College Physics With an Integrated Approach to Forces and Kinematics

Alan Giambattista, Betty McCarthy Richardson , Robert C. Richardson

Chapter 17

Electric Potential - all with Video Answers

Educators

Chapter Questions

Problem 1

Two point charges, $+5.0 \mu \mathrm{C}$ and $-2.0 \mu \mathrm{C}$, are separated by $5.0 \mathrm{~m}$. What is the electric potential energy?

Mayukh Banik

Mayukh Banik

Numerade Educator

Problem 2

A hydrogen atom has a single proton at its center and a single electron at a distance of approximately $0.0529 \mathrm{~nm}$ from the proton. (a) What is the electric potential energy in joules? (b) What is the significance of the sign of the answer?

Mohamed Mustafa

Mohamed Mustafa

Numerade Educator

Problem 3

How much work is done by an applied force that moves two charges of $6.5 \mu \mathrm{C}$ that are initially very far apart to a distance of $4.5 \mathrm{~cm}$ apart?

Mayukh Banik

Numerade Educator

Problem 4

The nucleus of a helium atom contains two protons that are approximately $1 \mathrm{fm}$ apart. How much work must be done by an external agent to bring the two protons from an infinite separation to a separation of $1.0 \mathrm{fm}$ ?

Mohamed Mustafa

Mohamed Mustafa

Numerade Educator

Problem 5

How much work does it
take for an external force to set up the arrangement of charged objects in the diagram on the corners of a right triangle when the three objects are initially very far away from each other?

Mayukh Banik

Numerade Educator

Problem 6

Two point charges $(+10.0 \mathrm{nC}$ and $-10.0 \mathrm{nC})$ are located $8.00 \mathrm{~cm}$ apart. For each problem, let $U=0$ when all of the charges are separated by infinite distances.
What is the potential energy for these two charges?

Mohamed Mustafa

Mohamed Mustafa

Numerade Educator

Problem 7

Two point charges $(+10.0 \mathrm{nC}$ and $-10.0 \mathrm{nC})$ are located $8.00 \mathrm{~cm}$ apart. For each problem, let $U=0$ when all of the charges are separated by infinite distances.
What is the potential energy if a third point charge $q=-4.2 \mathrm{nC}$ is placed at point $a ?$

Nafis Fuad

Numerade Educator

Problem 8

Two point charges $(+10.0 \mathrm{nC}$ and $-10.0 \mathrm{nC})$ are located $8.00 \mathrm{~cm}$ apart. For each problem, let $U=0$ when all of the charges are separated by infinite distances.
What is the potential energy if a third point charge $q=-4.2 \mathrm{nC}$ is placed at point $b$ ?

Mohamed Mustafa

Mohamed Mustafa

Numerade Educator

Problem 9

Two point charges $(+10.0 \mathrm{nC}$ and $-10.0 \mathrm{nC})$ are located $8.00 \mathrm{~cm}$ apart. For each problem, let $U=0$ when all of the charges are separated by infinite distances.
What is the potential energy if a third point charge $q=-4.2 \mathrm{nC}$ is placed at point $c$ ?

Mayukh Banik

Numerade Educator

Problem 10

Find the electric potential energy for the following array of charges: charge $q_{1}=+4.0 \mu \mathrm{C}$ is located at $(x, y)=(0.0,0.0) \mathrm{m} ;$ charge $q_{2}=+3.0 \mu \mathrm{C}$ is located
at $(4.0,3.0) \mathrm{m}$; and charge $q_{3}=-1.0 \mu \mathrm{C}$ is located at $(0.0,3.0) \mathrm{m}$.

Mohamed Mustafa

Mohamed Mustafa

Numerade Educator

Problem 11

In the diagram, how much work is done by the electric field as a third charge $q_{3}=+2.00 \mathrm{nC}$ is moved from infinity to point $a$ ?

Nafis Fuad

Numerade Educator

Problem 12

In the diagram, how much work is done by the electric field as a third charge $q_{3}=+2.00 \mathrm{n} \mathrm{C}$ is moved from infinity to point $b$ ?

Mohamed Mustafa

Mohamed Mustafa

Numerade Educator

Problem 13

In the diagram, how much work is done by the electric field as a third charge $q_{3}=+2.00 \mathrm{nC}$ is moved from point $a$ to point $b ?$

Nafis Fuad

Numerade Educator

Problem 14

In the diagram, how much work is done by the electric field as a third charge $q_{3}=+2.00 \mathrm{nC}$ is moved from point $b$ to point $c$ ?

Mohamed Mustafa

Mohamed Mustafa

Numerade Educator

Problem 15

A point charge $q=+3.0 \mathrm{nC}$ moves through a potential difference $\Delta V=V_{\mathrm{f}}-V_{\mathrm{i}}=+25 \mathrm{~V}$. What is the change in the electric potential energy?

Mayukh Banik

Numerade Educator

Problem 16

An electron is moved from point $A$, where the electric potential is $V_{A}=-240 \mathrm{~V}$, to point $B$, where the electric potential is $V_{B}=-360 \mathrm{~V}$. What is the change in the electric potential energy?

Mohamed Mustafa

Mohamed Mustafa

Numerade Educator

Problem 17

Find the electric field and the potential at the center of a square of side $2.0 \mathrm{~cm}$ with charges of $+9.0 \mu \mathrm{C}$ at each corner.

Mayukh Banik

Numerade Educator

Problem 18

Find the electric field and the potential at the center of a square of side $2.0 \mathrm{~cm}$ with a charge of $+9.0 \mu \mathrm{C}$ at one corner of the square and with charges of $-3.0 \mu \mathrm{C}$ at the remaining three corners.

