MP EM Ass 5: Applications of Gauss's Law

Assignment 5: Applications of Gauss's Law

Due: 8:00am on Wednesday, January 25, 2012

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

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The Electric Field of a Ball of Uniform Charge Density

A solid ball of radius has a uniform charge density .

Part A

What is the magnitude of the electric field at a distance from the center of the ball?

Hint A.1 Gauss's law

Hint not displayed

Hint A.2 Find

Hint not displayed

Express your answer in terms of , , , and .

ANSWER:

= Correct

Notice that this result is identical to that reached by applying Coulomb's law to a point charge centered at the

origin with . The field outside of a uniformly charged sphere does not depend on the size of the sphere,

only on its charge. A uniformly charged sphere generates an electric field as if all the charge were concentrated at

its center.

Part B

What is the magnitude of the electric field at a distance from the center of the ball?

Hint B.1 How does this situation compare to that of the field outside the ball?

Hint not displayed

Express your answer in terms of , , , and .

ANSWER:

= Correct

Part C

Let represent the electric field due to the charged ball throughout all of space. Which of the following

statements about the electric field are true?

Hint C.1 Plot the electric field

Hint not displayed

Check all that apply.

ANSWER:

.

.

.

The maximum electric field occurs when .

The maximum electric field occurs when .

The maximum electric field occurs as

Correct

± The Charge on a Thundercloud

In a thunderstorm, charge builds up on the water droplets or ice crystals in a cloud. Thus, the charge can be

considered to be distributed uniformly throughout the cloud. For the purposes of this problem, take the cloud to be

a sphere of diameter 1.00 kilometer. The point of this problem is to estimate the maximum amount of charge that

this cloud can contain, assuming that the charge builds up until the electric field at the surface of the cloud reaches

the value at which the surrounding air breaks down. This breakdown means that the air becomes highly ionized,

enabling it to conduct the charge from the cloud to the ground or another nearby cloud. The ionized air will then

emit light due to the recombination of the electrons and atoms to form excited molecules that radiate light. In

addition, the large current will heat up the air, resulting in its rapid expansion. These two phenomena account for

the appearance of lightning and the sound of thunder. Take the breakdown electric field of air to be

.

Part A

Estimate the total charge on the cloud when the breakdown of the surrounding air is reached.

Hint A.1 Use Gauss's law

Hint not displayed

Hint A.2 Evaluate Gauss's law

Hint not displayed

Express your answer numerically, to three significant figures, using .

ANSWER:

=

83.4

Correct Coulombs

Problem 22.56

A slab of insulating material has thickness and is oriented so that its faces are parallel to the yz-plane and given

by the planes and . The y- and z-dimensions of the slab are very large compared to and may be

treated as essentially infinite. The slab has a uniform positive charge density .

Part A

Explain why the electric field due to the slab is zero at the center of the slab .

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

ANSWER: My Answer:

Part B

Using Gauss's law, find the magnitude of the electric field due to the slab at the points .

Express your answer in terms of the variables , , , and .

ANSWER:

= Correct

Part C

What is the direction of the electric field due to the slab at the points ?

ANSWER:

+x-direction

-x-direction

Correct

Part D

Using Gauss's law, find the magnitude of the electric field due to the slab at the points .


ANSWER:

= Correct

Part E

What is the direction of the electric field due to the slab at the points ?

ANSWER:

+x-direction

-x-direction

Correct

A Conducting Shell around a Conducting Rod

An infinitely long conducting cylindrical rod with a positive

charge per unit length is surrounded by a conducting cylindrical shell (which is also infinitely long) with a charge

per unit length of and radius , as shown in the figure.

Part A

What is , the radial component of the electric field between the rod and cylindrical shell as a function of the

distance from the axis of the cylindrical rod?

Hint A.1 The implications of symmetry

Hint not displayed

Hint A.2 Apply Gauss' law

Hint not displayed

Hint A.3 Find the charge inside the Gaussian surface

Hint not displayed

Hint A.4 Find the flux

Hint not displayed

Express your answer in terms of , , and , the permittivity of free space.

ANSWER:

= Correct

Part B

What is , the surface charge density (charge per unit area) on the inner surface of the conducting shell?

Hint B.1 Apply Gauss's law

Hint not displayed

Hint B.2 Find the charge contribution from the surface

Hint not displayed

ANSWER:

=

Correct

Part C

What is , the surface charge density on the outside of the conducting shell? (Recall from the problem

statement that the conducting shell has a total charge per unit length given by .)

Hint C.1 What is the charge on the cylindrical shell?

Hint not displayed

ANSWER:

=

Correct

Part D

What is the radial component of the electric field, , outside the shell?

Hint D.1 How to approach the problem

Hint not displayed

Hint D.2 Find the charge within the Gaussian surface

Hint not displayed

Hint D.3 Find the flux in terms of the electric field

Hint not displayed

ANSWER:

= Correct

The Electric Field and Surface Charge at a Conductor

Learning Goal: To understand the behavior of the electric field at the surface of a conductor, and its relationship

to surface charge on the conductor.

