Physics for Scientist and Engineers: A Strategic Approach

Randall Knight

Chapter 30

Potential and Field - all with Video Answers

Educators

Chapter Questions

Problem 1

What is the potential difference between $x_{i}=10 \mathrm{cm}$ and $x_{f}=$ $30 \mathrm{cm}$ in the uniform electric field $E_{x}=1000 \mathrm{V} / \mathrm{m} ?$

Dading Chen

Dading Chen

Numerade Educator

Problem 2

What is the potential difference between $y_{1}=-5 \mathrm{cm}$ and $y_{t}=$ $5 \mathrm{cm}$ in the uniform electric field $\vec{E}=(20,000 \hat{\imath}-50,000 \hat{\jmath}) \mathrm{V} / \mathrm{m} ?$

Dading Chen

Numerade Educator

Problem 3

fIGURE EX30.3 a is a graph of $E_{x} .$ What is the potential difference between $x_{1}=1.0 \mathrm{m}$ and $x_{t}=3.0 \mathrm{m} ?$
(FIGURE CANT COPY)

Dading Chen

Numerade Educator

Problem 4

FIGURE EX30.4 is a graph of $E_{x} .$ The potential at the origin is $-50 \mathrm{V} .$ What is the potential at $x=3.0 \mathrm{m} ?$
(FIGURE CANT COPY)

Dading Chen

Numerade Educator

Problem 5

How much work does the charge escalator do to move $1.0 \mu C$ of charge from the negative terminal to the positive terminal of a $1.5 \mathrm{V}$ battery $?$

Dading Chen

Numerade Educator

Problem 6

How much work does the electric motor of a Van de Graaff generator do to lift a positive ion $(q=e)$ if the potential of the spherical clectrode is $1.0 \mathrm{MV} ?$

Dading Chen

Numerade Educator

Problem 7

What is the emf of a battery that does $0.60 \mathrm{J}$ of work to transfer 0.050 C of charge from the negative to the positive terminal?

Dading Chen

Numerade Educator

Problem 8

Light from the sun allows a solar cell to move electrons from the positive to the negative terminal, doing $2.4 \times 10^{-19} \mathrm{J}$ of work per electron. What is the emf of this solar cell?

Dading Chen

Numerade Educator

Problem 9

What are the magnitude and direction of the electric field at the dot.
(FIGURE CANT COPY)

Vishal Gupta

Numerade Educator

Problem 10

What are the magnitude and direction of the electric field at the dot.
(FIGURE CANT COPY)

Dading Chen

Numerade Educator

Problem 11

Figure EX30.11 is a graph of $V$ versus $x$. Draw the corresponding graph of $E_{x}$ versus $x$.
(FIGURE CANT COPY)

Dading Chen

Numerade Educator

Problem 12

Figure EX30.12 is a graph of $V$ versus $x$. Draw the corresponding graph of $E_{x}$ versus $x$.
(FIGURE CANT COPY)

Dading Chen

Numerade Educator

Problem 13

The electric potential in a region of uniform electric field is $-1000 \mathrm{V}$ at $x=-1.0 \mathrm{m}$ and $+1000 \mathrm{V}$ at $x=+1.0 \mathrm{m} .$ What is $E_{x} ?$

Dading Chen

Numerade Educator

Problem 14

The electric potential along the $x$ -axis is $V=100 x^{2}$ V, where $x$ is in meters. What is $E_{x}$ at (a) $x=0$ m and (b) $x=1$ m?

Dading Chen

Numerade Educator

Problem 15

The electric potential along the $x$ -axis is $V=50 x-100 / x \mathbf{V}$ where $x$ is in meters. What is $E_{x}$ at (a) $x=1.0 \mathrm{m}$ and (b) $x=2.0 \mathrm{m} ?$

Dading Chen

Numerade Educator

Problem 16

What is the potential difference $\Delta V_{34}$.
(FIGURE CANT COPY)

Dading Chen

Numerade Educator

Problem 17

Two $2.0 \mathrm{cm} \times 2.0 \mathrm{cm}$ square aluminum electrodes are spaced $0.50 \mathrm{mm}$ apart. The electrodes are connected to a $100 \mathrm{V}$ battery.
a. What is the capacitance?
b. What is the charge on each electrode?