Mohamed Mustafa

Mohamed Mustafa

Numerade Educator

Problem 19

A charge $Q=-50.0 \mathrm{nC}$ is located $0.30 \mathrm{~m}$ from point $A$ and $0.50 \mathrm{~m}$ from point $B$. (a) What is the potential at $A$ ? (b) What is the potential at $B ?$
(c) If a point charge $q$ is moved from $A$ to $B$ while
$Q$ is fixed in place, through what potential difference does it move? Does its potential increase or decrease?
(d) If $q=-1.0 \mathrm{nC}$, what is the change in electric potential energy as it moves from $A$ to $B ?$ Does the potential energy increase or decrease? (e) How much work is done by the electric field due to charge $Q$ as $q$ moves from $A$ to $B$ ?

Ajay Singhal

Numerade Educator

Problem 20

A charge of $+2.0 \mathrm{mC}$ is located at $x=0, y=0$ and a charge of $-4.0 \mathrm{mC}$ is located at $x=0, y=3.0 \mathrm{~m}$. What is the electric potential due to these charges a point with coordinates $x=4.0 \mathrm{~m}, y=0$ ?

Mohamed Mustafa

Mohamed Mustafa

Numerade Educator

Problem 21

The electric potential at a distance of $20.0 \mathrm{~cm}$ from a point charge is $+1.0 \mathrm{kV}$ (assuming $V=0$ at infinity).
(a) Is the point charge positive or negative? (b) At what distance is the potential $+2.0 \mathrm{kV} ?$ ( field and potential of point charge)

Mayukh Banik

Numerade Educator

Problem 22

A spherical conductor with a radius of $75.0 \mathrm{~cm}$ has an electric field of magnitude $8.40 \times 10^{5} \mathrm{~V} / \mathrm{m}$ just outside its surface. What is the electric potential just outside the surface, assuming the potential is zero far away from the conductor?

Mohamed Mustafa

Mohamed Mustafa

Numerade Educator

Problem 23

An array of four charges is arranged along the $x$ -axis at intervals of $1.0 \mathrm{~m}$. (a) If two of the charges are $+1.0 \mu \mathrm{C}$ and two are $-1.0 \mu \mathrm{C}$, draw a configuration of these charges that minimizes the potential at $x=0 .(\mathrm{b})$ If three of the charges are the same, $q=+1.0 \mu \mathrm{C}$, and the charge at the far right is $-1.0 \mu \mathrm{C}$, what is the potential at the origin?

Mayukh Banik

Numerade Educator

Problem 24

At a point $P$, a distance $R_{0}$ from a positive charge $Q_{0}$, the electric field has a magnitude $E_{0}=100 \mathrm{~N} / \mathrm{C}$ and the electric potential is $V_{0}=10 \mathrm{~V}$. The charge is now increased by a factor of three, becoming $3 Q_{0}$. (a) At what distance, $R_{\mathrm{E}}$, from the charge $3 Q_{0}$ will the electric field have the same value, $E=E_{0}$; and
(b) at what distance, $R_{\mathrm{V}}$, from the charge $3 Q_{0}$ will the electric potential have the same value, $V=V_{0}$ ?

Mohamed Mustafa

Mohamed Mustafa

Numerade Educator

Problem 25

Charges of $+2.0 \mathrm{nC}$ and $-1.0 \mathrm{nC}$
are located at opposite corners, $A$ and $C$, respectively, of a square which is $1.0 \mathrm{~m}$ on a side. What is the electric potential at a third corner, $B$, of the square (where there is no charge)?

Mayukh Banik

Numerade Educator

Problem 26

(a) Find the electric potential at points $a$ and $b$ for charges of $+4.2 \mathrm{nC}$ and $-6.4 \mathrm{nC}$ located as shown in the figure. (b) What is the potential difference $\Delta V$ for a trip from $a$ to $b$ ? (c) How much work must be done by an external agent to move a point charge of $+1.50 \mathrm{nC}$ from $a$ to $b ?$

Nafis Fuad

Numerade Educator

Problem 27

(a) Find the potential at points $a$ and $b$ in the diagram for charges $Q_{1}=+2.50 \mathrm{nC}$ and $Q_{2}=-2.50 \mathrm{nC}$. (b) How much work must be done by an external agent to bring a point charge $q$ from infinity to point $b$ ?

Nafis Fuad

Numerade Educator

Problem 28

(a) In the diagram, what are the potentials at points $a$ and $b$ ? Let $V=0$ at infinity. (b) What is the change in electric potential energy if a third charge $q_{3}=+2.00 \mathrm{n} \mathrm{C}$ is moved from
point $a$ to point $b$ ? (If you have done Problem 13, compare your answers.)

Zachary Brauchler

Zachary Brauchler

Numerade Educator

Problem 29

(a) In the diagram, what are the potentials at points $b$ and $c ?$ Let $V=0$ at infinity. (b) What is the change in electric potential energy if a third charge $q_{3}=+2.00 \mathrm{nC}$ is moved from point $b$ to point $c ?$ (If you have done Problem 14, compare your answers.)

Mayukh Banik

Numerade Educator

Problem 30

(a) In the diagram, what are the potentials at points $b$ and $c$ ? Let $V=0$ at infinity. (b) What is the change in electric potential energy if a third charge $q_{3}=+2.00 \mathrm{nC}$ is moved from point $b$ to point $c ?$ (If you have done Problem 14, compare your answers. $)$

Mayukh Banik

Numerade Educator

Problem 31

A uniform electric field has magnitude $240 \mathrm{~N} / \mathrm{C}$ and is directed to the right. A particle with charge $+4.2 \mathrm{nC}$ moves along the straight line from $a$ to $b$. (a) What is the electric force that acts on the particle? (b) What is the work done on the particle by the electric field? (c) What is the potential difference $V_{a}-V_{b}$ between points $a$ and $b$ ?