A conductor is placed in an external electrostatic field. The external field is uniform before the conductor is placed

within it. The conductor is completely isolated from any source of current or charge.

Part A

Which of the following describes the electric field inside this conductor?

ANSWER:

It is in the same direction as the original external field.

It is in the opposite direction from that of the original external field.

It has a direction determined entirely by the charge on its surface.

It is always zero.

Correct

The net electric field inside a conductor is always zero. If the net electric field were not zero, a current would

flow inside the conductor. This would build up charge on the exterior of the conductor. This charge would

oppose the field, ultimately (in a few nanoseconds for a metal) canceling the field to zero.

Part B

The charge density inside the conductor is:

ANSWER:

0

non-zero; but uniform

non-zero; non-uniform

infinite

Correct

You already know that there is a zero net electric field inside a conductor; therefore, if you surround any internal

point with a Gaussian surface, there will be no flux at any point on this surface, and hence the surface will

enclose zero net charge. This surface can be imagined around any point inside the conductor with the same result,

so the charge density must be zero everywhere inside the conductor. This argument breaks down at the surface of

the conductor, because in that case, part of the Gaussian surface must lie outside the conducting object, where

there is an electric field.

Part C

Assume that at some point just outside the surface of the conductor, the electric field has magnitude and is

directed toward the surface of the conductor. What is the charge density on the surface of the conductor at that

point?

Hint C.1 How to approach the problem

Hint not displayed

Hint C.2 Calculate the flux through the top of the cylinder

Hint not displayed

Hint C.3 Calculate the flux through the bottom of the box

Hint not displayed

Hint C.4 What is the charge inside the Gaussian surface?

Hint not displayed

Hint C.5 Apply Gauss's law

Hint not displayed

Express your answer in terms of and .

ANSWER:

=

Correct

The Charge Inside a Conductor

A spherical cavity is hollowed out of the interior of a neutral

conducting sphere. At the center of the cavity is a point charge, of positive charge .

Part A

What is the total surface charge on the interior surface of the conductor (i.e., on the wall of the cavity)?

Hint A.1 Gauss's law and properties of conductors

Hint not displayed

ANSWER:

=

Correct

Part B

What is the total surface charge on the exterior surface of the conductor?

Hint B.1 Properties of the conductor

Hint not displayed

ANSWER:

=

Correct

Part C

What is the magnitude of the electric field inside the cavity as a function of the distance from the point

charge? Let , as usual, denote .

Hint C.1 How to approach the problem

Hint not displayed

Hint C.2 Charge distributions and finding the electric field

Hint not displayed

ANSWER:

0

Correct

Part D

What is the electric field outside the conductor?

Hint D.1 How to approach the problem

Hint not displayed

Hint D.2 The distribution of

Hint not displayed

ANSWER:

zero

the same as the field produced by a point charge located at the center of the sphere

the same as the field produced by a point charge located at the position of the charge in the

cavity

Correct

Now a second charge, , is brought near the outside of the conductor. Which of the following quantities would

change?

Part E

The total surface charge on the wall of the cavity, :

Hint E.1 Canceling the field due to the charge

Hint not displayed

ANSWER:

would change

would not change

Correct

Part F

The total surface charge on the exterior of the conductor, :

Hint F.1 Canceling the field due to the charge

Hint not displayed

ANSWER:

would change

would not change

Correct

Part G

The electric field within the cavity, :

ANSWER:

would change

would not change

Correct

Part H

The electric field outside the conductor, :

ANSWER:

would change

would not change

Correct

Charge Distribution on a Conductor with a Cavity

A positive charge is brought close to a fixed neutral conductor that has a cavity. The cavity is neutral; that is, there

is no net charge inside the cavity.

Part A

Which of the figures best represents the charge distribution on the inner and outer walls of the conductor?

Hint A.1 Conductors have no internal field

Hint not displayed

Hint A.2 Charges on the cavity walls

Hint not displayed

ANSWER:

1

2

3

Correct

Problem 22.58

A nonuniform, but spherically symmetric, distribution of charge has a charge density given as follows:

for

for

where is a positive constant.

Part A

Find the total charge contained in the charge distribution.

Express your answer in terms of the variables , , , and appropriate constants.

ANSWER:

=

0

Correct

Part B

Obtain an expression for the electric field in the region .


ANSWER:

= 0

Correct

Part C

Obtain an expression for the electric field in the region .


ANSWER:

= Correct

Part D

Find the value of at which the electric field is maximum.


ANSWER:

=

Correct

Part E

Find the value of that maximum field.


ANSWER:

= Correct

Problem 22.62

A very long, solid insulating cylinder with radius has a cylindrical hole with radius bored along its entire

length. The axis of the hole is a distance from the axis of the cylinder, where

. The solid material of the cylinder has a uniform volume charge

density .

Part A

Find the magnitude and direction of the electric field inside the hole.


ANSWER:

= Correct

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MP EM Ass 5: Applications of Gauss's Law