# Chapter 19

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# Chapter 19 : Electric Potential and Electric Field - all with Video Solutions

### Problem 1

Find the ratio of speeds of an electron and a negative hydrogen ion (one having an extra electron) accelerated through the same voltage, assuming non-relativistic final speeds. Take the mass of the hydrogen ion to be $1.67 \times 10^{-27} \textrm{ kg}$.

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### Problem 2

An evacuated tube uses an accelerating voltage of 40 kV to accelerate electrons to hit a copper plate and produce x rays. Non-relativistically, what would be the maximum speed of these electrons?

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### Problem 3

A bare helium nucleus has two positive charges and a mass of $6.64 \times 10^{-27} \textrm{ kg}$ (a) Calculate its kinetic energy in joules at 2.00% of the speed of light. (b) What is this in electron volts? (c) What voltage would be needed to obtain this energy?

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### Problem 4

Singly charged gas ions are accelerated from rest through a voltage of 13.0 V. At what temperature will the average kinetic energy of gas molecules be the same as that given these ions?

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### Problem 5

The temperature near the center of the Sun is thought to be 15 million degrees Celsius ($1.5 \times 10^{7}\textrm{ }^\circ\textrm{C}$). Through what voltage must a singly charged ion be accelerated to have the same energy as the average kinetic energy of ions at this temperature?

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### Problem 6

(a) What is the average power output of a heart defibrillator that dissipates 400 J of energy in 10.0 ms? (b) Considering the high-power output, why doesn’t the defibrillator produce serious burns?

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### Problem 7

A lightning bolt strikes a tree, moving 20.0 C of charge through a potential difference of $1.00 \times 10^2 \textrm{ MV}$. (a) What energy was dissipated? (b) What mass of water could be raised from $15^\circ\textrm{C}$ to the boiling point and then boiled by this energy? (c) Discuss the damage that could be caused to the tree by the expansion of the boiling steam.

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### Problem 8

A $12.0\textrm{ V}$ battery operated bottle warmer heats 50.0 g of glass, $2.50\times 10^{2}\textrm{ g}$ of baby formula, and $2.00\times 10^{2}\textrm{ g}$ of aluminum from $20.0^\circ\textrm{ C}$ to $90.0^\circ\textrm{C}$. (a) How much charge is moved by the battery? (b) How many electrons per second flow if it takes 5.00 min to warm the formula? (Hint: Assume that the specific heat of baby formula is about the same as the specific heat of water.)

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### Problem 9

A battery-operated car utilizes a 12.0 V system. Find the charge the batteries must be able to move in order to accelerate the 750 kg car from rest to 25.0 m/s, make it climb a $2.00 \times 10^2 \textrm{ m}$ high hill, and then cause it to travel at a constant 25.0 m/s by exerting a $5.00 \times 10^2 \textrm{ N}$ force for an hour.

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### Problem 10

Fusion probability is greatly enhanced when appropriate nuclei are brought close together, but mutual Coulomb repulsion must be overcome. This can be done using the kinetic energy of high-temperature gas ions or by accelerating the nuclei toward one another. (a) Calculate the potential energy of two singly charged nuclei separated by $1.00\times 10^{-12}\textrm{ m}$ by finding the voltage of one at that distance and multiplying by the charge of the other. (b) At what temperature will atoms of a gas have an average kinetic energy equal to this needed electrical potential energy?

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### Problem 11

(a) Find the voltage near a 10.0 cm diameter metal sphere that has 8.00 C of excess positive charge on it. (b) What is unreasonable about this result? (c) Which assumptions are responsible?

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### Problem 14

What is the strength of the electric field between two parallel conducting plates separated by 1.00 cm and having a potential difference (voltage) between them of $1.50\times 10^{4}\textrm{ V}$?

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### Problem 15

The electric field strength between two parallel conducting plates separated by 4.00 cm is $7.50 \times 10^4 \textrm{ V/m}$. (a) What is the potential difference between the plates? (b) The plate with the lowest potential is taken to be at zero volts. What is the potential 1.00 cm from that plate (and 3.00 cm from the other)?