Mayukh Banik

Numerade Educator

Problem 32

In a region where there is an electric field, the electric forces do $+8.0 \times 10^{-19} \mathrm{~J}$ of work on an electron as it moves from point $X$ to point $Y$. (a) Which point, $X$ or $Y$, is at a higher potential? (b) What is the potential difference, $V_{Y}-V_{X}$, between point $Y$ and point $X$ ?

Zachary Brauchler

Zachary Brauchler

Numerade Educator

Problem 33

Suppose a uniform electric field of magnitude $100.0 \mathrm{~N} / \mathrm{C}$ exists in a region of space. How far apart are a pair of equipotential surfaces whose potentials differ by $1.0 \mathrm{~V} ?$

Mayukh Banik

Numerade Educator

Problem 34

Draw some electric field lines and a few equipotential surfaces outside a negatively charged hollow conducting sphere. What shape are the equipotential surfaces?

Zachary Brauchler

Zachary Brauchler

Numerade Educator

Problem 35

Draw some electric field lines and a few equipotential surfaces outside a positively charged conducting cylinder. What shape are the equipotential surfaces?

Mayukh Banik

Numerade Educator

Problem 36

It is believed that a large electric fish known as Torpedo occidentalis uses electricity to shock its victims. A typical fish can deliver a potential difference of $0.20 \mathrm{kV}$ for a duration of $1.5 \mathrm{~ms}$. This pulse delivers charge at a rate of $18 \mathrm{C} / \mathrm{s}$. (a) What is the rate at which work is done by the electric organs during a pulse? (b) What is the total amount of work done during one pulse?

Zachary Brauchler

Zachary Brauchler

Numerade Educator

Problem 37

A positive point charge is located at the center of a hollow $\quad$ spherical metal shell with zero net charge.
(a) Draw some electric field lines and sketch some equipotential surfaces for this arrangement. (b) Sketch graphs of the electric field magnitude and the potential as functions of $\dot{r}$

Linda Winkler

Numerade Educator

Problem 38

A positively charged oil drop is injected into a region of uniform electric field between two oppositely charged, horizontally oriented plates spaced $16 \mathrm{~cm}$ apart.
If the electric force on the drop is found to be $9.6 \times 10^{-16} \mathrm{~N}$ and the potential difference between the plates is $480 \mathrm{~V}$, what is the magnitude of the charge on the drop in terms of the elementary charge $e ?$ Ignore the small buoyant force on the drop.

Zachary Brauchler

Zachary Brauchler

Numerade Educator

Problem 39

A positively charged oil drop is injected into a region of uniform electric field between two oppositely charged, horizontally oriented plates spaced $16 \mathrm{~cm}$ apart.
If the mass of the drop is $1.0 \times 10^{-15} \mathrm{~kg}$ and it remains stationary when the potential difference between the plates is $9.76 \mathrm{kV}$, what is the magnitude of the charge on the drop? (Ignore the small buoyant force on the drop.)

Nafis Fuad

Numerade Educator

Problem 40

Point $P$ is at a potential of $500.0 \mathrm{kV}$ and point $S$ is at a potential of $200.0 \mathrm{kV}$. The space between these points is evacuated. When a charge of $+2 e$ moves from $P$ to $S$, by how much does its kinetic energy change?

Zachary Brauchler

Zachary Brauchler

Numerade Educator

Problem 41

An electron is accelerated from rest through a potential difference $\Delta V .$ If the electron reaches a speed of $7.26 \times 10^{6} \mathrm{~m} / \mathrm{s}$, what is the potential difference? Be sure to include the correct sign. (Does the electron move through an increase or a decrease in potential?)

Mayukh Banik

Numerade Educator

Problem 42

As an electron moves through a region of space, its speed decreases from $8.50 \times 10^{6} \mathrm{~m} / \mathrm{s}$ to $2.50 \times 10^{6} \mathrm{~m} / \mathrm{s}$.
The electric force is the only force acting on the electron. (a) Did the electron move to a higher potential or a lower potential? (b) Across what potential difference did the electron travel?

Zachary Brauchler

Zachary Brauchler

Numerade Educator

Problem 43

In the electron gun of Example $17.8$, if the potential difference between the cathode and anode is reduced to $6.0 \mathrm{kV}$, with what speed will the electrons reach the anode?

Mayukh Banik

Numerade Educator

Problem 44

In the electron gun of Example $17.8$, if the electrons reach the anode with a speed of $3.0 \times 10^{7} \mathrm{~m} / \mathrm{s}$, what is the potential difference between the cathode and the anode?

Zachary Brauchler

Zachary Brauchler

Numerade Educator

Problem 45

A beam of electrons of mass $m_{e}$ is deflected vertically by the uniform electric field between two oppositely charged, parallel metal plates. The plates are a distance $d$ apart and the potential difference between the plates is $\Delta V$. (a) What is the direction of the electric field between the plates? (b) If the $y$ -component of the electrons' velocity as they leave the region between the plates is $v_{y}$, derive an expression for the time it takes each electron to travel through the region between the plates in terms of $\Delta V, v_{v}, m_{\mathrm{e}}, d$, and $e$. (c) Does the electric potential energy of an electron increase, decrease, or stay constant while it moves between the plates? Explain.

Linda Winkler

Numerade Educator

Problem 46

An electron (charge $-e$ ) is projected horizontally into the space between two oppositely charged parallel plates. The electric field between the plates is $500.0 \mathrm{~N} / \mathrm{C}$ upward. If the vertical deflection of the electron as it leaves the plates has magnitude $3.0 \mathrm{~mm}$, how much has its kinetic energy increased due to the electric field? [Hint: First find the potential difference through which the electron moves.]

Narayan Hari

Numerade Educator

Problem 47

. An alpha particle (charge $+2 e$ ) moves through a potential difference $\Delta V=-0.50 \mathrm{kV}$. Its initial kinetic energy is $1.20 \times 10^{-16} \mathrm{~J}$. What is its final kinetic energy?

