Chapter 16 Part 1 (Potential)

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Are Coulomb Forces Conservative?

Learning Goal: To review the concept of conservative forces and to understand that electrostatic forces are, in fact, conservative.

As you may recall from mechanics, some forces have a very special property, namely, that the work done on an object does not depend on the object's trajectory; rather, it depends only on the initial and the final positions of the object.

Such forces are called conservative forces. If only conservative forces act within a closed system, the total amount of mechanical energy is conserved within the system (hence the term "conservative"). Such forces have a number of properties that simplify the solution of many problems.

You may also recall that a potential energy function can be defined with respect to a conservative force. This property of conservative forces will be of particular interest of us.

Not all forces that we deal with are conservative, of course. For instance, the amount of work done by a frictional force very much depends on the object's trajectory. Friction, therefore, is not a conservative force. In contrast, the gravitational force and the normal force are examples of conservative forces. What about electrostatic (Coulomb) forces? Are they conservative, and is there a potential energy function associated with them?

In this problem, you will be asked to use the given diagram to calculate the work done by the electric field on a particle of charge and see for yourself whether that work appears to be trajectory-independent. Recall that the force acting on a charged particle in an electric field is given by .

Recall that the work done on an object by a constant force is

,

where is the magnitude of the force acting on the object, is the magnitude of the displacement that the object undergoes, and is the angle between the vectors and .

Consider a uniform electric field and a rectangle ABCD, as shown in the figure. Sides AB and CD are parallel to and have length ; let be angle BAC.

Part A

Calculate the work done by the electrostatic force on a particle of charge as it moves from A to B.

Hint A.1 Find the angle

Hint not displayed

Express your answer in terms of some or all the variables , , , and .

ANSWER: =

Correct

The angle between the force and the displacement is zero here, so , and the general formula for work becomes .

Part B

Calculate the work done by the electrostatic force on the charged particle as it moves from B to C.

Express your answer in terms of some or all the variables , , , and .

ANSWER: = 0

Correct

Now the angle between the force and the displacement is 90, so , and the work done is zero.

Part C

Calculate the total amount of work done by the electrostatic force on the charged particle as it moves from A to B to C.

Express your answer in terms of some or all the variables , , , and .

ANSWER: =

Correct

Part D

Now assume that the particle "chooses" a different way of traveling. Calculate the total amount of work done by the electrostatic force on the charged particle as it moves from A to D to C.

Express your answer in terms of some or all the variables , , and .

ANSWER: = Answer not displayed

Part E

Part not displayed

Transition will be visible after you complete previous part(s).

Part F

Part not displayed

Transition will be visible after you complete previous part(s).

Part G

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Potential Energy of a Battery

Learning Goal: To understand electrical potential, electrical potential energy, and the relationship between them.

Electric potential and electric potential energy are related but different concepts. Be careful not to confuse the terms. Electrical potential energy is the potential energy that a charge has due to its position relative to other charges. The electric potential at a specific position is a measure of the amount of potential energy per unit charge a particle of net charge would have at that position. In other words, if a charge has an electric potential energy , the electric potential at the location of is

.

Recall that the gravitational potential energy () of an object of mass depends on where you define . The difference in gravitational potential energy between two points is the physically relevant quantity. Similarly, for electric potential energy, the important quantity is the change in electric potential energy: . This is why we often just measure the potential difference . When we say that the potential of a car battery is 12 , we mean that the potential difference between the positive and negative terminals of the battery is 12 .

Consider dropping a ball from rest. This ball moves from a state of high gravitational potential energy to one of low gravitational potential energy as it falls to the ground. Similarly, charges move from a state of high electric potential energy to one of low electric potential energy.

Part A

Mustang Sally just finished restoring her 1965 Ford Mustang car. To save money, she did not get a new battery. When she tries to start the car, she discovers that the battery is dead and she needs a jump start. While unhooking the jumper cables, the positive and negative cables almost touch and a spark jumps between the ends of the cables. This spark is caused by the movement of electrons through the air between the battery terminals. In what direction are the electrons traveling?