Dading Chen

Numerade Educator

Problem 18

You need to construct a 100 pF capacitor for a science project. You plan to cut two $L \times L$ metal squares and insert spacers between them. The thinnest spacers you have are 0.20 mm thick.

Dading Chen

Numerade Educator

Problem 19

A switch that connects a battery to a $10 \mu \mathrm{F}$ capacitor is closed. Several seconds later you find that the capacitor plates are charged to $\pm 30 \mu \mathrm{C}$. What is the emf of the battery?

Dading Chen

Numerade Educator

Problem 20

What is the emf of a battery that will charge a $2.0 \mu$ F capacitor to $\pm 48 \mu \mathrm{C} ?$

Dading Chen

Numerade Educator

Problem 21

Two electrodes connected to a $9.0 \mathrm{V}$ battery are charged to ±45 nC. What is the capacitance of the electrodes?

Dading Chen

Numerade Educator

Problem 22

A $6 \mu$ F capacitor, a $10 \mu F$ capacitor, and a $16 \mu F$ capacitor are connected in parallel. What is their equivalent capacitance?

Dading Chen

Numerade Educator

Problem 23

A $6 \mu$ F capacitor, a $10 \mu$ F capacitor, and a $16 \mu$ F capacitor are connected in series. What is their equivalent capacitance?

Dading Chen

Numerade Educator

Problem 24

You need a capacitance of $50 \mu \mathrm{F}$, but you don't happen to have
a $50 \mu \mathrm{F}$ capacitor. You do have a $30 \mu \mathrm{F}$ capacitor. What additional capacitor do you need to produce a total capacitance of $50 \mu$ F? Should you join the two capacitors in parallel or in series?

Dading Chen

Numerade Educator

Problem 25

You need a capacitance of $50 \mu \mathrm{F}$, but you don't happen to have a $50 \mu \mathrm{F}$ capacitor. You do have a $75 \mu \mathrm{F}$ capacitor. What additional capacitor do you need to produce a total capacitance of 50 \muF? Should you join the two capacitors in parallel or in series?

Dading Chen

Numerade Educator

Problem 26

What is the capacitance of the two metal spheres .
(FIGURE CANT COPY)

Dading Chen

Numerade Educator

Problem 27

Initially, the switch in Figure EX30.27 is open and the capacitor is uncharged. How much charge flows through the switch after the switch is closed?

Dading Chen

Numerade Educator

Problem 28

To what potential should you charge a $1.0 \mu \mathrm{F}$ capacitor to store $1.0 \mathrm{J}$ of energy?

Dading Chen

Numerade Educator

Problem 29

FIGURE EX30.29 shows $Q$ versus $t$ for a $2.0 \mu \mathrm{F}$ capacitor. Draw a graph showing $U_{\mathrm{c}}$ versus $t$.
(FIGURE CANT COPY)

Dading Chen

Numerade Educator

Problem 30

Capacitor 2 has half the capacitance and twice the potential difference as capacitor $1 .$ What is the ratio $U_{\mathrm{C} 1} / U_{\mathrm{C} 2} ?$

Dading Chen

Numerade Educator

Problem 31

$50 \mathrm{pJ}$ of energy is stored in a $2.0 \mathrm{cm} \times 2.0 \mathrm{cm} \times 2.0 \mathrm{cm}$
region of uniform electric field. What is the electric field strength?

Dading Chen

Numerade Educator

Problem 32

A 2.0 -cm-diameter parallel-plate capacitor with a spacing of $0.50 \mathrm{mm}$ is charged to $200 \mathrm{V}$. What are (a) the total energy stored in the electric field and (b) the energy density?

Dading Chen

Numerade Educator

Problem 33

Two 5.0 -cm-diameter metal disks are separated by a 0.20 -mm thick piece of paper.
a. What is the capacitance?
b. What is the maximum potential difference between the disks?