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### Problem 16

How far apart are two conducting plates that have an electric field strength of $4.50\times 10^{3}\textrm{ V/m}$ between them, if their potential difference is 15.0 kV?

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### Problem 17

(a) Will the electric field strength between two parallel conducting plates exceed the breakdown strength for air ($3.0 \times 10^6 \textrm{ V/m}$) if the plates are separated by 2.00 mm and a potential difference of $5.0 \times 10^3 \textrm{ V}$ is applied? (b) How close together can the plates be with this applied voltage?

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### Problem 18

The voltage across a membrane forming a cell wall is 80.0 mV and the membrane is 9.00 nm thick. What is the electric field strength? (The value is surprisingly large, but correct. Membranes are discussed in Capacitors and Dielectrics and Nerve Conduction—Electrocardiograms.) You may assume a uniform electric field.

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### Problem 19

Membrane walls of living cells have surprisingly large electric fields across them due to separation of ions. (Membranes are discussed in some detail in Nerve Conduction—Electrocardiograms.) What is the voltage across an 8.00 nm–thick membrane if the electric field strength across it is 5.50 MV/m? You may assume a uniform electric field.

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### Problem 20

Two parallel conducting plates are separated by 10.0 cm, and one of them is taken to be at zero volts. (a) What is the electric field strength between them, if the potential 8.00 cm from the zero volt plate (and 2.00 cm from the other) is 450 V? (b) What is the voltage between the plates?

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### Problem 21

Find the maximum potential difference between two parallel conducting plates separated by 0.500 cm of air, given the maximum sustainable electric field strength in air to be $3.0 \times 10^6 \textrm{ V/m}$

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### Problem 22

A doubly charged ion is accelerated to an energy of 32.0 keV by the electric field between two parallel conducting plates separated by 2.00 cm. What is the electric field strength between the plates?

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### Problem 23

An electron is to be accelerated in a uniform electric field having a strength of $2.00 \times 10^6 \textrm{ V/m}$. (a) What energy in keV is given to the electron if it is accelerated through 0.400 m? (b) Over what distance would it have to be accelerated to increase its energy by 50.0 GeV?

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### Problem 25

What is the potential $0.530 \times 10^{-10} \textrm{ m}$ from a proton (the average distance between the proton and electron in a hydrogen atom)?

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### Problem 26

(a) A sphere has a surface uniformly charged with 1.00 C. At what distance from its center is the potential 5.00 MV? (b) What does your answer imply about the practical aspect of isolating such a large charge?

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### Problem 27

How far from a $1.00 \textrm{ }\mu\textrm{C}$ point charge will the potential be 100 V? At what distance will it be $2.00 \times 10^2 \textrm{ V}$?

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### Problem 29

If the potential due to a point charge is $5.00 \times 10^2 \textrm{ V}$ at a distance of 15.0 m, what are the sign and magnitude of the charge?

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### Problem 30

In nuclear fission, a nucleus splits roughly in half. (a) What is the potential $2.00\times 10^{-14}\textrm{ m}$ from a fragment that has 46 protons in it? (b) What is the potential energy in MeV of a similarly charged fragment at this distance?

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### Problem 31

A research Van de Graaff generator has a 2.00-m- diameter metal sphere with a charge of 5.00 mC on it. (a) What is the potential near its surface? (b) At what distance from its center is the potential 1.00 MV? (c) An oxygen atom with three missing electrons is released near the Van de Graaff generator. What is its energy in MeV at this distance?

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### Problem 32

An electrostatic paint sprayer has a 0.200-m-diameter metal sphere at a potential of 25.0 kV that repels paint droplets onto a grounded object. (a) What charge is on the sphere? (b) What charge must a 0.100-mg drop of paint have to arrive at the object with a speed of 10.0 m/s?

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### Problem 33

In one of the classic nuclear physics experiments at the beginning of the 20th century, an alpha particle was accelerated toward a gold nucleus, and its path was substantially deflected by the Coulomb interaction. If the energy of the doubly charged alpha nucleus was 5.00 MeV, how close to the gold nucleus (79 protons) could it come before being deflected?

