Assignment 7: Calculating Potentials and Forces

Due: 8:00am on Wednesday, February 1, 2012

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

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Potential Difference and Potential near a Charged Sheet

Let and be two points near and on the same side of a charged sheet with surface charge

density . The electric field due to such a charged sheet has magnitude everywhere, and the field

points away from the sheet, as shown in the diagram.

Part A

What is the potential difference between points A and B?

Hint A.1 Formula for potential difference

Hint not displayed

Hint A.2 Calculating the line integral

Hint not displayed

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

ANSWER:

Correct

Note that the expression will not yield the correct potential if you apply it to two points on opposite

sides of the sheet. For example, the expression does not indicate that two points on opposite sides of the sheet and

the same distance from it are at the same potential ( ), which is clear from the symmetry of the

situation. If you take care in carrying out the integration to observe the change in the direction of the electric field

as you pass from one side of the sheet to the other, you will find that the potential difference between A and B is

actually given by

.

Recall that the potential difference, the quantity asked for in Part A, is a well-defined quantity for any situation.

The potential, however, is only defined once you pick a point as the zero-potential point. Different choices simply

change the potential by an additive constant, so the potential difference will stay the same, regardless of what point

you designate as having zero potential.

Part B

If the potential at is taken to be zero, what is the value of the potential at a point at some positive

distance from the surface of the sheet?

Hint B.1 How to approach the problem

Hint not displayed

ANSWER:

0

Correct

Part C

Now take the potential to be zero at instead of at infinity. What is the value of at point A some positive

distance from the sheet?

Hint C.1 How to approach the problem

Hint not displayed

ANSWER:

0

Correct

Note that the potential is zero everywhere on the sheet, that is, at every point whose y coordinate is zero. You

always have the freedom to choose a convenient location and reference potential with respect to which other

potentials are measured, since it is potential differences and not absolute potentials that actually matter when one

is doing something with charges in the real world. Potentials, however, are a useful calculational and

bookkeeping tool. For example, if there were four points of interest in an electrical unit, there would be six

possible potential differences, so it would be easier to keep track of the four potentials corresponding to the four

points instead of working with potential differences.

For the case of a charged sheet, it is clear that choosing the potential at the sheet to be zero is a more convenient

choice than choosing the potential to be zero far away from the sheet. In this way, the potentials of points near the

sheet remain finite. The opposite is true for a point charge.

Problem 23.70

A thin insulating rod is bent into a semicircular arc of radius , and a total electric charge is distributed

uniformly along the rod.

Part A

Calculate the potential at the center of curvature of the arc if the potential is assumed to be zero at infinity.

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

ANSWER:

= Correct

The charged ring is the building-block shape of dq to be used in the following cylinder and disk problems--ie build the

cylinder by stacking rings for example

Potential of a Charged Ring

A ring with radius and a uniformly distributed total charge lies in the xy plane, centered at the origin.

Part A

What is the potential due to the ring on the z axis as a function of ?

Hint A.1 How to approach the problem

Hint not displayed

Hint A.2 The potential due to a point charge

Hint not displayed

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

ANSWER:

= Correct

Part B

What is the magnitude of the electric field on the z axis as a function of , for ?

Hint B.1 Determine the direction of the field

Hint not displayed

Hint B.2 The relationship between electric field and potential

Hint not displayed

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

ANSWER:

| | = Correct

Notice that while the potential is a strictly decreasing function of , the electric field first increases till

and then starts to decrease.

Why does the electric field exhibit such a behavior?

Though the contribution to the electric field from each point on the ring strictly decreases as a function of , the

vector cancellation from points on opposite sides of the ring becomes very strong for small . on

account of these vector cancellations. On the other hand , even though all the individual 's point

in (almost) the same direction there, because the contribution to the electric field, per unit length of the ring

as .

Potential of a Charged Cylinder

A hollow cylinder of radius and height has a total charge uniformly distributed over its surface. The axis of

the cylinder coincides with the z axis, and the cylinder is centered at the origin, as shown in the figure.

Part A

What is the electric potential at the origin?

Hint A.1 How to approach the problem

Hint not displayed

Hint A.2 Potential due to a thin ring

Hint not displayed

ANSWER:

Correct

Part B

What is the potential in the limit as goes to zero?