Dading Chen

Numerade Educator

Problem 34

Two $5.0 \mathrm{mm} \times 5.0 \mathrm{mm}$ electrodes with a 0.10 -mm-thick sheet of Mylar between them are attached to a $9.0 \mathrm{V}$ battery. Without disconnecting the battery, the Mylar is withdrawn. (Very small spacers keep the electrode separation unchanged.) What are the charge, potential difference, and electric field (a) before and (b) after the Mylar is withdrawn?

Dading Chen

Numerade Educator

Problem 35

Two 5.0 -cm-diameter metal disks separated by a $0.50-\mathrm{mm}$ thick piece of Pyrex glass are charged to a potential difference of 1000 V. What are (a) the surface charge density on the disks and (b) the surface charge density on the glass?

Dading Chen

Numerade Educator

Problem 36

a. Which point, A or B, has a larger electric potential?
b. What is the potential difference between $A$ and $B ?$
(FIGURE CANT COPY)

Dading Chen

Numerade Educator

Problem 37

The electric field in a region of space is $E_{x}=-1000 x \mathrm{V} / \mathrm{m}$ where $x$ is in meters.
a. Graph $E_{x}$ versus $x$ over the region $-1 \mathrm{m} \leq x \leq 1 \mathrm{m}$.
b. What is the potential difference between $x_{1}=-20 \mathrm{cm}$ and $x_{f}=30 \mathrm{cm} ?$.

Dading Chen

Numerade Educator

Problem 38

The electric field in a region of space is $E_{x}=5000 x \mathrm{V} / \mathrm{m}$ where $x$ is in meters.
a. Graph $E_{x}$ versus $x$ over the region $-1 \mathrm{m} \leq x \leq 1 \mathrm{m}$.
b. Find an expression for the potential $V$ at position $x$. As a reference, let $V=0$ V at the origin.
c. Graph $V$ versus $x$ over the region $-1 \mathrm{m} \leq x \leq 1 \mathrm{m}$.

Dading Chen

Numerade Educator

Problem 39

An infinitely long cylinder of radius $R$ has linear charge density $\lambda$. The potential on the surface of the cylinder is $V_{0},$ and the electric field outside the cylinder is $E_{r}=\lambda / 2 \pi \epsilon_{0} r .$ Find the potential relative to the surface at a point that is distance $r$ from the axis, assuming $r>R$.

Dading Chen

Numerade Educator

Problem 40

FIGURE EX30.40 shows $E_{x}$ the $x$ -component of the electric field, as a function of position along the $x$ -axis. Find and graph $V$ versus $x$ over the region $0 \mathrm{cm} \leq$ $x \leq 3 \mathrm{cm} .$ As a reference, let $V=0 \mathrm{V}$ at $x=3 \mathrm{cm}$.
(FIGURE CANT COPY)

Dading Chen

Numerade Educator

Problem 41

Figure EX30.41 is an edge view of three charged metal electrodes. Draw a graph of (a) $E_{x}$ versus $x$ and (b) $V$ versus $x$ over the region $0 \leq x \leq 3 \mathrm{cm}$.
(FIGURE CANT COPY)

Dading Chen

Numerade Educator

Problem 42

FIGURE EX30.42 shows a graph of $V$ versus $x$ in a region of space. The potential is independent of $y$ and $z$
a. Draw a graph of $E_{x}$ versus $x$
b. Draw a contour map of the potential in the $x y$ -plane in the square-shaped region $-3 \mathrm{m} \leq x \leq 3 \mathrm{m}$ and $-3 \mathrm{m} \leq y \leq$ 3 m. Show and label the $-10 \mathrm{V},-5 \mathrm{V}, 0 \mathrm{V},+5 \mathrm{V},$ and +10 V equipotential surfaces.
c. Draw electric field vectors on your contour map of part b.
(FIGURE CANT COPY)

Dading Chen

Numerade Educator

Problem 43

Use the on-axis potential of a charged disk from Chapter 29 to find the on-axis electric field of a charged disk.