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### Problem 34

(a) What is the potential between two points situated 10 cm and 20 cm from a $3.0\textrm{ }\mu\textrm{C}$ point charge? (b) To what location should the point at 20 cm be moved to increase this potential difference by a factor of two?

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### Problem 35

(a) What is the final speed of an electron accelerated from rest through a voltage of 25.0 MV by a negatively charged Van de Graaff terminal? (b) What is unreasonable about this result? (c) Which assumptions are responsible?

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### Problem 36

(a) Sketch the equipotential lines near a point charge $+q$. Indicate the direction of increasing potential. (b) Do the same for a point charge $-3q$.

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### Problem 38

Figure 19.34 shows the electric field lines near two charges $q_1$ and $q_2$ , the first having a magnitude four times that of the second. Sketch the equipotential lines for these two charges, and indicate the direction of increasing potential.

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### Problem 40

Sketch the equipotential lines in the vicinity of two opposite charges, where the negative charge is three times as great in magnitude as the positive. See Figure 19.34 for a similar situation. Indicate the direction of increasing potential.

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### Problem 42

Sketch the equipotential lines surrounding the two conducting plates shown in Figure 19.36, given the top plate is positive and the bottom plate has an equal amount of negative charge. Be certain to indicate the distribution of charge on the plates. Is the field strongest where the plates are closest? Why should it be?

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### Problem 43

(a) Sketch the electric field lines in the vicinity of the charged insulator in Figure 19.37. Note its non-uniform charge distribution. (b) Sketch equipotential lines surrounding the insulator. Indicate the direction of increasing potential.

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### Problem 44

The naturally occurring charge on the ground on a fine day out in the open country is –1.00 nC/m2 . (a) What is the electric field relative to ground at a height of 3.00 m? (b) Calculate the electric potential at this height. (c) Sketch electric field and equipotential lines for this scenario.

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### Problem 45

The lesser electric ray (Narcine bancroftii) maintains an incredible charge on its head and a charge equal in magnitude but opposite in sign on its tail (Figure 19.38). (a) Sketch the equipotential lines surrounding the ray. (b) Sketch the equipotentials when the ray is near a ship with a conducting surface. (c) How could this charge distribution be of use to the ray?

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### Problem 48

What charge is stored on a parallel plate capacitor with metal plates, each of area $1.00\textrm{ m}^2$, separated by 1.00 mm, when $3.00\times 10^{3}\textrm{ V}$ is applied across it?

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### Problem 49

Calculate the voltage applied to a $2.00 \textrm{ }\mu\textrm{F}$ capacitor when it holds $3.10 \textrm{ }\mu\textrm{C}$ of charge.

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### Problem 54

(a) What is the capacitance of a parallel plate capacitor having plates of area $1.50\textrm{ m}^2$ that are separated by 0.0200 mm of neoprene rubber? (b) What charge does it hold when 9.00 V is applied to it?

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### Problem 55

A prankster applies 450 V to an $80.0 \textrm{ }\mu\textrm{F}$ capacitor and then tosses it to an unsuspecting victim. The victim’s finger is burned by the discharge of the capacitor through 0.200 g of flesh. What is the temperature increase of the flesh? Is it reasonable to assume no phase change?

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### Problem 56

(a) A certain parallel plate capacitor has plates of area $4.00\textrm{ m}^2$, separated by 0.0100 mm of nylon, and stores 0.170 C of charge. What is the applied voltage? (b) What is unreasonable about this result? (c) Which assumptions are responsible or inconsistent?

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### Problem 58

Suppose you want a capacitor bank with a total capacitance of 0.750 F and you possess numerous 1.50 mF capacitors. What is the smallest number you could hook together to achieve your goal, and how would you connect them?

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### Problem 59

What total capacitances can you make by connecting a $5.00 \textrm{ }\mu\textrm{F}$ and an $8.00 \textrm{ }\mu\textrm{F}$ capacitor together?

