Solutions] Mastering Physics HW11

HW11 Due: 11:59pm on Monday, September 28, 2009 Note: To understand how points are awarded, read your instructor's Grading Policy. [Return to Standard Assignment View]

In Parts A-F of "Electric Force and Potential: Spherical Symmetry", the point charge is positive and the zero of potential is understood to be an infinite distance from the point charge. GBA

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 electric potential

and its associated

? 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 Hint E.1

How to approach the problem Hint not displayed

located at the origin (

)?

ANSWER:

A B C D E F Correct

Part F Which curve could indicate the electric potential due to a positive charge Hint F.1

located at the origin (

)?

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 ANSWER:

located at the origin (

)?

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

HINT: In both of these ranking tasks, at least two of the six items should be ranked equally. You do this by putting those two items in the same column in the ranking box. GBA

Electric Potential Ranking Task In the figurethere are two point charges,

and

. There are

also six positions, labeled A through F, at various distances from the two point charges. You will be asked about the electric potential at the different points (A through F).

Part A Rank the locations A to F on the basis of the electric potential at each point. Rank positive electric potentials as larger than negative electric potentials. Hint A.1

Definition of electric potential Hint not displayed

Hint A.2

Conceptualizing electric potential Hint not displayed

Rank the locations from largest to smallest potential. To rank items as equivalent, overlap them. ANSWER:

View Correct

Change in Electric Potential Ranking Task In the diagram below, there are two charges of

and

and six points (a through f) at various distances from the two

charges. You will be asked to rank changes in the electric potential along paths between pairs of points.

Part A Using the diagram to the left, rank each of the given paths on the basis of the change in electric potential. Rank the largestmagnitude positive change (increase in electric potential) as largest and the largest-magnitude negative change (decrease in electric potential) as smallest. Hint A.1

Change in electric potential Hint not displayed

Hint A.2

Determine the algebraic sign of the change in potential

Hint A.3

Conceptualizing changes in electric potential

Hint not displayed

Hint not displayed Rank from largest to smallest. To rank items as equivalent, overlap them. ANSWER: View Correct

Think carefully about the sign of your answer for Part (c) of Problem 23.66. Make sure that you understand the physical reason for this sign. GBA

Problem 23.66 A charge

is placed at the origin of an xy-coordinate system, and a charge

positive x-axis at

. A third charge

is placed on the

is now placed at the point

,

.

Part A Calculate the potential at the point

,

due to the first two charges. Let the potential be zero far from the

charges. ANSWER:

-300 V Correct

Part B Calculate the potential at the point ANSWER:

,

due to the first two charges.

419 V Correct

Part C If the third charge moves from the point

,

to the point

,

, calculate the work

done on it by the field of the first two charges. ANSWER:

−4.31×10 −6 J Correct

NOTE: In "Potential of a Charged Ring", "Potential of a Charged Annulus", and "Potential of a Finite Rod", you MAY type your answers in terms of k. Do not believe any MP error responses that read "The correct answer does not depend on the variable: k". Due to a bug in these three problems, MP sometimes gives this response when the error is something else entirely. If you get this response, you should regard it as "Try Again". GBA

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 Hint A.1

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

How to approach the problem

The formula for the electric potential produced by a static charge distribution involves the amount of charge and the distance from the charge to the position where the potential is measured. All points on the ring are equidistant from a given point on the z axis. This enables you to calculate the electric potential simply, without doing an integral. Hint A.2

The potential due to a point charge

If you incorporate the symmetry of the problem, you will need only to know the formula for potential of a point charge: , where is the potential at distance from the point charge, is the magnitude of the charge, and

is

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

, ,

, and

or

.

ANSWER: = Correct

Part B What is the magnitude of the electric field Hint B.1

on the z axis as a function of , for

?

Determine the direction of the field

By symmetry, the electric field has only one Cartesian component. In what direction does the electric field point? ANSWER:

Correct

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 . these vector cancellations. On the other hand

, even though all the individual

same direction there, because the contribution to the electric field, per unit length of the ring

on account of 's point in (almost) the as

.

For the "Potential of a Charged Annulus", I do not recommend following the MP hints. From our work with electric field integration, you should know how to write an integral over an annulus; and in this case the integral is simpler to write since potential is a scalar rather than a vector. GBA

Potential of a Charged Annulus An annular ring with a uniform surface charge density The annulus has an inner radius

and outer radius

sits in the xy plane, with its center at the origin of the coordinate axes. .

Part A If you can find symmetries in a physical situation, you can often greatly simplify your calculations. In this part you will find a symmetry in the annular ring before calculating the potential along the axis through the ring's center in Part B. Consider three sets of points: points lying on the vertical line A; those on circle B; and those on the horizontal line C, as shown in the figure. Which set of points makes the same contribution toward the potential calculated at any point along the axis of the annulus?

Hint A.1

Definition of the potential due to a point charge Hint not displayed

ANSWER:

points on line A points on circle B points on line C Correct

Part B By exploiting the above symmetry, or otherwise, calculate the electric potential distance

at a point on the axis of the annulus a

from its center.

Hint B.1

How to exploit the angular symmetry of the problem

The total potential

at a point on the axis of the annulus can be written as ,

where

is the distance from a point on the annulus to the point at which the potential is to be determined. However, on

account of the angular symmetry of this problem, it is more convenient to write this integral in terms of polar coordinates: .

The integral over

is easy and should be done first, since the integrand has no dependence on . This will put the integral

in the form ,

where

is the area of a thin annular slice of thickness

Hint B.2

Find the area of an annular slice

What is

, the area of a thin annular slice of thickness

Express your answer in terms of ANSWER:

and

and radius .

and radius ?

.

= Correct

Now do the integral over . Don't forget that

Hint B.3

is a function of . You will need to use a variable substitution.

Doing the integral Hint not displayed

Hint B.4

A formula for the integral Hint not displayed

Express your answer in terms of some or all of the variables

,

,

, and

. Use

.

ANSWER: = Correct

It is interestering to note that the potential at any point on the axis of a disk of radius

can be obtained from the

expression above by setting

and

. Doing so, one obtains .

Conversely, the annulus can be thought of as the superposition of two disks, one with charge density and the other with charge density

and radius

. In the region from the center to

and radius

,

, the opposite charge densities

cancel out, so the net charge distribution would be just like that of the annulus. Moreover, by adding the potentials due to these two disks, using the formula above, you would recover the potential of the annulus. It is also instructive to look at the general behavior of these potentials as a function of the parameters. Clearly, the potential increases with increasing charge densities, as well as with increasing areas (if the charge density is held constant), which intuitively seems reasonable. However, if the distance increases, it is not clear whether the potential should grow, since

appears in both terms, of which one is subtracted from the other. If you are far from the disk, the

disk looks like a point, and the potential should drop off, just like the potential due to a point charge. Indeed, on account of the negative second term in the expressions, this is the case. Try some values or check that the derivative of is indeed negative. You can also check that the above expression actually reduces to the potential due to a point charge for .

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

Hint A.1

above the midpoint of the rod on the y axis?

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 to

,

. Thus, for

, the logarithm approaches

, in which case the result reduces

. 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.

Score Summary: Your score on this assignment is 98.3%. You received 49.17 out of a possible total of 50 points.