Keshav Singh

Numerade Educator

Problem 44

a. Use the methods of Chapter 29 to find the potential at distance $x$ on the axis of the charged rod shown in FIGURE EX30.44.
b. Use the result of part a to find the electric field at distance $x$ on the axis of a rod.
(FIGURE CANT COPY)

Dading Chen

Numerade Educator

Problem 45

Determine the magnitude and direction of the electric field at points 1 and 2 in FIGURE EX30.45
(FIGURE CANT COPY)

Dading Chen

Numerade Educator

Problem 46

Shows a set of equipotential lines and five labeled points.
a. From measurements made on this figure with a ruler, using the scale on the figure, estimate the electric field strength $E$ at the five points indicated.
b. Trace the figure on your paper, then show the electric field vectors $\vec{E}$ at the five points.

Dading Chen

Numerade Educator

Problem 47

The electric potential in a region of space is $V=\left(150 x^{2}-\right.$ $\left.200 y^{2}\right) V,$ where $x$ and $y$ are in meters. What are the strength and direction of the electric field at $(x, y)=(2.0 \mathrm{m}, 2.0 \mathrm{m}) ?$ Give the direction as an angle cw or ccw (specify which) from the positive $x$ -axis.

Dading Chen

Numerade Educator

Problem 48

The electric potential in a region of space is $V=200 /$ $\sqrt{x^{2}+y^{2}},$ where $x$ and $y$ are in meters. What are the strength and direction of the electric field at $(x, y)=(2.0 \mathrm{m}, 1.0 \mathrm{m}) ?$ Give the direction as an angle cw or ccw (specify which) from the positive $x$ -axis.

Dading Chen

Numerade Educator

Problem 49

Shows the electric potential at points on a $5.0 \mathrm{cm} \times 5.0 \mathrm{cm}$ grid.
a Reproduce this figure on your paper, then draw the $50 \mathrm{V}$ $75 \mathrm{V},$ and $100 \mathrm{V}$ equipotential surfaces.
b. Estimate the electric field (strength and direction) at the points $A, B, C,$ and $D$
c. Draw the electric field vectors at points $A, B, C,$ and $D$ on your diagram.
$$\begin{array}{ccccc}0 & 25 & 50 & 50 & 50 \\25 & 50 & 75 & 100 & 50 \\50 & \begin{array}{c}c \\
a\end{array} & 75 & 100 & 100 \cdot^{B} & 50 \\25 & 50 & 75 & 100 & 50 \\0 & 25 & 50 & 50 & 50\end{array}$$

Dading Chen

Numerade Educator

Problem 50

Metal sphere 1 has a positive charge of $6.0 \mathrm{nC}$. Metal sphere
2. which is twice the diameter of sphere 1 , is initially uncharged. The spheres are then connected together by a long, thin metal wire. What are the final charges on each sphere?

Dading Chen

Numerade Educator

Problem 51

The metal spheres in FIGURE EX30.51 are charged to $\pm 300 \mathrm{V}$. Draw this figure on your paper, then draw a plausible contour map of the potential, showing and labeling the $-300 \mathrm{V}$ $-200 \mathrm{V},-100 \mathrm{V}, \ldots, 300 \mathrm{V}$ equipotential surfaces.
(FIGURE CANT COPY)

Dading Chen

Numerade Educator

Problem 52

The potential at the center of a 4.0 -cm-diameter copper sphere is $500 \mathrm{V},$ relative to $V=0 \mathrm{V}$ at infinity. How much excess charge is on the sphere?

Dading Chen

Numerade Educator

Problem 53

To see why charge density and electric field are larger at the sharp comers of a conductor, consider two metal spheres of radii $r_{1}=R$ and $r_{2}=2 R,$ both charged to the same potential $V_{0}$
a. What is the ratio $\eta_{1} / \eta_{2}$ of their surface charge densities?
b. What is the ratio $E_{1} / E_{2}$ of the electric field strengths at their surfaces?