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### Problem 62

(a) An $8.00\textrm{ }\mu\textrm{F}$ capacitor is connected in parallel to another capacitor, producing a total capacitance of $5.00\textrm{ }\mu\textrm{F}$ . What is the capacitance of the second capacitor? (b) What is unreasonable about this result? (c) Which assumptions are unreasonable or inconsistent?

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### Problem 63

(a) What is the energy stored in the $10.0\textrm{ }\mu\textrm{F}$ capacitor of a heart defibrillator charged to $9.00 \times 10^3 \textrm{ V}$? (b) Find the amount of stored charge.

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### Problem 64

In open heart surgery, a much smaller amount of energy will defibrillate the heart. (a) What voltage is applied to the $8.00\textrm{ }\mu\textrm{F}$ capacitor of a heart defibrillator that stores 40.0 J of energy? (b) Find the amount of stored charge.

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### Problem 66

Suppose you have a 9.00 V battery, a $2.00\textrm{ }\mu\textrm{F}$ capacitor, and a $7.40\textrm{ }\mu\textrm{F}$ capacitor. (a) Find the charge and energy stored if the capacitors are connected to the battery in series. (b) Do the same for a parallel connection.

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### Problem 67

A nervous physicist worries that the two metal shelves of his wood frame bookcase might obtain a high voltage if charged by static electricity, perhaps produced by friction. (a) What is the capacitance of the empty shelves if they have area $1.00 \times 10^2 \textrm{ m}^2$ and are 0.200 m apart? (b) What is the voltage between them if opposite charges of magnitude 2.00 nC are placed on them? (c) To show that this voltage poses a small hazard, calculate the energy stored.

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### Problem 68

Show that for a given dielectric material the maximum energy a parallel plate capacitor can store is directly proportional to the volume of dielectric ( $\textrm{Volume} = A \cdot d$). Note that the applied voltage is limited by the dielectric strength.

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### Problem 70

(a) On a particular day, it takes $9.60\times 10^{3}\textrm{ J}$ of electric energy to start a truck’s engine. Calculate the capacitance of a capacitor that could store that amount of energy at 12.0 V. (b) What is unreasonable about this result? (c) Which assumptions are responsible?

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## Test Prep for AP® Courses

### Problem 1 (AP)

An electron is placed in an electric field of $12.0 \textrm{ N/C}$ to the right. What is the resulting force on the electron?
1. $1.33 \times 10^{-20} \textrm{ N right}$
2. $1.33 \times 10^{-20} \textrm{ N left}$

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### Problem 9 (AP)

What is the internal energy of a system consisting of two point charges, one 2.0 μC, and the other −3.0 μC, placed 1.2 m away from each other?
1. $-3.8 \times 10^{-2} \textrm{ J}$
2. $-4.5 \times 10^{-2} \textrm{ J}$
3. $4.5 \times 10^{-2} \textrm{ J}$
4. $3.8 \times 10^{-2} \textrm{ J}$

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### Problem 11 (AP)

A system has 2.00 μC charges at (50 cm, 0) and (−50 cm, 0) and a −1.00 μC charge at (0, 70 cm). As the y-coordinate of the −1.00 μC charge increases, the potential energy ___. As the y-coordinate of the −1.00 μC charge decreases, the potential energy ___.
1. increases, increases
2. increases, decreases
3. decreases, increases
4. decreases, decreases

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### Problem 12 (AP)

A system of three point charges has a 1.00 μC charge at the origin, a −2.00 μC charge at x=30 cm, and a 3.00 μC charge at x=70 cm. What happens to the total potential energy of this system if the −2.00 μC charge and the 3.00 μC charge trade places?

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### Problem 13 (AP)

Take a square configuration of point charges, two positive and two negative, all of the same magnitude, with like charges sharing diagonals. What will happen to the internal energy of this system if one of the negative charges becomes a positive charge of the same magnitude?
1. increase
2. decrease
3. no change
4. not enough information

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### Problem 14 (AP)

Take a square configuration of point charges, two positive and two negative, all of the same magnitude, with like charges sharing diagonals. What will happen to the internal energy of this system if the sides of the square decrease in length?