Mayukh Banik

Numerade Educator

Problem 48

In 1911 , Ernest Rutherford discovered the nucleus of the atom by observing the scattering of helium nuclei from gold nuclei. If a helium nucleus with a mass of $6.68 \times 10^{-27} \mathrm{~kg}$, a charge of $+2 e$, and an initial velocity of $1.50 \times 10^{7} \mathrm{~m} / \mathrm{s}$ is projected head-on toward a gold nucleus with a charge of $+79 e$, how close will the helium atom come to the gold nucleus before it stops and turns around? (Assume the gold nucleus is held in place by other gold atoms and does not move.)

Zachary Brauchler

Zachary Brauchler

Numerade Educator

Problem 49

The figure shows a graph of electric potential versus position along the $x$ -axis. A proton is originally at point $A$, moving in the positive $x$ -direction. How much kinetic energy does it need to have at point $A$ in order to be able to reach point $E$ (with no forces acting on the electron other than those due to the indicated potential)? Points $B, C$, and $D$ have to be passed on the way.

Mayukh Banik

Numerade Educator

Problem 50

Repeat Problem 49 for an electron rather than a proton.

Zachary Brauchler

Zachary Brauchler

Numerade Educator

Problem 51

A $2.0-\mu \mathrm{F}$ capacitor is connected to a $9.0$ - $\mathrm{V}$ battery. What is the magnitude of the charge on each plate?

Mayukh Banik

Numerade Educator

Problem 52

The plates of a $15.0-\mu \mathrm{F}$ capacitor have net charges of $+0.75 \mu \mathrm{C}$ and $-0.75 \mu \mathrm{C}$, respectively.
(a) What is the potential difference between the plates? (b) Which plate is at the higher potential?

Zachary Brauchler

Zachary Brauchler

Numerade Educator

Problem 53

If a capacitor has a capacitance of $10.2 \mu \mathrm{F}$ and we wish to lower the potential difference across the plates by $60.0 \mathrm{~V}$, what magnitude of charge will we have to remove from each plate?

Mayukh Banik

Numerade Educator

Problem 54

A parallel plate capacitor has a capacitance of $2.0 \mu \mathrm{F}$ and plate separation of $1.0 \mathrm{~mm}$. (a) How much potential difference can be placed across the capacitor before dielectric breakdown of air occurs $\left(E_{\max }=3 \times 10^{6} \mathrm{~V} / \mathrm{m}\right) ?$
(b) What is the magnitude of the greatest charge the capacitor can store before breakdown?

Zachary Brauchler

Zachary Brauchler

Numerade Educator

Problem 55

A parallel plate capacitor is charged by connecting it to a $12-V$ battery. The battery is then disconnected from the capacitor. The plates are then pulled apart so the spacing between the plates is increased. What is the effect (a) on the electric field between the plates? (b) on the potential difference between the plates?

Mayukh Banik

Numerade Educator

Problem 56

A parallel plate capacitor has a capacitance of $1.20 \mathrm{nF}$. There is a charge of magnitude $0.800 \mu \mathrm{C}$ on each plate.
(a) What is the potential difference between the plates?
(b) If the plate separation is doubled, while the charge is kept constant, what will happen to the potential difference?

Zachary Brauchler

Zachary Brauchler

Numerade Educator

Problem 57

A parallel plate capacitor is connected to a $12-\mathrm{V}$ battery. While the battery remains connected, the plates are pushed together so the spacing is decreased. What is the effect on (a) the potential difference between the plates?
(b) the electric field between the plates? (c) the magnitude of charge on the plates?

Mayukh Banik

Numerade Educator

Problem 58

A parallel plate capacitor has a capacitance of $1.20 \mathrm{nF}$ and is connected to a 12-V battery. (a) What is the magnitude of the charge on each plate? (b) If the plate separation is doubled while the plates remain connected to the battery, what happens to the charge on each plate and the electric field between the plates?

Zachary Brauchler

Zachary Brauchler

Numerade Educator

Problem 59

A variable capacitor is made of two parallel semicircular plates with air between them. One plate is fixed in place and the other can be rotated. The electric field is zero everywhere except in the region where the plates overlap. When the plates are directly across from one another, the capacitance is $0.694 \mathrm{pF}$. (a) What is the capacitance when the movable plate is rotated so that only one half its area is across from the stationary plate? (b) What is the capacitance when the movable plate is rotated so that two thirds of its area is across from the stationarv plate?

Mayukh Banik

Numerade Educator

Problem 60

A shark is able to detect the presence of electric fields as small as $1.0 \mu \mathrm{V} / \mathrm{m}$. To get an idea of the magnitude of this field, suppose you have a parallel plate capacitor connected to a 1.5-V battery. How far apart must the parallel plates be to have an electric field of $1.0 \mu \mathrm{V} / \mathrm{m}$ between the plates?

Zachary Brauchler

Zachary Brauchler

Numerade Educator

Problem 61

Two metal spheres have charges of equal magnitude, $3.2 \times 10^{-14} \mathrm{C}$, but opposite sign. If the potential difference between the two spheres is $4.0 \mathrm{mV}$, what is the capacitance?

Nafis Fuad

Numerade Educator

Problem 62

Suppose you were to wrap the Moon in aluminum foil and place a charge $Q$ on it. What is the capacitance of the Moon in this case? [

Zachary Brauchler

Zachary Brauchler

Numerade Educator

Problem 63

A tiny hole is made in the center of the negatively and positively charged plates of a capacitor, allowing a beam of electrons to pass through and emerge from the far side. If $40.0 \mathrm{~V}$ are applied across the capacitor plates and the electrons enter through the hole in the negatively charged plate with a speed of $2.50 \times 10^{6} \mathrm{~m} / \mathrm{s}$, what is the speed of the electrons as they emerge from the hole in the positive plate?