Hint A.1 Another way to think about the movement of charge

Hint not displayed

ANSWER: The electrons are traveling from the negative to the positive either terminal to the other the positive to the negative

Correct terminal.

The positive terminal is at a higher potential than the negative terminal. Unless provided with energy, positive charges will flow from a high to a low potential, and negatively charged electrons will flow from a low to a high potential. The table below summarizes this movement.

Direction of motion

high to low potential

low to high potential

Since potential difference is the energy per unit charge, it is measured in units of energy divided by charge. Specifically, potential difference is generally measured in volts (whose symbol is ). One volt is equal to one joule per coulomb: .

Part B

There is a 12 potential difference between the positive and negative ends of the jumper cables, which are a short distance apart. An electron at the negative end ready to jump to the positive end has a certain amount of potential energy. On what quantities does this electrical potential energy depend?

Hint B.1 The expression for electric potential energy

Hint not displayed

ANSWER: the distance between the ends of the cables

the potential difference between the ends of the cables

the charge on the electron

the distance and the potential difference

the distance and the charge

the potential difference and the charge

the potential difference, charge, and distance

Correct

Part C

Assume that two of the electrons at the negative terminal have attached themselves to a nearby neutral atom. There is now a negative ion with a charge at this terminal. What are the electric potential and electric potential energy of the negative ion relative to the electron?

ANSWER: The electric potential and the electric potential energy are both twice as much.

The electric potential is twice as much and the electric potential energy is the same.

The electric potential is the same and the electric potential energy is twice as much.

The electric potential and the electric potential energy are both the same.

The electric potential is the same and the electric potential energy is increased by the mass ratio of the oxygen ion to the electron.

The electric potential is twice as much and the electric potential energy is increased by the mass ratio of the oxygen ion to the electron.

Correct

Part D

What is the electric potential energy of an electron at the negative end of the cable, relative to the positive end of the cable? In other words, assume that the electric potential of the positive terminal is 0 and that of the negative terminal is . Recall that .

Enter your answer numerically in joules.

ANSWER: =

Correct

Part E

At the negative terminal of the battery the electron has electric potential energy. What happens to this energy as the electron jumps from the negative to the positive terminal?

ANSWER: It disappears.

It is converted to kinetic energy.

It heats the battery.

It increases the potential of the battery.

Correct

Just as gravitational potential energy is converted to kinetic energy when something falls, electrical potential energy is converted to kinetic energy when a charge goes from a high potential energy state to a low potential energy state.

Part F

If you wanted to move an electron from the positive to the negative terminal of the battery, how much work would you need to do on the electron?

Hint F.1 Formula for work

Hint not displayed

Enter your answer numerically in joules.

ANSWER: =

Correct

Because moving a negative charge from the positive to the negative terminal of the battery would increase its electric potential energy, it would take positive work to move the charge. This is simliar to lifting a ball upward. You do positive work on the ball to increase its gravitational potential energy.

Energy Stored in a Charge Configuration

Four point charges, A, B, C, and D, are placed at the corners of a square with side length . Charges A, B, and C have charge , and D has charge .

Throughout this problem, use in place of .

Part A

If you calculate , the amount of work it took to assemble this charge configuration if the point charges were initially infinitely far apart, you will find that the contribution for each charge is proportional to . In the space provided, enter the numeric value that multiplies the above factor, in .

Hint A.1 How to approach the problem

Hint not displayed

Hint A.2 Electric potential and potential energy

Hint not displayed

Hint A.3 Work required to place charge A

Hint not displayed

Hint A.4 Work required to place charge B

Hint not displayed

Hint A.5 Work required to place charge C

Hint not displayed

Hint A.6 Find the work required to place charge D

Hint not displayed

ANSWER: = 0

Correct

The hints led you through the problem by adding one charge at a time. A little thought shows that this is equivalent to simply adding the energies of all possible pairs:

.