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### Problem 15 (AP)

A system has 2.00 μC charges at (50 cm, 0) and (−50 cm, 0) and a −1.00 μC charge at (0, 70 cm), with a velocity in the –y-direction. When the −1.00 μC charge is at (0, 0) the potential energy is at a ___ and the kinetic energy is ___.
1. maximum, maximum
2. maximum, minimum
3. minimum, maximum
4. minimum, minimum

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### Problem 17 (AP)

A negatively charged massive particle is dropped from above the two plates in Figure 19.7 into the space between them. Which best describes the trajectory it takes?
1. A rightward-curving parabola
2. A leftward-curving parabola
3. A rightward-curving section of a circle
4. A leftward-curving section of a circle

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### Problem 18 (AP)

Two massive particles with identical charge are launched into the uniform field between two plates from the same launch point with the same velocity. They both impact the positively charged plate, but the second one does so four times as far as the first. What sign is the charge? What physical difference would give them different impact points (quantify as a relative percent)? How does this compare to the gravitational projectile motion case?

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### Problem 19 (AP)

Two plates are lying horizontally, but stacked with one 10.0 cm above the other. If the upper plate is held at +100 V, what is the magnitude and direction of the electric field between the plates if the lower is held at +50.0 V? -50.0 V?
1. 500 V/m, 1500 V/m, down
2. 500 V/m, 1500 V/m, up
3. 1500 V/m, 500 V/m, down
4. 1500 V/m, 500 V/m, up

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### Problem 20 (AP)

Two parallel conducting plates are 15 cm apart, each with an area of $0.75\textrm{ m}^2$. The left one has a charge of -0.225 C placed on it, while the right has a charge of 0.225 C. What is the magnitude and direction of the electric field between the two?

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### Problem 21 (AP)

Consider three parallel conducting plates, with a space of 3.0 cm between them. The leftmost one is at a potential of +45 V, the middle one is held at ground, and the rightmost is at a potential of -75 V. What is the magnitude of the average electric field on an electron traveling between the plates? (Assume that the middle one has holes for the electron to go through.)
1. 1500 V/m
2. 2500 V/m
3. 4000 V/m
4. 2000 V/m

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### Problem 22 (AP)

A new kind of electron gun has a rear plate at −25.0 kV, a grounded plate 2.00 cm in front of that, and a +25.0 kV plate 4.00 cm in front of that. What is the magnitude of the average electric field?

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### Problem 23 (AP)

A certain electric potential isoline graph has isolines every 5.0 V. If six of these lines cross a 40 cm path drawn between two points of interest, what is the (magnitude of the average) electric field along this path?
1. 750 V/m
2. 150 V/m
3. 38 V/m
4. 75 V/m

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### Problem 24 (AP)

Given a system of two parallel conducting plates held at a fixed potential difference, describe what happens to the isolines of the electric potential between them as the distance between them is changed. How does this relate to the electric field strength?

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### Problem 25 (AP)

How would Figure 19.15 be different with two positive charges replacing the two negative charges?
1. The equipotential lines would have positive values.
2. It would actually resemble Figure 19.14.
3. no change
4. not enough information

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### Problem 26 (AP)

Consider two conducting plates, placed on adjacent sides of a square, but with a 1-m space between the corner of the square and the plate. These plates are not touching, not centered on each other, but are at right angles. Each plate is 1 m wide. If the plates are held at a fixed potential difference $\Delta V$, draw the equipotential lines for this system.

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### Problem 27 (AP)

As isolines of electric potential get closer together, the electric field gets stronger. What shape would a hill have as the isolines of gravitational potential get closer together?
1. constant slope
2. steeper slope
3. shallower slope
4. a U-shape

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### Problem 29 (AP)

How much work is necessary to keep a positive point charge in orbit around a negative point charge?
1. A lot; this system is unstable.
2. Just a little; the isolines are far enough apart that crossing them doesn’t take much work.
3. None; we’re traveling along an isoline, which requires no work.
4. There’s not enough information to tell.

