Assignment 6: Electric Potential

Due: 8:00am on Friday, January 27, 2012

Note: To understand how points are awarded, read your instructor's Grading Policy.

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The first problem is ungraded/optional for practice if you need some extra help. It is a tutorial on electric potential

energy, electric potential, and force using the gravitational field as a more familiar comparison. Do some, all, or none.

Electric Potential Energy versus Electric Potential

Learning Goal: To understand the relationship and differences between electric potential and electric potential

energy.

In this problem we will learn about the relationships between electric force , electric field , potential energy

, and electric potential . To understand these concepts, we will first study a system with which you are already

familiar: the uniform gravitational field.

Gravitational Force and Potential Energy

First we review the force and potential energy of an object of mass in a uniform gravitational field that points

downward (in the direction), like the gravitational field near the earth's surface.

Part A

Find the force on an object of mass in the uniform gravitational field when it is at height .

Express in terms of , , , and .

ANSWER:

=

Correct

Because we are in a uniform field, the force does not depend on the object's location. Therefore, the variable

does not appear in the correct answer.

Part B

Now find the gravitational potential energy of the object when it is at an arbitrary height . Take zero

potential to be at position . Keep in mind that the potential energy is a scalar, not a vector.

Express in terms of , , and .

ANSWER:

=

Correct

Part C

In what direction does the object accelerate when released with initial velocity upward?

ANSWER:

upward

downward

upward or downward depending on its mass

downward only if the ratio of to initial velocity is large enough

Correct

Electric Force and Potential Energy

Now consider the analogous case of a particle with charge placed in a uniform electric field of strength ,

pointing downward (in the direction)

Part D

Find , the electric force on the charged particle at height .

Hint D.1 Relationship between force and electric field

Hint not displayed

Express in terms of , , , and .

ANSWER:

=

Correct

Part E

Now find the potential energy of this charged particle when it is at height . Take zero potential to be at

position .

Express in terms of , , and .

ANSWER:

=

Correct

Part F

In what direction does the charged particle accelerate when released with upward initial velocity?

ANSWER:

upward

downward

upward or downward depending on its charge

downward only if the ratio of to initial velocity is large enough

Correct

Electric Field and Electric Potential

The electric potential is defined by the relationship where is the electric potential energy of a particle

with charge .

Part G

Find the electric potential of the uniform electric field . Note that this field is not pointing in the same

direction as the field in the previous section of this problem. Take zero potential to be at position .

Express in terms of , , and .

ANSWER:

= Correct

The SI unit for electric potential is the volt ( ). The volt is a derived unit, which means that it can be written in

terms of other SI units. In terms of the fundamental units of length, mass, time and charge, the volt can be

expressed as follows:

Part H

The electric field can be derived from the electric potential, just as the electrostatic force can be determined from

the electric potential energy. The relationship between electric field and electric potential is , where

is the gradient operator:

.

The partial derivative means the derivative of with respect to , holding all other variables constant.

Consider again the electric potential corresponding to the field . This potential depends on the z

coordinate only, so and .

Find an expression for the electric field in terms of the derivative of .

Express your answer as a vector in terms of the unit vectors , , and/or . Use for the derivative of with

respect to .

ANSWER:

= Correct

Part I

A positive test charge will accelerate toward regions of ________ electric potential and ________ electric

potential energy.

Hint I.1 Direction of the electric field

Hint not displayed

Hint I.2 Formula for the force on a charge in an electric field

Hint not displayed

Hint I.3 Formula for electric potential energy

Hint not displayed

Choose the appropriate answer combination to fill in the blanks correctly.

ANSWER:

higher; higher

higher; lower

lower; higher

lower; lower

Correct

Part J

A negative test charge will accelerate toward regions of ________ electric potential and ________ electric

potential energy.

Hint J.1 Direction of the electric field

Hint not displayed

Hint J.2 Formula for the force on a charge in an electric field

Hint not displayed

Hint J.3 Formula for electric potential energy

Hint not displayed

Choose the appropriate answer combination to fill in the blanks correctly.