Mayukh Banik

Numerade Educator

Problem 64

A spherical conductor of radius $R$ carries a total charge
Q. (a) Show that the magnitude of the electric field just outside the sphere is $E=\sigma / \epsilon_{0}$, where $\sigma$ is the charge per unit area on the conductor's surface. (b) Construct an argument to show why the electric field at a point $\bar{P}$ just outside any conductor in electrostatic equilibrium has magnitude $E=\sigma / \epsilon_{0}$, where $\sigma$ is the local surface charge density.

Linda Winkler

Numerade Educator

Problem 65

A $6.2$ -cm by $2.2-\mathrm{cm}$ parallel plate capacitor has the plates separated by a distance of $2.0 \mathrm{~mm}$. (a) When $4.0 \times 10^{-11} \mathrm{C}$ of charge is placed on this capacitor, what is the electric field between the plates? (b) If a dielectric with dielectric constant of $5.5$ is placed between the plates while the charge on the capacitor stays the same, what is the electric field in the dielectric?

Mayukh Banik

Numerade Educator

Problem 66

Before a lightning strike can occur, the breakdown limit for damp air must be reached. If this occurs for an electric field of $3.33 \times 10^{5} \mathrm{~V} / \mathrm{m}$, what is the maximum possible height above the Earth for the bottom of a thundercloud, which is at a potential $1.00 \times 10^{8} \mathrm{~V}$ below Earth's surface potential, if there is to be a lightning strike? Two cows, with approximately $1.8 \mathrm{~m}$ between their front

Zachary Brauchler

Zachary Brauchler

Numerade Educator

Problem 67

Two cows, with approximately $1.8 \mathrm{~m}$ between their front and hind legs, are standing under a tree during a thunderstorm. See the diagram with Conceptual Question 16 .
(a) If the equipotential surfaces about the tree just after a lightning strike are as shown, what is the average electric field between Cow A's front and hind legs? (b) Which cow is more likely to be killed? Explain.

Mayukh Banik

Numerade Educator

Problem 68

A parallel plate capacitor has a charge of $0.020 \mu \mathrm{C}$ on each plate with a potential difference of $240 \mathrm{~V}$. The parallel plates are separated by $0.40 \mathrm{~mm}$ of bakelite. What is the capacitance of this capacitor?

Zachary Brauchler

Zachary Brauchler

Numerade Educator

Problem 69

Two metal spheres are separated by a distance of $1.0 \mathrm{~cm}$ and a power supply maintains a constant potential difference of $900 \mathrm{~V}$ between them. The spheres are brought closer to one another until a spark flies between them. If the dielectric strength of dry air is $3.0 \times 10^{6} \mathrm{~V} / \mathrm{m}$, what is the distance between the spheres at this time?

Mayukh Banik

Numerade Educator

Problem 70

To make a parallel plate capacitor, you have available two flat plates of aluminum (area $120 \mathrm{~cm}^{2}$ ), a sheet of paper (thickness $=0.10 \mathrm{~mm}, \kappa=3.5$ ), a sheet of glass (thickness $=2.0 \mathrm{~mm}, \kappa=7.0$ ), and a slab of paraffin (thickness $=10.0 \mathrm{~mm}, \kappa=2.0$ ). (a) What is the largest capacitance possible using one of these dielectrics?
(b) What is the smallest?

Zachary Brauchler

Zachary Brauchler

Numerade Educator

Problem 71

A capacitor can be made from two sheets of aluminum foil separated by a sheet of waxed paper. If the sheets of aluminum are $0.30 \mathrm{~m}$ by $0.40 \mathrm{~m}$ and the waxed paper, of slightly larger dimensions, is of thickness $0.030 \mathrm{~mm}$ and dielectric constant $\kappa=2.5$, what is the capacitance of this capacitor?

Mayukh Banik

Numerade Educator

Problem 72

In capacitive electrostimulation, electrodes are placed on opposite sides of a limb. A potential difference is applied to the electrodes, which is believed to be beneficial in treating bone defects and breaks. If the capacitance is measured to be $0.59 \mathrm{pF}$, the electrodes are $4.0 \mathrm{~cm}^{2}$ in area, and the limb is $3.0 \mathrm{~cm}$ in diameter, what is the (average) dielectric constant of the tissue in the limb?

Zachary Brauchler

Zachary Brauchler

Numerade Educator

Problem 73

A certain capacitor stores $450 \mathrm{~J}$ of energy when it holds $8.0 \times 10^{-2} \mathrm{C}$ of charge. What is (a) the capacitance of this capacitor and (b) the potential difference across the plates?

Mayukh Banik

Numerade Educator

Problem 74

What is the maximum electric energy density possible in dry air without dielectric breakdown occurring?

Zachary Brauchler

Zachary Brauchler

Numerade Educator

Problem 75

A parallel plate capacitor has a charge of $5.5 \times 10^{-7} \mathrm{C}$ on one plate and $-5.5 \times 10^{-7} \mathrm{C}$ on the other. The distance between the plates is increased by $50 \%$ while the charge on each plate stays the same. What happens to the energy stored in the capacitor?

