Electric Potential Energy versus Electric Potential

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E ð lisfræ ð i 2, vor 2007

23a. Electric Potential

Assignment is due at 2:00am on Wednesday, January 31, 2007

Credit for problems submitted late will decrease to 0% after the deadline has passed.

The wrong answer penalty is 2% per part. Multiple choice questions are penalized as described in the online help.

The unopened hint bonus is 2% per part.

You are allowed 4 attempts per answer.

Electric Potential and Properties

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: =

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 .

Express in terms of , , and . Note that because potential energy is a scalar, and not a vector, there will be no unit vector in the answer.

ANSWER: =

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

Electric Force and Potential Energy

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

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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: =

Part E

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

Express (a scalar quantity) in terms of , , and .

ANSWER: =

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

Electric Field and Electric Potential

The electric potential is defined by the relationship with charge .

, where is the electric potential energy of a particle

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: =

Part H

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. What are the dimensions of the volt in terms of the fundamental SI units?

Express your answer in terms of the standard abbreviations for the fundamental SI units:

(meters), (kilograms), (seconds), and (coulombs)

ANSWER:

volts =

Part I

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

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gradient operator:

.

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

Consider again the electric potential coordinate only, so and .

corresponding to the field . This potential depends on the z

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 derivative of with respect to .

ANSWER: =

for the

Part J

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

Part J.1 Direction of the electric field

In what direction do electric fields point?

ANSWER: from regions of higher electric potential to lower electric potential from regions of lower electric potential to higher electric potential

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

The force on a charge in an electric field is given by

.

This is similar to the equation for the force on a mass in a uniform gravitational field.

Hint J.3 Formula for electric potential energy

The electric potential energy of a charge at electric potential is given by

.

This is similar to the equation , for the gravitational potential energy of a particle with mass .

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

ANSWER: higher; higher higher; lower lower; higher lower; lower

Part K

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

Part K.1 Direction of the electric field

Part not displayed

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


Hint K.3 Formula for electric potential energy


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

ANSWER: higher; higher higher; lower lower; higher lower; lower

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: Answer not displayed

Part B


Part C


Charges moving in Electric Potential

The Fate of an Electron in a Uniform Electric Field

In this problem we will study the behavior of an electron in a uniform electric field. Consider a uniform electric field (magnitude ) as shown in the figure within a parallel plate capacitor in vacuum.

First, let us review the relationship between an electric field and its associated electric potential . For now, ignore the electron located between the plates.

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Part A

Calculate the electric potential inside the capacitor as a function of height . Take the potential at the bottom plate to be zero.

Hint A.1 Relationship of field and potential


Hint A.2 Limits of integration


Express in terms of and .

ANSWER: =

Now an electron of mass and charge (where is a positive quantity) is placed within the electric field (see the figure) at height .

Part B

Calculate the electon's potential energy , neglecting gravitational potential energy.

Hint B.1 Definition of electric potential energy


Express your answer in terms of , , and .

ANSWER: =

Part C

The electron, having been held at height , is now released from rest. Calculate its speed (i.e., ) when it reaches the top plate.

Part C.1 How to approach the problem


Part C.2 Find the initial energy


Part C.3 Find the final energy


Express in terms of , , , , , and other given quantities and constants.

ANSWER:

=

Now we consider the effect of changing either the charge or the mass of the charged particle that is released from rest at height .

Part D

Imagine a particle that has three times the mass of the electron. All other quantities given above remain the same.

What is the ratio of the kinetic energy that this heavy particle would have when it reaches the upper plate to the kinetic energy that the electron would have? That is, what is ?

Hint D.1 Energy conservation

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In the motion of both particles, total energy is conserved. Does the potential energy of the particles depend on their mass?

ANSWER: 3

1 none of the above: the heavy particle will never reach the upper plate

Part E

Imagine a third particle, which we will call a cyberon. It has three times the mass of an electron ( ). It has a positive charge that is three times the magnitude ( ) of the charge on an electron. What is the ratio of the speed that the cyberon would have when it reaches the upper plate after being released from rest at position to the speed that the electron would have? That is, what is ?

ANSWER: 3

1 none of the above: the cyberon will never reach the upper plate

Because it has positive instead of negative charge, the cyberon will accelerate downward, toward the lower plate.

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: Answer not displayed

Part B


Energy in Configuration of Charges

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 .

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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

The Coulomb force is conservative. If we define the potential energy of the system to be zero when the charges are infinitely far apart, the amount of work needed to place any one charge in a configuration is equal to its electric potential energy. Imagine moving charge A, then B, then C, and finally D into place. Find the work required to add each charge to the configuration by calculating the potential energy of each just after it is added.

Add the work required for each charge to find the total work required.

Hint A.2 Electric potential and potential energy

Recall that the electric potential at a point at distance from a charge is , where . Note that this equation implicitly defines the electric potential to be zero at . The electric potential energy of a charge is equal to , where is the electric potential at the position of the charge before the charge is placed there. To find the potential at a point due to multiple charges, sum the potentials at that point due to each charge.

Part A.3 Work required to place charge A

What is , the work required to assemble the charge distribution shown in the figure?

Part A.3.a Find the potential at the location of charge A


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

ANSWER: = 0

Part A.4 Work required to place charge B

What is , the amount of work required to add charge B to the configuration, as shown in the figure?

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Part A.4.a Find the potential at the location of charge B


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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

Summary 3 of 5 problems complete (56.77% avg. score)

14.19 of 15 points

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Electric Potential Energy versus Electric Potential

Electric Potential Energy versus Electric Potential

Electric Potential, Potential Energy, and Force

The Fate of an Electron in a Uniform Electric Field

Bouncing Electrons

Energy Stored in a Charge Configuration

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Electric Potential Energy versus Electric Potential