Assignment 2: Electric Fields

Due: 8:00am on Friday, January 13, 2012

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

[Switch to Standard Assignment View]

The Electric Field at a Point Due to Two Point Charges

A point charge is at the point meters, meters, and a second point charge

is at the point meters, .

Part A

Calculate the magnitude of the net electric field at the origin due to these two point charges.

Hint A.1 How to approach the problem

Hint not displayed

Hint A.2 Calculate the x component of the field created by

Hint not displayed

Hint A.3 Calculate the y component of

Hint not displayed

Hint A.4 Calculate the magnitude of the field created by the first charge

Hint not displayed

Hint A.5 Calculate the x component of

Hint not displayed

Hint A.6 Calculate the y component of

Hint not displayed

Hint A.7 Putting it all together

Hint not displayed

Express your answer in newtons per coulomb to three significant figures.

ANSWER:

= 131

Correct

Part B

What is the direction, relative to the negative x axis, of the net electric field at the origin due to these two point

charges.

Hint B.1 How to approach the problem

Hint not displayed

Express your answer in degrees to three significant figures.

ANSWER:

= 12.6

Correct up from the negative x axis

The Trajectory of a Charge in an Electric Field

An charge with mass and charge is emitted from the origin, . A large, flat screen is located at

. There is a target on the screen at y position , where . In this problem, you will examine two

different ways that the charge might hit the target. Ignore gravity in this problem.

Part A

Assume that the charge is emitted with velocity in the positive x direction. Between the origin and the screen,

the charge travels through a constant electric field pointing in the positive y direction. What should the magnitude

of the electric field be if the charge is to hit the target on the screen?

Hint A.1 How to approach the problem

Hint not displayed

Hint A.2 Find the equation of motion in the x direction

Hint not displayed

Hint A.3 Find the equation of motion in the y direction

Hint not displayed

Hint A.4 Combine Your Results

Hint not displayed

Hint A.5 Find

Hint not displayed

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

ANSWER:

= Correct

Part B

Now assume that the charge is emitted with velocity in the positive y direction. Between the origin and the

screen, the charge travels through a constant electric field pointing in the positive x direction. What should the

magnitude of the electric field be if the charge is to hit the target on the screen?

Hint B.1 How to approach the problem

Hint not displayed

Hint B.2 Find the equation of motion in the y direction

Hint not displayed

Hint B.3 Find the equation of motion in the x direction

Hint not displayed

Hint B.4 Combine your results

Hint not displayed

Hint B.5 Find

Hint not displayed

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

ANSWER:

= Correct

The equations of motion for this part are identical to the equations of motion for the previous part, with and

interchanged. Thus it is no surprise that the answers to the two parts are also identical, with and

interchanged.

An integral useful in the next problem can be found in Appendix B of our text

The Electric Field Produced by a Finite Charged Wire

A charged wire of negligible thickness has length units and

has a linear charge density . Consider the electric field at the point , a distance above the midpoint of the

wire.

Part A

The field points along one of the primary axes. Which one?

Hint A.1 Consider opposite ends of the wire

Hint not displayed

ANSWER:

Correct

Part B

What is the magnitude of the electric field at point ? Throughout this part, express your answers in terms of

the constant , defined by .

Hint B.1 How to approach the problem

Hint not displayed

Hint B.2 Find the field due to an infinitesimal segment

Hint not displayed

Hint B.3 A necessary integral

Hint not displayed

Express your answer in terms of , , , and .

ANSWER:

= Correct

In part d below you might want to remind yourself of the spring equation, F = ma = -kx

Charged Ring

Consider a uniformly charged ring in the xy plane, centered at the origin. The ring has radius and positive charge

distributed evenly along its circumference.

Part A

What is the direction of the electric field at any point on the z axis?

Hint A.1 How to approach the problem

Hint not displayed

ANSWER:

parallel to the x axis

parallel to the y axis

parallel to the z axis

in a circle parallel to the xy plane

Correct

Part B

What is the magnitude of the electric field along the positive z axis?

Hint B.1 Formula for the electric field

Hint not displayed

Hint B.2 Simplifying with symmetry

Hint not displayed

Hint B.3 Integrating around the ring

Hint not displayed

Use in your answer, where .

