\right)^{2}}
$$
The electrodes are of finite size. However, you can perform the calculation on the assumption that they are infinitely small, disregarding the fact that $E$ and $J$ would then be infinite at their surfaces.
You can use the principle of superposition as follows. The current in the ground is the sum of a radial distribution emanating from $C_{1}$ plus another radial distribution converging on $C_{2}$. Thus, at a point $r_{1}, r_{2}$,
$$
\boldsymbol{E}=\frac{\boldsymbol{r}_{1}}{2 \pi \sigma r_{1}^{2}}-\frac{I \hat{r}_{2}}{2 \pi \sigma r_{2}^{2}}
$$

Dominador Tan

Numerade Educator

02:36

Problem 7

A spherical shell of uniform conductivity has inner and outer radii $R_{1}$ and $R_{2}$, respectively. It has copper electrodes plated on the inner and outer surfaces.
Show that the resistance is $\left(1 / R_{1}-1 / R_{2}\right) / 4 \pi \sigma$

Vidhi Bhatt

Numerade Educator

02:36

Problem 8

A square film of Nichrome, an alloy of nickel and chromium, has copper electrodes deposited on two opposite edges.
Show that the resistance between the electrodes depends only on the thickness of the film and on its conductivity, as long as the film is square. This surface resistance is expressed in ohms per square.

Mahipal Kumawat

Numerade Educator

05:35

Problem 9

A rectangular plate $A B C D$ has a thickness $s$ and a conductivity $\sigma$. With conducting electrodes on edges $A B$ and $C D$, the resistance is $R_{1}$. With electrodes on $B C$ and $D A$, the resistance is $R_{2}$.
Show that $R_{1} R_{2}=1 /\left(\sigma^{2} s^{2}\right)$.
This equation also applies to any region bounded by equipotentials and lines of current flow. We shall use this theorem in Prob. 33.5. It was first proved by Isukada.\dagger

Sanat Mukherjee

Numerade Educator

01:10

Problem 10

A battery feeds a resistance $R$ as in Fig. 4-6. The battery acts as a pump, forcing conduction electrons toward the negative electrode. The battery is cylindrical, of length $s$ and cross-sectional area $\mathscr{A}$, with electrodes at each end. Then $|\boldsymbol{\nabla} V|=V / \mathrm{s}$ and, inside the battery, $J=\sigma\left(E_{p}-|\boldsymbol{\nabla} V|\right)$, where $E_{p}$ is the "pumping field."
Find the output voltage as a function of the current. Set $R=s / \sigma \mathcal{A}$ as the output resistance of the battery.

Varsha Aggarwal

Numerade Educator

06:38

Problem 11

A simple model for the drift of a conduction electron is the following. The electron describes a ballistic trajectory for a while, under the action of the ambient electric field, and then the electron suffers an impact. Its velocity just after the impact is unrelated to its velocity before the impact, and we set it equal to zero. The electron then starts out on another ballistic trajectory, and the process repeats itself. Let the mean time between the collisions be $\Delta t$ and the effective mass (Sec. 4.3.4) be $m^{*}$.
Find the mobility in terms of $\Delta t$ and $m^{*}$.

Eduard Sanchez

Numerade Educator

01:28

Problem 12

We derived Eq. $4-40$ on the assumption that the charge carriers are electrons. Suppose the carriers are holes. Then the charge changes sign, and both terms on the right are positive. If that is so, the complementary function, which one obtains on disregarding the forcing term $e E_{m} j \omega t$, is
$$
v_{d}=v_{d 0} \exp \left(\frac{e}{m^{*} M}\right) t
$$
Then $v_{d}$ increases exponentially with time, which is absurd. Show that, if the charge carriers are holes, then
$$
m^{*} \frac{d v_{d}}{d t}=+e E_{m} \exp j \omega t-\frac{e}{\mathcal{M}} v_{d}
$$
and that
$$
v_{d}=+\frac{M}{1+j \omega m^{*} \mathcal{M} / e} E_{m} \exp j \omega t, \quad \sigma=\frac{N e M}{1+j \omega m^{*} M / e}
$$

Dominador Tan

Numerade Educator

01:56

Problem 13

Figure $4-7$ shows the principle of operation of a resistojet used as a thruster for correcting the trajectory or the attitude of a satellite.
Assuming complete conversion of the electric energy to kinetic energy, calculate the thrust for a power input of 3 kilowatts and a flow of $0.6$ gram of hydrogen per second.

Ajay Singhal

Numerade Educator