Mayukh Banik

Numerade Educator

Problem 76

A large parallel plate capacitor has plate separation of $1.00 \mathrm{~cm}$ and plate area of $314 \mathrm{~cm}^{2}$. The capacitor is connected across a voltage of $20.0 \mathrm{~V}$ and has air between the plates. How much work is done on the capacitor as the plate separation is increased to $2.00 \mathrm{~cm} ?$

Zachary Brauchler

Zachary Brauchler

Numerade Educator

Problem 77

Figure $17.31 \mathrm{~b}$ shows a thundercloud before a lightning strike has occurred. The bottom of the thundercloud and the Earth's surface might be modeled as a charged parallel plate capacitor. The base of the cloud, which is roughly parallel to the Earth's surface, serves as the negative plate and the region of Earth's surface under the cloud serves as the positive plate. The separation between the cloud base and the Earth's surface is small compared to the length of the cloud. (a) Find the capacitance for a thundercloud of base dimensions $4.5 \mathrm{~km}$ by $2.5 \mathrm{~km}$ located $550 \mathrm{~m}$ above the Earth's surface. (b) Find the energy stored in this capacitor if the charge magnitude is $18 \mathrm{C}$.

Mayukh Banik

Numerade Educator

Problem 78

A parallel plate capacitor of capacitance $6.0 \mu \mathrm{F}$ has the space between the plates filled with a slab of glass with $\kappa=3.0 .$ The capacitor is charged by attaching it to a 1.5-V battery. After the capacitor is disconnected from the battery, the dielectric slab is removed. Find (a) the capacitance, (b) the potential difference, (c) the charge on the plates, and (d) the energy stored in the capacitor after the glass is removed.

Zachary Brauchler

Zachary Brauchler

Numerade Educator

Problem 79

A parallel plate capacitor is composed of two square plates, $10.0 \mathrm{~cm}$ on a side, separated by an air gap of $0.75 \mathrm{~mm}$. (a) What is the charge on this capacitor when there is a potential difference of $150 \mathrm{~V}$ between the plates? (b) What energy is stored in this capacitor?

Mayukh Banik

Numerade Educator

Problem 80

The capacitor of Problem 79 is initially charged to a $150-\mathrm{V}$ potential difference. The plates are then physically separated by another $0.750 \mathrm{~mm}$ in such a way that none of the charge can leak off the plates. Find (a) the new capacitance and (b) the new energy stored in the capacitor. Explain the result using conservation of energy.

Zachary Brauchler

Zachary Brauchler

Numerade Educator

Problem 81

Capacitors are used in many applications where you need to supply a short burst of energy. A $100.0-\mu \mathrm{F}$ capacitor in an electronic flash lamp supplies an average power of $10.0 \mathrm{~kW}$ to the lamp for $2.0 \mathrm{~ms}$. (a) To what potential difference must the capacitor initially be charged? (b) What is its initial charge?

Mayukh Banik

Numerade Educator

Problem 82

A parallel plate capacitor has a charge of $0.020 \mu \mathrm{C}$ on each plate with a potential difference of $240 \mathrm{~V}$. The parallel plates are separated by $0.40 \mathrm{~mm}$ of air. What energy is stored in this capacitor?

Zachary Brauchler

Zachary Brauchler

Numerade Educator

Problem 83

A parallel plate capacitor has a capacitance of $1.20 \mathrm{nF}$. There is a charge of $0.80 \mu \mathrm{C}$ on each plate. How much work must be done by an external agent to double the plate separation while keeping the charge constant?

Mayukh Banik

Numerade Educator

Problem 84

A defibrillator is used to restart a person's heart after it stops beating. Energy is delivered to the heart by discharging a capacitor through the body tissues near the heart. If the capacitance of the defibrillator is $9 \mu \mathrm{F}$ and the energy delivered is to be $300 \mathrm{~J}$, to what potential difference must the capacitor be charged?

Zachary Brauchler

Zachary Brauchler

Numerade Educator

Problem 85

A defibrillator consists of a $15-\mu \mathrm{F}$ capacitor that is charged to $9.0 \mathrm{kV}$. (a) If the capacitor is discharged in $2.0 \mathrm{~ms}$, how much charge passes through the body tissues? (b) What is the average power delivered to the tissues?

Mayukh Banik

Numerade Educator

Problem 86

The bottom of a thundercloud is at a potential of $-1.00 \times 10^{8} \mathrm{~V}$ with respect to Earth's surface. If a charge of $-25.0 \mathrm{C}$ is transferred to the Earth during a lightning strike, find the electric potential energy released. (Assume that the system acts like a capacitor-as charge flows, the potential difference decreases to zero.)

Zachary Brauchler

Zachary Brauchler

Numerade Educator

Problem 87

(a) If the bottom of a thundercloud has a potential of $-1.00 \times 10^{9} \mathrm{~V}$ with respect to Earth and a charge of $-20.0 \mathrm{C}$ is discharged from the cloud to Earth during a lightning strike, how much electric potential energy is released? (Assume that the system acts like a capacitoras charge flows, the potential difference decreases to zero.) (b) If a tree is struck by the lightning bolt and $10.0 \%$ of the energy released vaporizes sap in the tree, about how much sap is vaporized? (Assume the sap to have the same latent heat as water.) (c) If $10.0 \%$ of the energy released from the lightning strike could be stored and used by a homeowner who uses $400.0 \mathrm{~kW} \cdot \mathrm{hr}$ of electricity per month, for how long could the lightning bolt supply electricity to the home?

Mayukh Banik

Numerade Educator

Problem 88

Charges of $-12.0 \mathrm{nC}$ and $-22.0 \mathrm{nC}$ are separated by $0.700 \mathrm{~m}$. What is the potential midway between the two charges?

Zachary Brauchler

Zachary Brauchler

Numerade Educator

Problem 89

Two point charges $(+10.0 \mathrm{nC}$ and $-10.0 \mathrm{nC}$ ) are located $8.00 \mathrm{~cm}$ apart.
(a) What is the electric potential energy when a point charge of $-4.2 \mathrm{nC}$ is placed at points $a, b$, and $c$ in turn? Let $U=0$ when the $-4.2 \mathrm{nC}$ charge is far away (but the other two are still in place). (b) How much work would an external force have to do to move the point charge from $b$ to $a$ ?