ANSWER:

higher; higher

higher; lower

lower; higher

lower; lower

Correct

A charge in an electric field will experience a force in the direction of decreasing potential energy. Since the

electric potential energy of a negative charge is equal to the charge times the electric potential ( ), the

direction of decreasing electric potential energy is the direction of increasing electric potential.

Electric Potential, Potential Energy, and Force

Learning Goal: To review relationships among electric potential, electric potential energy, and force on a test

charge

This problem is a review of the relationship between an electric field , its associated electric potential , the

electric potential energy , and the direction of force on a test charge.

Part A

Electric field lines always begin at _______ charges (or at infinity) and end at _______ charges (or at infinity).

One could also say that the lines we use to represent an electric field indicate the direction in which a _______

test charge would initially move when released from rest. Which of the following fills in the three missing words

correctly?

ANSWER:

(positive; negative; negative)

(positive; negative; positive)

(negative; positive; negative)

(negative; positive; positive)

Correct

Note that the electric field vector is everywhere tangent to the electric field lines. Like electric field lines, the

electric field vector generally points away from positive charges and toward negative charges.

Part B

Would a positive test charge released from rest move toward a region of higher or lower electric potential

(compared to the electric potential at the point where it is released)?

Hint B.1 Potential, field, and force

Hint not displayed

ANSWER:

higher electric potential

lower electric potential

Correct

Part C

Now imagine that the sign of our test particle is changed from positive to negative, but the electric potential

remains the same. Which of the following statements is correct?

Hint C.1 Direction of field and force

Hint not displayed

Hint C.2 Direction of force and potential energy gradient

Hint not displayed

ANSWER:

The direction of the force will change and it will point to regions of higher potential energy.

The direction of the force will not change and it will point to regions of higher potential

energy.

The direction of the force will not change and it will point to regions of lower potential

energy.

The direction of the force will change and it will point to regions of lower potential energy.

Correct

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:

= Correct

Since and , it is clear that . It appears that the work done by the

electrostatic force on the particle is the same for both paths that begin at point A and end at point C. We now

have a reasonable suspicion that this force may, in fact, be conservative. Let us check some more.

Part E

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

Hint E.1 Find the distance between A and C

Hint not displayed

Hint E.2 Find the angle

Hint not displayed

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

ANSWER:

= Correct

Though we have not proved it, it can be shown that the Coulomb force is indeed conservative. This implies that

the amount of work done by the electrostatic force on the charged particle as it moves in a curved path from

A to F to C is also equal to .

With the knowledge that the Coulomb force is conservative, and again referring to the diagram, answer the

following questions. These questions are meant to highlight some important properties of conservative forces.

Part F

Find the amount of work done by the electrostatic force on the charged particle as it moves along the straight

path from B to A.

Hint F.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 180 here, so , and the general formula for work

becomes .

Note that .

The amount of work done by the electrostatic force on the charged particle as it moves from A to B to A is

equal to

Part G

Find the amount of work done by the electrostatic force on the charged particle as it moves from A to B

to C to D to A.

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

ANSWER:

= 0

Correct

Another important property of conservative forces, which can be very helpful in problem solving, is that the total

work done by a conservative force over a closed path is zero.

Electric Force and Potential: Spherical Symmetry

Learning Goal: To understand the electric potential and electric field of a point charge in three dimensions

Consider a positive point charge , located at the origin of three-dimensional space.

Throughout this problem, use in place of .

Part A

Due to symmetry, the electric field of a point charge at the origin must point _____ from the origin.

Answer in one word.

ANSWER:

away

Correct

Part B

Find , the magnitude of the electric field at distance from the point charge .

Express your answer in terms of , , and .

ANSWER:

= Correct

Part C

Find , the electric potential at distance from the point charge .

Express your answer in terms of , , and .

ANSWER:

= Correct

Part D

Which of the following is the correct relationship between the magnitude of a radial electric field and its

associated electric potential ? More than one answer may be correct for the particular case of a point charge

at the origin, but you should choose the correct general relationship.

ANSWER:

Correct

Now consider the figure, which shows several functions of the variable .

