Chapter 10, Electric Fields Viii Video Solutions, Electromagnetic Fields and Waves: Including Electric Circuits

Problem 1

The dielectric of a parallel-plate capacitor has a relative permittivity $\epsilon_{r}$ and a conductivity $\sigma_{\mathrm{co}}$. The conductivity of the dielectric is much less than that of the plates, which makes $\boldsymbol{E}$ uniform between the plates.
(a) Assume surface charge densities $\sigma_{\mathrm{ch}}$ on the plates, and use Gauss's law to relate $E$ to $\sigma_{\mathrm{ch}}$. Then show that the impedance of the capacitor is the same as that of a resistor and a capacitor in parallel.
(b) Show that when the capacitor is disconnected, the charge on the capacitor decreases by a factor of $e$ in $\epsilon / \sigma_{\mathrm{co}}$ seconds. This is the relaxation time of the capacitor.

Vidhi Bhatt

Numerade Educator

02:47

Problem 2

The dielectric of a parallel-plate capacitor is made up of two parts, as in Fig. 10-8.
(a) Find the impedance $Z$.
(b) Call a conductivity $\sigma_{\mathrm{co}}$ and a surface charge density $\sigma_{\mathrm{ch}} .$ Calculate the surface charge density on the interface.

Salamat Ali

Numerade Educator

02:19

Problem 3

A parallel-plate capacitor has plates of area $\mathscr{A}$ separated by a distance $s$.
Its dielectric has a conductivity $\sigma=a+b x$, where $x$ is the distance to one plate, and a uniform relative permittivity $\epsilon$,
(a) Calculate the resistance $R$ of the capacitor.
(b) Show that with a steady voltage $V$ applied to the electrodes, there is a uniform volume density of free charge.
(c) Sketch lines of $\boldsymbol{E}$ for $b>0$. The field is not uniform.
(d) With an alternating voltage across the electrodes,
$$
I=\mathscr{A}\left(\sigma E+\frac{\partial D}{\partial t}\right)=\mathscr{A J}_{t}
$$
Show that $\boldsymbol{\nabla} \cdot J_{t}=\partial J_{t} / \partial x=0$. Then $J_{t}$ is independent of $x$.
(e) Show that $E=J_{t} /\left(a+b x+j \omega \epsilon, \epsilon_{0}\right)$.
(f) Now find the impedance $Z=V / I$. The real part of $Z$ is not the $R$ that we found above. However, if $\omega=0$, we revert to $R$, as expected. See Prob. 10-1.
(g) Find $\rho_{f}$.

Manik Pulyani

Numerade Educator

03:36

Problem 4

One particular ceramic capacitor is cylindrical, with three electrodes as in Fig. 10-9. The ceramic disks each have a diameter of 21 millimeters and a thickness of $0.5$ millimeter. The nominal capacitance is $0.05$ microfarad within a range of $-20 \%$ to $+80 \%$. What is the approximate value of $\epsilon_{r} ?$

Ren Jie Tuieng

Numerade Educator

03:05

Problem 5

Figure $10-10$ shows a schematic diagram of an instrument that has been used to plot the hysteresis curves of ferroelectric materials. An oscillator applies an alternating voltage $V$ to two capacitors in series, the parallelplate capacitor $C_{x}$ containing the material and a normal capacitor $C \gg C_{x}$. The voltage across $C$ goes to the $Y$ input of an $X Y$ recorder, while the voltage across $R_{2} \ll R_{1}$ goes to the $X$ input. The ferroelectric sample is a few millimeters thick, and $V \approx 10$ kilovolts, $f \approx 10^{-2}$ hertz.
Explain why the voltage across $C$ is proportional to the $D$ in the sample contained in $C_{x}$, while that across $R_{2}$ is proportional to $E$. The recorder draws essentially zero current at its $X$ and $Y$ terminals.

James Kiss

Numerade Educator

01:30

Problem 6

A parallel-plate capacitor whose dielectric is nonlinear is connected to a power supply. The voltage $V$ increases slightly by $d V$, and an extra charge $d Q$ flows into the capacitor. Show that the density of stored energy increases by $E d D$.

Mahipal Kumawat

Numerade Educator

03:27

Problem 7

Show that the area of the hysteresis loop for a ferroelectric material is equal to the energy dissipated per cubic meter and per cycle.

James Kiss

Numerade Educator

05:08

Problem 8

Show that, at the interface between two conductors,
$$
\left(\boldsymbol{E}_{1}-\boldsymbol{E}_{2}\right) \times \hat{\boldsymbol{n}}=0, \quad\left(\boldsymbol{J}_{1}-\boldsymbol{J}_{2}\right) \times \hat{\boldsymbol{n}}=\frac{d \sigma_{\mathrm{ch}}}{d t}
$$
where $\hat{n}$ is a unit vector that is normal to the interface and points away from conductor 1 , and where $\sigma_{c h}$ is the surface charge density on the interface.