David Morabito

Numerade Educator

Problem 90

If an electron moves from one point at a potential of $-100.0 \mathrm{~V}$ to another point at a potential of $+100.0 \mathrm{~V}$ how much work is done by the electric field?

Zachary Brauchler

Zachary Brauchler

Numerade Educator

Problem 91

A van de Graaff generator has a metal sphere of radius $15 \mathrm{~cm}$. To what potential can it be charged before the electric field at its surface exceeds $3.0 \times 10^{6} \mathrm{~N} / \mathrm{C}$ (which is sufficient to break down dry air and initiate a spark)?

Mayukh Banik

Numerade Educator

Problem 92

Find the potential at the sodium ion, $\mathrm{Na}^{+}$, which is surrounded by two chloride ions, $\mathrm{Cl}$, and a calcium ion. $\mathrm{Ca}^{2+}$, in water as shown in the diagram. The effective charge of the positive sodium ion in water is $2.0 \times 10^{-21} \mathrm{C}$, of the negative chlorine ion is $-2.0 \times 10^{-21} \mathrm{C}$, and of the positive calcium ion is $4.0 \times 10^{-21} \mathrm{C}$.

Vishal Gupta

Numerade Educator

Problem 93

An infinitely long conducting cylinder sits near an infinite conducting sheet (side view in the diagram). The cylinder and sheet have equal and opposite charges; the cylinder is positive. (a) Sketch some electric field lines.
(b) Sketch some equipotential surfaces.

Mayukh Banik

Numerade Educator

Problem 94

Two parallel plates are $4.0 \mathrm{~cm}$ apart. The bottom plate is charged positively and the top plate is charged negatively, producing a uniform electric field of $5.0 \times 10^{4} \mathrm{~N} / \mathrm{C}$ in the region between the plates. What is the time required for an electron, which starts at rest at the upper plate, to reach the lower plate? (Assume a vacuum exists between the plates.)

Zachary Brauchler

Zachary Brauchler

Numerade Educator

Problem 95

The potential difference across a cell membrane is $-90 \mathrm{mV}$. If the membrane's thickness is $10 \mathrm{~nm}$, what is the magnitude of the electric field in the membrane? Assume the field is uniform.

Mayukh Banik

Numerade Educator

Problem 96

A beam of electrons traveling with a speed of $3.0 \times 10^{7} \mathrm{~m} / \mathrm{s}$
enters a uniform, downward electric field of magnitude $2.0 \times 10^{4} \mathrm{~N} / \mathrm{C}$ between the deflection plates of an oscilloscope. The initial velocity of the electrons is perpendicular to the field. The plates are $6.0 \mathrm{~cm}$ long.
(a) What is the direction and magnitude of the change in velocity of the electrons while they are between the plates? (b) How far are the electrons deflected in the $\pm y$ -direction while between the plates?

Zachary Brauchler

Zachary Brauchler

Numerade Educator

Problem 97

A negatively charged particle of mass $5.00 \times 10^{-19} \mathrm{~kg}$ is moving with a speed of $35.0 \mathrm{~m} / \mathrm{s}$ when it enters the region between two parallel capacitor plates. The initial velocity of the charge is parallel to the plate surfaces and in the positive $x$ -direction. The plates are square with a side of $1.00 \mathrm{~cm}$ and the voltage across the plates is $3.00 \mathrm{~V}$. If the particle is initially $1.00 \mathrm{~mm}$ from both plates, and it just barely clears the positive plate after traveling $1.00 \mathrm{~cm}$ through the region between the plates, how many excess electrons are on the particle? You may ignore gravitational and edge effects.

Vishal Gupta

Numerade Educator

Problem 98

(a) Show that it was valid to ignore the gravitational force in Problem 97. (b) What are the components of velocity of the particle when it emerges from the plates?

Zachary Brauchler

Zachary Brauchler

Numerade Educator

Problem 99

Refer to Problem 97 . One capacitor plate has an excess of electrons and the other has a matching deficit of electrons. What is the number of excess electrons?

Mayukh Banik

Numerade Educator

Problem 100

A parallel plate capacitor has a charge of $0.020 \mu \mathrm{C}$ on each plate with a potential difference of $240 \mathrm{~V}$. The parallel plates are separated by $0.40 \mathrm{~mm}$ of air.
(a) What is the capacitance for this capacitor? (b) What is the area of a single plate? (c) At what voltage will the air between the plates become ionized? Assume a dielectric strength of $3.0 \mathrm{kV} / \mathrm{mm}$ for air.

Zachary Brauchler

Zachary Brauchler

Numerade Educator

Problem 101

A $200.0-\mu \mathrm{F}$ capacitor is placed across a $12.0-\mathrm{V}$ battery. When a switch is thrown, the battery is removed from the capacitor and the capacitor is connected across a heater that is immersed in $1.00 \mathrm{~cm}^{3}$ of water. Assuming that all the energy from the capacitor is delivered to the water, what is the temperature change of the water?

Mayukh Banik

Numerade Educator

Problem 102

A cell membrane has a surface area of $1.0 \times 10^{-7} \mathrm{~m}^{2}$, a dielectric constant of $5.2$, and a thickness of $7.5 \mathrm{~nm}$. The membrane acts like the dielectric in a parallel plate capacitor; a layer of positive ions on the outer surface and a layer of negative ions on the inner surface act as the capacitor plates. The potential difference between the "plates" is $90.0 \mathrm{mV}$. (a) How much energy is stored in this capacitor? (b) How many positive ions are there on the outside of the membrane? Assume that all the ions are singly charged (charge $+e$ ).