Note that this is not equivalent to adding the potential energies of each charge. Adding the potential energies will give you double the correct answer because you will be counting each charge twice.

Part B

Which of the following figures depicts a charge configuration that requires less work to assemble than the configuration in the problem introduction? Assume that all charges have the same magnitude .

ANSWER: figure a

figure b

figure c

Correct

Ionic Potentials across Cell Membranes Conceptual Question

In its resting state, the membrane surrounding a neuron is permeable to potassium ions but not permeable to sodium ions. Thus, positive K ions can flow through the membrane in an attempt to equalize K concentration, but Na ions cannot. This leads to an excess of Na ions outside of the cell. If the space outside the cell is defined as zero electric potential, then the electric potential of the interior of the cell is negative. This resting potential is typically about 80 . A schematic of this situation is shown in the figure.

In response to a stimulus, the membrane can become permeable to Na ions. As Na ions rush into the cell, the interior of the cell reaches an electric potential of about 40 . This process is termed depolarization. In response to depolarization, the membrane again becomes impermeable to Na ions, and the K ions flow out of the interior of the cell (driven by the positive electric potential inside of the cell), reestablishing the resting potential. This is termed repolarization. Only a small percentage of the available Na and K ions participate in each depolarization/repolarization cycle, so the cell can respond to many stimuli in succession without depleting its "stock" of available Na and K ions. A graph of an electric potential inside a cell vs. time is shown in the next figure for a single depolarization/repolarization cycle.

Part A

During the resting phase, what is the electric potential energy of a typical Na ion outside of the cell?

Hint A.1 The electron volt

Hint not displayed

ANSWER: 40

+40

80

+80

0

Correct

Part B

During the resting phase, what is the electrical potential energy of a typical K ion inside of the cell?

Hint B.1 The electron volt

Hint not displayed

ANSWER: 40

+40

80

+80

0

Correct

Part C

During depolarization, what is the work done (by the electric field) on the first few Na ions that enter the cell?

Hint C.1 The electron volt

Hint not displayed

Hint C.2 Algebraic sign of the work

Hint not displayed

Hint C.3 Magnitude of the work

Hint not displayed

ANSWER: 40

+40

80

+80

120

+120

0

Correct

Part D

During repolarization, what is the work done (by the electric field) on the first few K ions that exit the cell?

Hint D.1 The electron volt

Hint not displayed

Hint D.2 Algebraic sign of the work

Hint not displayed

Hint D.3 Magnitude of the work

Hint not displayed

ANSWER: 40

+40

80

+80

120

+120

0

Correct

Electric Potential and Potential Energy

A particle with charge 6.40×10−19 is placed on the x axis in a region where the electric potential due to other charges increases in the +x direction but does not change in the y or z direction.

Part A

The particle, initially at rest, is acted upon only by the electric force and moves from point a to point b along the x axis, increasing its kinetic energy by 1.28×10−18 . In what direction and through what potential difference does the particle move?

Hint A.1 How to approach the problem

Hint not displayed

Hint A.2 Electric potential

Hint not displayed

Hint A.3 Find the change in potential energy of the particle

Hint not displayed

ANSWER: The particle moves to the left through a potential difference of 2.00 .

The particle moves to the left through a potential difference of 2.00 .

The particle moves to the right through a potential difference of 2.00 .

The particle moves to the right through a potential difference of 2.00 .

The particle moves to the left through a potential difference of 20.0 .

The particle moves to the right through a potential difference of 20.0 .

Correct

Thus, if no forces other than the electric force act on a positively charged particle, the particle always moves toward a point at lower potential.

Part B

If the particle moves from point b to point c in the y direction, what is the change in its potential energy, ?