Part E

Which curve could indicate the magnitude of the electric field due to a charge located at the origin ( )?

Hint E.1 How to approach the problem

Hint not displayed

ANSWER:

A

B

C

D

E

F

Correct

Part F

Which curve could indicate the electric potential due to a positive charge located at the origin ( )?

Hint F.1 How to approach the problem

Hint not displayed

ANSWER:

A

B

C

D

E

F

Correct

Part G

Which curve could indicate the electric potential due to a negative charge located at the origin ( )?

ANSWER:

A

B

C

D

E

F

Correct

Part H

For either a positive or a negative charge, the electric field points from regions of ______ electric potential.

ANSWER:

higher to lower

lower to higher

Correct

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

Bouncing Electrons

Two electrons, each with mass and charge , are released from positions very far from each other. With respect

to a certain reference frame, electron A has initial nonzero speed toward electron B in the positive x direction,

and electron B has initial speed toward electron A in the negative x direction. The electrons move directly

toward each other along the x axis (very hard to do with real electrons). As the electrons approach each other, they

slow due to their electric repulsion. This repulsion eventually pushes them away from each other.

Part A

Which of the following statements about the motion of the electrons in the given reference frame will be true at

the instant the two electrons reach their minimum separation?

ANSWER:

Electron A is moving faster than electron B.

Electron B is moving faster than electron A.

Both electrons are moving at the same (nonzero) speed in opposite directions.

Both electrons are moving at the same (nonzero) speed in the same direction.

Both electrons are momentarily stationary.

Correct

If at a given moment the electrons are still moving toward each other, then they will be closer in the next instant.

If at a given moment the electrons are moving away from each other, then they were closer in the previous

instant. The electrons will be traveling in the same direction at the same speed at the moment they reach their

minimum separation. Only in a reference frame in which the total momentum is zero (the center of momentum

frame) would the electrons be stationary at their minimum separation.

Part B

What is the minimum separation that the electrons reach?

Hint B.1 How to approach the problem

Hint not displayed

Hint B.2 Find the initial energy

Hint not displayed

Hint B.3 Find the final energy

Hint not displayed

Hint B.4 Find the initial momentum

Hint not displayed

Hint B.5 Find the final momentum

Hint not displayed

Hint B.6 Some math help

Hint not displayed

Express your answer in term of , , , and (where ).

ANSWER:

=

Correct

An experienced physicist might approach this problem by considering the system of electrons in a reference

frame in which the initial momentum is zero. In this frame the initial speed of each electron is . Try solving the

problem this way. Make sure that you obtain the same result for , and decide for yourself which approach is

easier.

Exercise 23.34

A ring of diameter 7.50 is fixed in place and carries a charge of 5.70 uniformly spread over its

circumference.

Part A

How much work does it take to move a tiny 4.00 charged ball of mass 2.00 from very far away to the center

of the ring?

ANSWER:

= 5.47

Correct

Part B

Is it necessary to take a path along the axis of the ring?

ANSWER:

Yes

No

Correct

Part C

Why?

Essay answers are limited to about 500 words (3800 characters maximum, including spaces).

ANSWER: My Answer:

Part D

If the ball is slightly displaced from the center of the ring, what will it do?

Essay answers are limited to about 500 words (3800 characters maximum, including spaces).

ANSWER: My Answer:

Part E

What is the maximum speed it will reach?

ANSWER:

=

74.0

Correct

Exercise 23.43

The electric field at the surface of a charged, solid, copper sphere with radius 0.250 is 4200 , directed

toward the center of the sphere. .

Part A

What is the potential at the center of the sphere, if we take the potential to be zero infinitely far from the sphere?

ANSWER:

= -1050

Correct

Problem 23.62

A small sphere with mass 2.80 hangs by a thread between two large parallel vertical plates 5.00 apart

. The plates are insulating and have uniform surface charge

densities and . The charge on the sphere is = 9.60×10−6

.

Part A

What potential difference between the plates will cause the thread to assume an angle of 30.0 with the vertical?

ANSWER:

= 82.5

Correct

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