Zachary Brauchler

Zachary Brauchler

Numerade Educator

Problem 103

An axon has the outer part of its membrane positively charged and the inner part negatively charged. The membrane has a thickness of $4.4 \mathrm{~nm}$ and a dielectric constant $\kappa=5 .$ If we model the axon as a parallel plate capacitor whose area is $5 \mu \mathrm{m}^{2}$, what is its capacitance?

Mayukh Banik

Numerade Educator

Problem 104

An electron beam is deflected upward through $3.0 \mathrm{~mm}$ while traveling in a vacuum between two deflection plates $12.0 \mathrm{~mm}$ apart. The potential difference between the deflecting plates is $100.0 \mathrm{kV}$ and the kinetic energy of each electron as it enters the space between the plates is $2.0 \times 10^{-15} \mathrm{~J}$. What is the kinetic energy of each electron when it leaves the space between the plates?

Nafis Fuad

Numerade Educator

Problem 105

A point charge $q=-2.5 \mathrm{nC}$ is initially at rest adjacent to the negative plate of a capacitor. The charge per unit area on the plates is $4.0 \mu \mathrm{C} / \mathrm{m}^{2}$ and the space between the plates is $6.0 \mathrm{~mm}$. (a) What is the potential difference between the plates? (b) What is the kinetic energy of the point charge just before it hits the positive plate, assuming no other forces act on it?

Mayukh Banik

Numerade Educator

Problem 106

An alpha particle (helium nucleus, charge $+2 e$ ) starts from rest and travels a distance of $1.0 \mathrm{~cm}$ under the influence of a uniform electric field of magnitude $10.0 \mathrm{kV} / \mathrm{m}$. What is the final kinetic energy of the alpha particle?

Zachary Brauchler

Zachary Brauchler

Numerade Educator

Problem 107

The inside of a cell membrane is at a potential of $90.0 \mathrm{mV}$ lower than the outside. How much work does the electric field do when a sodium ion (Na') with a charge of $+e$ moves through the membrane from outside to inside?

Mayukh Banik

Numerade Educator

Problem 108

Draw some electric field lines and a few equipotential surfaces outside a positively charged metal cube.

Linda Winkler

Numerade Educator

Problem 109

A parallel plate capacitor is attached to a battery that supplies a constant voltage. While the battery remains attached to the capacitor, the distance between the parallel plates increases by $25 \%$. What happens to the energy stored in the capacitor?

Mayukh Banik

Numerade Educator

Problem 110

A parallel plate capacitor is attached to a battery that supplies a constant voltage. While the battery is still attached, a diclectric of dielectric constant $\kappa=3.0$ is inserted so that it just fits between the plates. What is the energy stored in the capacitor after the dielectric is inserted in terms of the energy $U_{0}$ before the dielectric was inserted?

Zachary Brauchler

Zachary Brauchler

Numerade Educator

Problem 111

(a) Calculate the capacitance per unit length of an axon of radius $5.0 \mu \mathrm{m}$ (see Fig. 17.14). The membrane acts as an insulator between the conducting fluids inside and outside the neuron. The membrane is $6.0 \mathrm{~nm}$ thick and has a dielectric constant of $7.0$. (Note: The membrane is thin compared with the radius of the axon, so the axon can be treated as a parallel plate capacitor.)
(b) In its resting state (no signal being transmitted), the potential of the fluid inside is about $85 \mathrm{mV}$ lower than the outside. Therefore, there must be small net charges $\pm Q$ on either side of the membrane. Which side has positive charge? What is the magnitude of the charge density on the surfaces of the membrane?

Mayukh Banik

Numerade Educator

Problem 112

A $4.00-\mu \mathrm{F}$ air gap capacitor is connected to a $100.0$ - $\mathrm{V}$ battery until the capacitor is fully charged. The battery is removed and then a dielectric of dielectric constant $6.0$ is inserted between the plates without allowing any charge to leak off the plates. (a) Find the energy stored in the capacitor before and after the dielectric is inserted. [Hint: First find the new capacitance and potential difference.] (b) Does an external agent have to do positive work to insert the dielectric or to remove the dielectric? Explain.

Zachary Brauchler

Zachary Brauchler

Numerade Educator

Problem 113

It has only been fairly recently that $1.0$ -F capacitors have been readily available. A typical 1.0-F capacitor can withstand up to $5.00 \mathrm{~V}$. To get an idea why it isn't easy to make a 1.0-F capacitor, imagine making a 1.0-F parallel plate capacitor using titanium dioxide $(\kappa=90.0$, breakdown strength $4.00 \mathrm{kV} / \mathrm{mm}$ ) as the dielectric.
(a) Find the minimum thickness of the titanium dioxide such that the capacitor can withstand $5.00 \mathrm{~V}$. (b) Find the area of the plates so that the capacitance is $1.0 \mathrm{~F}$.

Mayukh Banik

Numerade Educator

Problem 114

The potential difference across a cell membrane from outside to inside is initially at $-90 \mathrm{mV}$ (when in its resting phase). When a stimulus is applied, $\mathrm{Na}^{+}$ ions are allowed to move into the cell such that the potential changes to $+20 \mathrm{mV}$ for a short amount of time.
(a) If the membrane capacitance per unit area is$1 \mu \mathrm{F} / \mathrm{cm}^{2}$, how much charge moves through a membrane of area $0.05 \mathrm{~cm}^{2} ?$ (b) The charge on $\mathrm{Na}^{+}$ is $+e$. How many ions move through the membrane?

Zachary Brauchler

Zachary Brauchler

Numerade Educator

Problem 115

A parallel plate capacitor is connected to a battery. The space between the plates is filled with air. The electric field strength between the plates is $20.0 \mathrm{~V} / \mathrm{m}$. Then. with the battery still connected, a slab of dielectric $(\kappa=4.0)$ is inserted between the plates. The thickness of the dielectric is half the distance between the plates. Find the electric field inside the dielectric.

Mayukh Banik

Numerade Educator