ANSWER:

= Correct

Notice that this expression is valid for both positive and negative charges as well as for points located on the

positive and negative z axis. If the charge is positive, the electric field should point outward. For points on the

positive z axis, the field points in the positive z direction, which is outward from the origin. For points on the

negative z axis, the field points in the negative z direction, which is also outward from the origin. If the charge is

negative, the electric field should point toward the origin. For points on the positive z axis, the negative sign from

the charge causes the electric field to point in the negative z direction, which points toward the origin. For points

on the negative z axis, the negative sign from the z coordinate and the negative sign from the charge cancel, and

the field points in the positive z direction, which also points toward the origin. Therefore, even though we

obtained the above result for postive and , the algebraic expression is valid for any signs of the parameters. As

a check, it is good to see that if is much greater than the magnitude of is approximately , independent

of the size of the ring: The field due to the ring is almost the same as that due to a point charge at the origin.

Part C

Imagine a small metal ball of mass and negative charge . The ball is released from rest at the point

and constrained to move along the z axis, with no damping. If , what will be the ball's subsequent

trajectory?

ANSWER:

repelled from the origin

attracted toward the origin and coming to rest

oscillating along the z axis between and

circling around the z axis at

Correct

Part D

The ball will oscillate along the z axis between and in simple harmonic motion. What will be the

angular frequency of these oscillations? Use the approximation to simplify your calculation; that is,

assume that .

Hint D.1 Simple harmonic motion

Hint not displayed

Hint D.2 Find the force on the charge

Hint not displayed

Express your answer in terms of given charges, dimensions, and constants.

ANSWER:

=

Correct

Careful application of the result from Example 21.11 in text is helpful in the next (finite sheets) problem

What About Finite Sheets?

Frequently in physics, one makes simplifying approximations. A common one in electricity is the notion of infinite

charged sheets. This approximation is useful when a problem deals with points whose distance from a finite

charged sheet is small compared to the size of the sheet. In this problem, you will look at the electric field from

two finite sheets and compare it to the results for infinite sheets to get a better idea of when this approximation is

valid.

Consider two thin disks, of negligible thickness, of radius oriented perpendicular to the x axis such that the x

axis runs through the center of each disk. The disk centered at

has positive charge density , and the disk centered at has negative charge density , where the

charge density is charge per unit area.

Part A

What is the magnitude of the electric field at the point on the x axis with x coordinate ?

Hint A.1 How to approach the problem

Hint not displayed

Hint A.2 The magnitude of the electric field due to a single disk

Hint not displayed

Hint A.3 Determine the general form of the electric field between the disks

Hint not displayed

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

ANSWER:

=

Correct

Notice that as approaches , this expression approaches , the result for two infinite sheets. Also, note

that the minimum value of the electric field, which corresponds in this case to the greatest deviation from the

result for two infinite sheets, occurs halfway between the disks (i.e., at ).

Part B

For what value of the ratio of plate radius to separation between the plates does the electric field at the point

on the x axis differ by 1 percent from the result for infinite sheets?

Hint B.1 Percent difference

Hint not displayed

Express your answer to two significant figures.

ANSWER:

= 50

Correct

As mentioned above, this is the point on the x axis where the deviation from the result for two infinite sheets is

greatest. A common component of electrical circuits called a capacitor is usually made from two thin charged

sheets that are separated by a small distance. In such a capacitor, the ratio is far greater than 50. Based on

your result, you can see that the infinite sheet approximation is quite good for a capacitor.

This applet shows the electric field lines from a pair of finite plates (viewed edge-on). You can adjust the surface

charge density. You can also move the test charge around and increase or decrease its charge to see what sort of

force it would experience. Notice that the deviation from uniform electric field only becomes noticeable near the

edges of the capacitor plates.

Problem 21.96 is from the old 12th edition

Problem 21.96

Positive charge is uniformly distributed around a semicircle of radius .

Part A

Find the magnitude of the electric field at the center of curvature P.

Express your answer in terms of the given quantities and appropriate constants.

ANSWER:

= Correct

Part B

What is the direction of the electric field at the center of curvature P.

ANSWER:

downward

upward

Correct

Problem 21.99

Two 1.20 nonconducting wires meet at a right angle. One segment carries 4.00 of charge distributed

uniformly along its length, and the other carries 4.00 distributed uniformly along it, as shown in the figure

.

Part A

Find the magnitude of the electric field these wires produce at point , which is 60.0 from each wire.

ANSWER:

=

1.00×105

Correct

Part B

Find the direction of the electric field these wires produce at point , which is 60.0 from each wire.(Suppose

that the -axis directed vertically.)

ANSWER:

= 135

Correct countercockwise from the -axis

Part C

If an electron is released at , what is the magnitude of the net force that these wires exert on it?

ANSWER:

=

1.60×10−14

Correct

Part D

If an electron is released at , what is the direction of the net force that these wires exert on it?(Suppose that the

-axis directed vertically.)

ANSWER:

= 315

Correct countercockwise from the -axis