Dading Chen

Numerade Educator

Problem 54

Two $2.0 \mathrm{cm} \times 2.0 \mathrm{cm}$ metal electrodes are spaced $1.0 \mathrm{mm}$ apart and connected by wires to the terminals of a $9.0 \mathrm{V}$ battery.
a. What are the charge on each electrode and the potential difference between them? The wires are disconnected, and insulated handles are used to pull the plates apart to a new spacing of $2.0 \mathrm{mm}$.
b. What are the charge on each electrode and the potential difference between them?

Dading Chen

Numerade Educator

Problem 55

Two $2.0 \mathrm{cm} \times 2.0 \mathrm{cm}$ metal electrodes are spaced $1.0 \mathrm{mm}$ apart and connected by wires to the terminals of a $9.0 \mathrm{V}$ battery.
a. What are the charge on each electrode and the potential difference between them? While the plates are still connected to the battery, insulated handles are used to pull them apart to a new spacing of $2.0 \mathrm{mm}$.
b. What are the charge on each electrode and the potential difference between them?

Dading Chen

Numerade Educator

Problem 56

A spherical capacitor with a 1.0 mm gap between the spheres has a capacitance of $100 \mathrm{pF}$. What are the diameters of the two

Dading Chen

Numerade Educator

Problem 57

Find expressions for the equivalent capacitance of (a) $N$ identical capacitors $C$ in parallel and (b) $N$ identical capacitors $C$ in series.

Dading Chen

Numerade Educator

Problem 58

What is the equivalent capacitance of the three capacitors in FIGURE EX30.58.
(FIGURE CANT COPY)

Dading Chen

Numerade Educator

Problem 59

What is the equivalent capacitance of the three capacitors in FIGURE EX30.59.
(FIGURE CANT COPY)

Dading Chen

Numerade Educator

Problem 60

What are the charge on and the potential difference across each capacitor in FiGuRE P30.50?
(FIGURE CANT COPY)

Dading Chen

Numerade Educator

Problem 61

What are the charge on and the potential difference across each capacitor in FIGURE P30.61?
(FIGURE CANT COPY)

Dading Chen

Numerade Educator

Problem 62

What are the charge on and the potential difference across each capacitor in FIGURE P30.62?
(FIGURE CANT COPY)

Dading Chen

Numerade Educator

Problem 63

You have three $12 \mu$ F capacitors. Draw diagrams showing how you could arrange all three so that their equivalent capacitance is (a) $4.0 \mu \mathrm{F},$ (b) $8.0 \mu \mathrm{F},$ (c) $18 \mu \mathrm{F},$ and (d) $36 \mu \mathrm{F}$.

Vishal Gupta

Numerade Educator

Problem 64

What is the capacitance of the three concentric metal spherical shells in FIGURE P30.64? Hint: Can you think of this as a combination of capacitors?
(FIGURE CANT COPY)

Dading Chen

Numerade Educator

Problem 65

Six identical capacitors with capacitance $C$ are connected as shown in FIGURE P30.65
a. What is the equivalent capacitance of these six capacitors?
b. What is the potential difference between points a and b?
(FIGURE CANT COPY)

Dading Chen

Numerade Educator

Problem 66

What is the capacitance of the two electrodes in FIGURE P30.66? Hint: Can you think of this as a combination of capacitors?
(FIGURE CANT COPY)

Dading Chen

Numerade Educator

Problem 67

Initially, the switch inFIGURE P30.67is in position $\mathrm{A}$ and capacitors $C_{2}$ and $C_{3}$ are uncharged. Then the switch is flipped to position $\mathbf{B}$. Afterward, what are the charge on and the potential difference across each capacitor?
(FIGURE CANT COPY)

Dading Chen

Numerade Educator

Problem 68

A battery with an emf of 60 $\mathrm{V}$ is connected to the two capacitors shown in FIGURE P30.68. Afterward, the charge on capacitor 2 is $450 \mu C$. What is the capacitance of capacitor $2 ?$
(FIGURE CANT COPY)

Dading Chen

Numerade Educator

Problem 69

Capacitors $C_{1}=10 \mu \mathrm{F}$ and $C_{2}=20 \mu \mathrm{F}$ are each charged to
10 $\mathrm{V}$, then disconnected from the battery without changing the charge on the capacitor plates. The two capacitors are then connected in parallel, with the positive plate of $C_{1}$ connected to the negative plate of $C_{2}$ and vice versa. Afterward, what are the charge on and the potential difference across each capacitor?