Hint B.1 How to approach the problem

Hint not displayed

ANSWER: 1.28×10−18

1.28×10−18

0

Correct

Every time a charged particle moves along a line of constant potential, its potential energy remains constant and the electric field does no work on the particle.

Electric Potential Energy of Three Point Charges

Part A

Three equal point charges, each with charge 1.25 , are placed at the vertices of an equilateral triangle whose sides are of length 0.450 . What is the electric potential energy of the system? (Take as zero the potential energy of the three charges when they are infinitely far apart.)

Hint A.1 How to approach the problem

Hint not displayed

Hint A.2 Find the electric potential energy of one pair

Hint not displayed

Hint A.3 How many interactions are there?

Hint not displayed

Use = 8.85×10−12 for the permittivity of free space.

ANSWER: = Answer not displayed

Speed of an Electron in an Electric Field

Two stationary positive point charges, charge 1 of magnitude 4.00 and charge 2 of magnitude 1.95 , are separated by a distance of 54.0 . An electron is released from rest at the point midway between the two charges, and it moves along the line connecting the two charges.

Part A

What is the speed of the electron when it is 10.0 from charge 1?

Hint A.1 How to approach the problem

Hint not displayed

Hint A.2 Calculate the potential at the midpoint

Hint not displayed

Hint A.3 Calculate the initial potential energy

Hint not displayed

Hint A.4 Calculate the initial kinetic energy

Hint not displayed

Hint A.5 Calculate the final potential energy

Hint not displayed

Hint A.6 Putting it all together

Hint not displayed

Express your answer in meters per second.

ANSWER: = Answer not displayed

Moving a Charge

A point charge with charge is held stationary at the origin. A second point charge with charge moves from the point (, 0) to the point (, ).

Part A

How much work is done by the electrostatic force on the moving point charge?

Hint A.1 How to approach the problem

Hint not displayed

Hint A.2 Calculate the initial electric potential energy of the system of charges

Hint not displayed

Hint A.3 Calculate the final electric potential energy

Hint not displayed

Express your answer in joules. Use for Coulomb's constant ().

ANSWER: = Answer not displayed

C Exercise 16.1

A pair of parallel plates is charged by a 18- battery.

Part A

If the electric field between the plates is 1200 , how far apart are the plates?

Express your answer using two significant figures.

ANSWER: = 1.5

Correct

C Exercise 16.2

A pair of parallel plates is charged by a 6.0- battery.

Part A

How much work is required to move a particle with a charge of -4.7 from the positive to the negative plate?

Express your answer using two significant figures.

ANSWER: = Answer not displayed

C Exercise 16.3

Part A

If it takes +2.4×10−5 to move a positively charged particle between two charged parallel plates, what is the charge on the particle if the plates are connected to a 24- battery?

Express your answer using two significant figures.

ANSWER: = 1.0

Correct

Part B

Was it moved from the negative to the positive plate or from the positive to the negative plate?

ANSWER: from the negative to the positive plate

from the positive to the negative plate

Correct

C Exercise 16.4

An electron is accelerated by a uniform electric field (1600 ) pointing vertically upward.

Part A

Determine the direction of electron's velocity after it moves 0.10 from rest.

ANSWER: to the left

to the right

upward

downward

Answer not displayed

Part B

Use Newton's laws to determine the value of electron's velocity after it moves 0.10 from rest.

Express your answer using two significant figures.

ANSWER: = Answer not displayed

C Exercise 16.12

It takes +7.0 of work to move two charges from a large distance apart to 2.2 from one another.

Part A

If the charges have the same magnitude how large is each charge?

Express your answer using two significant figures.

ANSWER: = Answer not displayed

Part B

What can you tell about their signs?

ANSWER: both charges are positive

both charges are negative

signs of the charges are opposite

signs of the charges are the same

nothing

Answer not displayed

C Exercise 16.15

Part A

Compute the energy necessary to bring together (from a very large distance) the charges = 4.2 , = 5.0 and = -3.8 in the configuration shown in the figure .