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### Problem 30 (AP)

Consider two conducting plates, placed on adjacent sides of a square, but with a 1-m space between the corner of the square and the plate. These plates are not touching, not centered on each other, but are at right angles. Each plate is 1 m wide. If the plates are held at a fixed potential difference ΔV, sketch the path of both a positively charged object placed between the near ends, and a negatively charged object placed near the open ends.

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### Problem 31 (AP)

Two parallel plate capacitors are otherwise identical, except the second one has twice the distance between the plates of the first. If placed in otherwise identical circuits, how much charge will the second plate have on it compared to the first?
1. four times as much
2. twice as much
3. the same
4. half as much

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### Problem 33 (AP)

A parallel plate capacitor with adjustable-size square plates is placed in a circuit. How does the charge on the capacitor change as the length of the sides of the plates is increased?
1. it grows proportional to $\textrm{length}^2$
2. it grows proportional to length
3. it shrinks proportional to length
4. it shrinks proportional to $\textrm{length}^2$

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### Problem 35 (AP)

A student was changing one of the dimensions of a square parallel plate capacitor and measuring the resultant charge in a circuit with a battery. However, the student forgot which dimension was being varied, and didn’t write it or any units down. Given the table, which dimension was it?
 Dimension 1 1.1 1.2 1.3 Charge ($\mu\textrm{C}$) 0.5 0.61 0.71 0.86
1. The distance between the plates
2. The area
3. The length of a side
4. Both the area and the length of a side

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### Problem 36 (AP)

In an experiment in which a circular parallel plate capacitor in a circuit with a battery has the radius and plate separation grow at the same relative rate, what will happen to the total charge on the capacitor?

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### Problem 37 (AP)

Consider a parallel plate capacitor, with no dielectric material, attached to a battery with a fixed voltage. What happens when a dielectric is inserted into the capacitor?
1. Nothing changes, except now there is a dielectric in the capacitor.
2. The energy in the system decreases, making it very easy to move the dielectric in.
3. You have to do work to move the dielectric, increasing the energy in the system.
4. The reversed polarity destroys the battery.

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### Problem 38 (AP)

Consider a parallel plate capacitor with no dielectric material. It was attached to a battery with a fixed voltage to charge up, but now the battery has been disconnected. What happens to the energy of the system and the dielectric material when a dielectric is inserted into the capacitor?

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### Problem 39 (AP)

What happens to the energy stored in a circuit as you increase the number of capacitors connected in parallel? Series?
1. increases, increases
2. increases, decreases
3. decreases, increases
4. decreases, decreases

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### Problem 41 (AP)

Consider a parallel plate capacitor with metal plates, each of square shape of 1.00 m on a side, separated by 1.00 mm. What is the energy of this capacitor with $3.00 \times 10^3 \textrm{ V}$ applied to it?
1. $3.98 \times 10^{-2} \textrm{ J}$
2. $5.08 \times 10^{14} \textrm{ J}$
3. $1.33 \times 10^{-5} \textrm{ J}$
4. $1.69 \times 10^{11} \textrm{ J}$

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### Problem 42 (AP)

Consider a parallel plate capacitor with metal plates, each of square shape of 1.00 m on a side, separated by 1.00 mm. What is the internal energy stored in this system if the charge on the capacitor is 30.0 μC?

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### Problem 43 (AP)

Consider a parallel plate capacitor with metal plates, each of square shape of 1.00 m on a side, separated by 1.00 mm. If the plates grow in area while the voltage is held fixed, the capacitance ___ and the stored energy ___.
1. decreases, decreases
2. decreases, increases
3. increases, decreases
4. increases, increases

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### Problem 44 (AP)

Consider a parallel plate capacitor with metal plates, each of square shape of 1.00 m on a side, separated by 1.00 mm. What happens to the energy of this system if the area of the plates increases while the charge remains fixed?

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