Hint B.1 How to take the limit

Hint not displayed

Express your answer in terms of , , and .

ANSWER:

= Correct

Note that this expression is the same as that for the potential at the center of a charged a ring! The reason for this

is that if the radius of the cylinder is much larger than the length of the cylinder, the cylinder looks and behaves

much like a ring.

Potential of a Charged Disk

A disk of radius has a total charge uniformly distributed over its surface. The disk has negligible thickness and

lies in the xy plane. Throughout this problem, you may use the

variable in place of .

Part A

What is the electric potential on the z axis as a function of , for ?

Hint A.1 How to approach the problem

Hint not displayed

Hint A.2 Find the potential due to a ring

Hint not displayed

Hint A.3 A useful antiderivative

Hint not displayed

Express your answer in terms of , , and . You may use instead of .

ANSWER:

= Correct

Part B

What is the magnitude of the electric field on the axis, as a function of , for ?

Hint B.1 Direction of the electric field

Hint not displayed

Hint B.2 Electric field from potential

Hint not displayed

Express your answer in terms of some or all of the variables , , and . You may use instead of .

ANSWER:

= Correct

Since the magnitude of the electric field (and potential) must be symmetric about the plane, the general

expression for the magnitude of the electric field on the z axis for all is

.

Note the use of instead of .

Potential of a Finite Rod

A finite rod of length has total charge , distributed uniformly along its length. The rod lies on the x -axis and is

centered at the origin. Thus one endpoint is located at , and the other is located at . Define the

electric potential to be zero at an infinite distance away from the rod. Throughout this problem, you may use the

constant in place of the expression .

Part A

What is , the electric potential at point A (see the figure), located a distance above the midpoint of the rod on

the y axis?

Hint A.1 How to approach the problem

Hint not displayed

Hint A.2 Find the electric potential of a section of the rod

Hint not displayed

Hint A.3 A helpful integral

Hint not displayed

Express your answer in terms of , , , and .

ANSWER: =

Correct

If , this answer can be approximated as

.

For , . For this problem, this means that the logarithm can be further approximated as

, and the expression for potential reduces to . This is what we expect, because it means

that from far away, the potential due to the charged rod looks like that due to a point charge.

Part B

What is , the electric potential at point , located at distance from one end of the rod (on the x axis)?

Hint B.1 How to approach the problem

Hint not displayed

Hint B.2 Find the distance from point B to a segment of the rod

Hint not displayed

Give your answer in terms of , , , and .

ANSWER:

= Correct

This result can be written as

.

As before, for , . Thus, for , the logarithm approaches , in which case the result

reduces to . This is what we expect, because it means that from far away, the potential due to the charged rod

looks like that due to a point charge.

Problem 23.64

A Geiger counter detects radiation such as alpha particles by using the fact that the radiation ionizes the air along

its path. A thin wire lies on the axis of a hollow metal cylinder and is insulated from it

. A large potential difference is established between the wire and

the outer cylinder, with the wire at higher potential; this sets up a strong electric field directed radially outward.

When ionizing radiation enters the device, it ionizes a few air molecules. The free electrons produced are

accelerated by the electric field toward the wire and, on the way there, ionize many more air molecules. Thus a

current pulse is produced that can be detected by appropriate electronic circuitry and converted to an audible

"click." Suppose the radius of the central wire is 145 and the radius of the hollow cylinder is 1.80 .

Part A

What potential difference between the wire and the cylinder produces an electric field of at a

distance of 1.20 from the axis of the wire? (Assume that the wire and cylinder are both very long in

comparison to their radii.)

ANSWER:

= 1160

Correct

Charged Mercury Droplets

A uniformly charged spherical droplet of mercury with electric potential breaks into identical spherical

droplets, each with electric potential . The small droplets are far enough apart form one another that they do

not interact significantly.

Part A

Find , the ratio of , the electric potential of the initial drop, to , the electric potential of one of the

smaller drops.

Hint A.1 How to approach the problem

Hint not displayed

Hint A.2 Find the charge on the small droplets

Hint not displayed

Hint A.3 Find the radius of a small droplet

Hint not displayed

The ratio should be dimensionless and should depend only on

ANSWER:

=

Correct