Dading Chen

Numerade Educator

Problem 70

An isolated $5.0 \mu \mathrm{F}$ parallel-plate capacitor has $4.0 \mathrm{mC}$ of charge. An extermal force changes the distance between the electrodes until the capacitance is $2.0 \mu \mathrm{F}$. How much work is done by the external force?

Dading Chen

Numerade Educator

Problem 71

A parallel-plate capacitor is constructed from two $10 \mathrm{cm} \times$ $10 \mathrm{cm}$ electrodes spaced $1.0 \mathrm{mm}$ apart. The capacitor plates are charged to ±10 nC, then disconnected from the battery.
a. How much energy is stored in the capacitor?
b. Insulating handles are used to pull the capacitor plates apart until the spacing is 2.0 mm. Now how much energy is stored in the capacitor?
c. Energy must be conserved. How do you account for the difference between a and b?

Dading Chen

Numerade Educator

Problem 72

What is the energy density in the electric field at the surface of a 1.0-cm-diameter sphere charged to a potential of $1000 \mathrm{V} ?$

Dading Chen

Numerade Educator

Problem 73

The flash unit in a camera uses a $3.0 \mathrm{V}$ battery to charge a capacitor. The capacitor is then discharged through a flashlamp. The discharge takes $10 \mu \mathrm{s}$, and the average power dissipated in the flashlamp is $10 \mathrm{W}$. What is the capacitance of the capacitor?

Dading Chen

Numerade Educator

Problem 74

You need to melt a $0.50 \mathrm{kg}$ block of ice at $-10^{\circ} \mathrm{C}$ in a hurry. The stove isn't working, but you do have a 50 V battery. It occurs to you that you could build a capacitor from a couple of pieces of sheet metal that are nearby, charge the capacitor with the battery, then discharge it through the block of ice. If you use square sheets spaced $2.0 \mathrm{mm}$ apart, what must the dimensions of the sheets be to accomplish your goal? Is this feasible?

Dading Chen

Numerade Educator

Problem 75

Derive Equation 30.34 for the induced surface charge density on the dielectric in a capacitor.

Dading Chen

Numerade Educator

Problem 76

The radiation detector known as a Geiger counter uses a closed, hollow, cylindrical tube with an insulated wire along its axis. Suppose a Geiger tube, as it's called, has a 1.0 -mm-diameter wire in a tube with a 25 mm inner diameter. The tube is filled with a low-pressure gas whose dielectric strength is $1.0 \times$ $10^{6} \mathrm{V} / \mathrm{m} .$ What is the maximum potential difference between the wire and the tube?

Dading Chen

Numerade Educator

Problem 77

A vacuum-insulated parallel-plate capacitor with plate separation $d$ has capacitance $C_{0} .$ What is the capacitance if an insulator with dielectric constant $\kappa$ and thickness $d / 2$ is slipped between the electrodes?

Dading Chen

Numerade Educator

Problem 78

Given the equation(s) used to solve.
a problem. For each of these, you are to
a. Write a realistic problem for which this is the correct equation(s).
b. Finish the solution of the problem.
$$2 z \mathrm{V} / \mathrm{m}=-\frac{d V}{d z}$$

Dading Chen

Numerade Educator

Problem 79

Given the equation(s) used to solve.
a problem. For each of these, you are to
a. Write a realistic problem for which this is the correct equation(s).
b. Finish the solution of the problem.
$$\begin{aligned}&400 \mathrm{nC}=(100 \mathrm{V}) C\\&C=\frac{\left(8.85 \times 10^{-12} \mathrm{C}^{2} / \mathrm{Nm}^{2}\right)(0.10 \mathrm{m} \times 0.10 \mathrm{m})}{d}\end{aligned}$$