Express your answer using two significant figures.

ANSWER: = Answer not displayed

C Exercise 16.7 (IE)

Part A

At one-third the original distance from a positive point charge, by what factor is the electric potential changed?

ANSWER: 1/3

9

1/9

3

Correct

Part B

Why?

ANSWER: My Answer:

Part C

How far from a 4.5 - charge is a point with an electric potential value of 20 ?

Express your answer using two significant figures.

ANSWER: = Answer not displayed

Part D

How much of a change in potential would occur if the point were moved to three times that distance?

Express your answer using two significant figures.

ANSWER: = Answer not displayed

Electric Fields and Equipotential Surfaces

The dashed lines in the diagram represent cross sections of equipotential surfaces drawn in 1- increments.

Part A

What is the work done by the electric force to move a 1- charge from A to B?

Hint A.1 Find the potential difference between A and B

Hint not displayed

Hint A.2 Potential difference and work

Hint not displayed

Express your answer in joules.

ANSWER: = 0

Correct

Part B

What is the work done by the electric force to move a 1- charge from A to D?

Hint B.1 Find the potential difference between A and D

Hint not displayed

Hint B.2 Potential difference and work

Hint not displayed

Express your answer in joules.

ANSWER: = 1

Correct

Part C

The magnitude of the electric field at point C is

Hint C.1 Electric field and equipotential surfaces

Hint not displayed

ANSWER: greater than the magnitude of the electric field at point B.

less than the magnitude of the electric field at point B.

equal to the magnitude of the electric field at point B.

unknown because the value of the electric potential at point C is unknown.

Correct

C Exercise 16.20

A uniform electric field of 10 points vertically upward.

Part A

How far apart are the equipotential planes that differ by 300 ?

Express your answer using two significant figures.

ANSWER: = Answer not displayed

C Exercise 16.22

Part A

Determine the potential 3.5 from the negative plate of a pair of parallel plates separated by 24.5 and connected to a 24- battery.

Express your answer using two significant figures.

ANSWER: = Answer not displayed

C Exercise 16.24

Part A

If the radius of the equipotential surface of a point charge is 19.0 and is at a potential of +2.10 (compared to zero at infinity), what are the magnitude and sign of the point charge?

ANSWER: = Answer not displayed

C Exercise 16.25 (IE)

Part A

The equipotential surfaces in the neighborhood of a positive point charge are spheres. Which sphere is associated with the higher electric potential?

ANSWER: the smaller one

the larger one

they are associated with the same potential

Correct

Part B

Calculate the amount of work (in electron-volts) it would take to move an electron from 13.1 to 16.3 away from a +4.30 point charge.

ANSWER: = Answer not displayed

C Exercise 16.30

Part A

Calculate the voltage required to accelerate a beam of protons initially at rest if they have a kinetic energy of 2.9 .

Express your answer using two significant figures.

ANSWER: = Answer not displayed

Part B

Calculate their speed if they have a kinetic energy of 2.9 .

Express your answer using two significant figures.

ANSWER: = Answer not displayed

Part C

Calculate the voltage required to accelerate a beam of protons initially at rest if they have a kinetic energy of 4.9 .

Express your answer using two significant figures.

ANSWER: = Answer not displayed

Part D

Calculate their speed if they have a kinetic energy of 4.9 .

Express your answer using two significant figures.

ANSWER: = Answer not displayed

Part E

Calculate the voltage required to accelerate a beam of protons initially at rest if they have a kinetic energy of 8.0×10−16 .

Express your answer using two significant figures.

ANSWER: = Answer not displayed

Part F

Calculate their speed if they have a kinetic energy of 8.0×10−16 .

Express your answer using two significant figures.

ANSWER: = Answer not displayed

Score Summary:

Your score on this assignment is 33%.

You received 69.36 out of a possible total of 210 points.

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