James Kiss

Numerade Educator

04:42

Problem 9

Investigate the possibility of propelling a small vehicle with an electric motor fed by a charged capacitor. Consider only the problem of energy storage.
(a) Show that the maximum energy density in the dielectric of a parallel-plate capacitor is $\epsilon a^{2} / 2$, where $a$ is the dielectric strength of the insulator, or the maximum $E$ before breakdown. A good dielectric to use would be Mylar, which has a dielectric strength of $1.6 \times 10^{8}$ volts/meter when in the form of thin sheets and a relative permittivity of $3.2$. The energy density would be about 10 times less with electrolytic capacitors.
(b) Calculate the energy density and the approximate size and mass of the capacitor that you would need to operate a 1 -kilowatt motor for 1 hour.

Khoobchandra Agrawal

Numerade Educator

01:56

Problem 10

Figure $9-4$ shows the $E$ and $D$ fields of a bar electret.
(a) Show that the lines of $\boldsymbol{E}$ do not bend at the cylindrical surface, but that the lines of $\boldsymbol{D}$ do bend.
(b) Show that the inverse is true at the end faces.

Penny Riley

Numerade Educator

01:20

Problem 11

(a) A dipole of fixed dipole moment $p$ orients itself in a uniform electric field $E$. Show that its potential energy is $-p E$, assuming that the potential energy is zero when the dipole axis is perpendicular to the field.
(b) A nonpolar molecule acquires a dipole moment in a uniform electric field that gradually increases from zero to $E$. Show that its potential energy is $+p E / 2$.

Josh Broderick Phillips

Numerade Educator

07:03

Problem 12

There exist accelerators for neutral molecules that operate as follows. Figure $10-11$ shows a pair of spheres that carry charges $+Q$ and $-Q$, and a molecule of dipole moment $p$. The molecule accelerates toward the spheres until it reaches their midpoint. At that instant the spheres are discharged, and the molecule continues on its way at a constant velocity. Such accelerators serve to study the processes that occur during molecular collisions.
(a) Find the kinetic energy acquired by the molecule if the distance $x$ in the figure is initially much larger than the distance $D$ between the pair of electrodes. Consider the spheres as point charges, and apply the principle of conservation of energy. Assume that $p$ is constant, and refer to Prob. 10-11. The dipole moment in fact increases as the molecule approaches, so that we have underestimated the energy.
(b) In one particular accelerator the electrode voltages are $\pm 40$ kilovolts, their radius is $0.25$ millimeter, and $D=1.00$ millimeter. Calculate the approximate value of the kinetic energy of a molecule in electronvolts for $p=2 \times 10^{-29}$ coulomb-meter and for 700 stages. That accelerator has a length of 10 meters.

Athiru Pathiraja

Numerade Educator

04:42

Problem 13

Electrostatic clamps are used for holding workpieces while they are being machined, for holding silicon wafers during electron beam microfabrication, etc. They comprise an insulated conducting plate maintained at a potential of several thousand volts and covered with a thin insulating sheet. The workpiece or the wafer rests on the sheet and is grounded. It is advisable to apply a film of oil to the sheet to prevent sparking.
One particular type operates at 3000 volts and has holding power of 2 atmospheres $\left(2 \times 10^{5}\right.$ pascals). If the insulator is Mylar $\left(\epsilon_{r}=3.2\right)$, what is its thickness?

Khoobchandra Agrawal

Numerade Educator

02:14

Problem 14

A certain capacitor consists of two polished circular aluminum plates, 237 millimeters in diameter, separated by a sheet of plastic $0.762$ millimeter thick, with a relative permittivity of $3.0$. Thin films of air subsist between the electrodes and the plastic. This reduces the capacitance below the rated value, and there is no way of clamping the plates mechanically with sufficient force.
(a) Someone suggests that the electric force alone might be sufficient to clamp the plates, at the operating voltage of 60 kilovolts. What is your opinion?
(b) Show that, if the complete capacitor is submerged in an oil with $\epsilon_{r} \approx 3$, the force is 3 times less. See the next problem.

Mayukh Banik

Numerade Educator

03:50

Problem 15

In Prob. 10-14 we found that the electric clamping force on a capacitor is larger by a factor of $\epsilon_{r}$ when there are air films between the electrodes and the dielectric, for a given applied voltage, or a given $E$. This is paradoxical. For a given $E$, the energy density is $\epsilon_{r}$ times larger in a dielectric than in air. Then the force should be $\epsilon_{r}$ times larger when there are no air films.
You can explain this paradox by considering the three capacitors of Fig. 10-12. In (2) the air film is much thinner than the dielectric. The electrode spacing is $s$ in all three capacitors. For each case find the surface charge densities $\sigma$ on the electrodes as well as $D, E$, and $\mathscr{E}$ in the air and in the dielectric.

Sheh Lit Chang

University of Washington

02:55

Problem 16

Electric forces on conductors immersed in liquid dielectrics are larger than in air by the factor $\epsilon_{r}$ if the voltages are the same. They are smaller than in air by the same factor $\epsilon$, if the charges are the same. Can you justify these general statements?
This is not in contradiction with Probs. $10-14$ and $10-15$, where we had a solid dielectric with thin films of air or oil next to the electrodes.

Dading Chen

Numerade Educator