Dading Chen

Numerade Educator

Problem 80

Given the equation(s) used to solve.
a problem. For each of these, you are to
a. Write a realistic problem for which this is the correct equation(s).
b. Finish the solution of the problem.
$$\left(\frac{1}{3 \mu \mathrm{F}}+\frac{1}{6 \mu \mathrm{F}}\right)^{-1}+C=4 \mu \mathrm{F}$$

Dading Chen

Numerade Educator

Problem 81

The electric potential in a region of space is $V=100\left(x^{2}-\right.$ $y^{2}$ ) $V,$ where $x$ and $y$ are in meters.
a. Draw a contour map of the potential, showing and labeling the $-400 \mathrm{V},-100 \mathrm{V}, 0 \mathrm{V},+100 \mathrm{V},$ and $+400 \mathrm{V}$ equipotential surfaces.
b. Find an expression for the electric field $\vec{E}$ at position $(x, y)$
c. Draw the electric field lines on your diagram of part a.

Dading Chen

Numerade Educator

Problem 82

An electric dipole at the origin consists of two charges $\pm q$ spaced distance $s$ apart along the $y$ -axis.
a. Find an expression for the potential $V(x, y)$ at an arbitrary point in the $x y$ -plane. Your answer will be in terms of $q, s, x$ and y.
b. Use the binomial approximation to simplify your result of part a when $s \ll x$ and $s \ll y .$
c. Assuming $s \ll x$ and $y,$ find expressions for $E_{x}$ and $E_{y}$, the components of $\overrightarrow{\boldsymbol{E}}$ for a dipole.
d. What is the on-axis field $\vec{E}$ ? Does your result agree with Equation $27.11 ?$
c. What is the field $\vec{E}$ on the bisecting axis? Does your result agree with Equation $27.12 ?$

Dading Chen

Numerade Educator

Problem 83

Charge is uniformly distributed with charge density $\rho$ inside a very long cylinder of radius $R$. Find the potential difference between the surface and the axis of the cylinder.

Dading Chen

Numerade Educator

Problem 84

Consider a uniformly charged sphere of radius $R$ and total charge $Q$. The electric field $E_{\text {out }}$ outside the sphere $(r \geq R)$ is simply that of a point charge $Q$. In Chapter $28,$ we used Gauss's law to find that the electric field $E_{\mathrm{i}}$ inside the sphere $(r \leq R)$ is radially outward with field strength $$E_{\mathrm{ia}}=\frac{1}{4 \pi \epsilon_{0}} \frac{Q}{R^{3}} r$$,
a. Graph $E$ versus $r$ for $0 \leq r \leq 3 R$
b. The electric potential $V_{\text {out }}$ outside the sphere is that of a point charge $Q$. Find an expression for the electric potential $V_{\text {in }}$ at position $r$ inside the sphere. As a reference, let $V_{\text {in }}=V_{\text {cut }}$ at the surface of the sphere.
c. What is the ratio $V_{\text {conse }} / V_{\text {rutices }} ?$
d. Graph $V$ versus $r$ for $0 \leq r \leq 3 R$.

Keshav Singh

Numerade Educator

Problem 85

High-frequency signals are often transmitted along a coaxial cable, such as the one shown in FIGURE P30.85. For example, the cable TV hookup coming into your home is a coaxial cable. The signal is carried on a wire of radius $R_{1}$ while the outer conductor of radius $R_{2}$ is grounded (i.e., at $V=0 \mathrm{V}$ ). An insulating them, and an insulating plastic coating goes around the outside.
a. Find an expression for the capacitance per meter of a coaxial cable. Assume that the insulating material between the cylinders is air.
b. Evaluate the capacitance per meter of a cable having $R_{1}=$ $0.50 \mathrm{mm}$ and $R_{2}=3.0 \mathrm{mm}$
(FIGURE CANT COPY)

Keshav Singh

Numerade Educator

Problem 86

Each capacitor in FIGURE P30.86 has capacitance $\mathcal{C}$. What is the equivalent capacitance between points a and b?
(FIGURE CANT COPY)

Dading Chen

Numerade Educator