Phys320 Week 4 Homework Answers 6l6u2u

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Assignment 4 Due: 12:00pm on Tuesday, April 28, 2009 Note: To understand how points are awarded, read your instructor's Grading Policy.

Placing Charges Conceptual Question Description: Simple conceptual question about placing various charges around a central charge such that the free body diagram given is correct. Below are free-body diagrams for three electric charges that lie in the same plane. Their relative positions are unknown. Part A Along which of the lines (A to H) in the figure should charge 2 be placed so that the free-body diagrams of charge 1 and charge 2 are consistent?

Hint A.1

How to approach the problem

Newton’s 3rd law states that the forces exerted by a pair of objects on each other are always equal in magnitude and opposite in direction. Identifying the forces that correspond to 3rd-law pairs in the free-body diagrams will enable you to place the particles in their proper relative position. Hint A.2

Placing charge 2

The two forces acting on charge 2 correspond to the forces exerted on it by charge 1 and charge 3. This means that one of these forces must pair with a force on charge 1 of equal magnitude and opposite direction and the other must pair with a force on charge 3 of equal magnitude and opposite direction. Also note that charge 2 should be repelled by charge 1, since both are negative. Therefore, the vector that represents the force of charge 1 on charge 2 must point away from charge 1. This information is all you need to place charge 2 in its correct position. ANSWER:

Part B Along which of the lines (A to H) in the figure should charge 3 be placed so that the free-body diagrams of charge 1, charge 2, and charge 3 are consistent?

ANSWER:

D

Part C Along which of the lines (A to H) in the figure should charge 2 be placed so that the free-body diagrams of charge 1 and charge 2 are consistent?

ANSWER:

H

Part D Along which lines (A to H) in the figure should charge 3 be placed so that the free-body diagrams of charge 1, charge 2, and charge 3 are consistent?

ANSWER:

F

The Sign of the Charge Description: Short conceptual problem on the static charge of glass marbles that have been rubbed with silk. This problem does not take into the effects of polarization within an insulator. This problem is based on Young/Geller Conceptual Analysis 17.1. Part A A glass marble is rubbed against a piece of silk. As a result the piece of fabric acquires extra electrons. What happens to the glass marble? Hint A.1

How to approach the problem

The key concept in this problem is the fact that electric charge is never created, but it is simply transferred from one object to another. Therefore, if the piece of silk acquired extra electrons after the glass marble was rubbed against it, these electrons must have been taken from the glass marble, leaving the marble with a deficiency of electrons. Since electrons have a negative charge, this should give you an indication of the type of electric charge the marble has after the transfer. Hint A.2

Electrostatic interaction between charges

Two like charges, either both positive or both negative, repel each other. A positive and a negative charge attract each other. Check all that apply. ANSWER:

The marble has lost the same number of electrons acquired by the piece of silk. The marble has acquired the same number of electrons acquired by the piece of silk. The marble acquires a positive charge and repels the piece of silk. The marble acquires a positive charge and attracts the piece of silk.

The marble acquires a negative charge and attracts the piece of silk. The marble acquires a negative charge and repels the piece of silk. This was a simple example of electrostatic interactions. When you rub a piece of glass against a piece of silk, the glass acquires a positive charge and the silk acquires a negative charge because some electrons were transferred from the glass to the silk in the rubbing process. The silk acquires the same net charge as the glass, but with the opposite sign. This charge distribution causes the silk and the glass marble to be attracted to one another. Part B Two glass marbles (1 and 2), each ed by a nylon thread, are rubbed against a piece of silk and then are placed near a third glass marble (3), also ed by a similar thread. Assuming that marble 3 has not been in with the piece of fabric, which of the following statements best describes the situation when the three marbles are brought together? Ignore the effects of polarization. Hint B.1

How to approach the problem

As was stated in the previous part, glass marbles acquire a positive charge when they are rubbed against a piece of silk. Therefore, marbles 1 and 2 have both the same charge and repel each other. What charge does marble 3 have? Hint B.2

Charged and uncharged objects

When an object has an excess of electrons, it has a negative net charge. When an object has a deficiency of electrons, it has a positive net charge. When an object has the same number of electrons and protons, then it has a zero net charge and it is said to be neutral. Typically, uncharged or neutral objects do not interact with electrically charged objects. (Interactions between neutral and charged objects are observed when polarization occurs; however, this phenomenon is ignored in this problem.) Most of the objects in everyday life are neutral; they acquire a nonzero net charge only if subjected to processes that involve a transfer of electrons. Rubbing a glass marble with a piece of silk is an example of such a process. Thus, both marbles 1 and 2 have a nonzero electric charge. Marble 3, which hasn't been in with the fabric, remains neutral. ANSWER:

Marbles 1 and 2 attract each other, but no interaction occurs with marble 3. Both marbles 1 and 2 attract marble 3. The three marbles will repel each other. Marbles 1 and 2 repel each other, but no interaction occurs with marble 3.

As you have seen here, electrostatic interactions occur between charged objects. Objects with like charges repel each other, whereas objects with opposite charges attract each other.

Magnitude and Direction of Electric Fields Description: Short quantitative problem for which students must find the charge producing an electric field and sum the electric fields produced by two charges. This problem is based on Young/Geller Conceptual Analysis 17.6. A small object A, electrically charged, creates an electric field. At a point P located 0.250

directly north of A, the field has a value of 40.0

directed to the south.

Part A What is the charge of object A? Hint A.1

How to approach the problem

Recall that the electric field at a point P due to a point charge is proportional to the magnitude of the charge and inversely proportional to the square of the distance of P from the charge. Furthermore, the direction of the field is determined by the sign of the charge. Hint A.2

Find an expression for the charge

Which of the following expressions gives the correct magnitude of charge Hint A.2.1

that produces an electric field of magnitude

at a distance

from the charge? In the following expressions

Magnitude of the electric field of a point charge

Given a point charge , the magnitude of the electric field

at a distance

from the charge is given by

,

where the constant of proportionality is ANSWER:

= 8.99×109

.

is a constant that has units of

.

Hint A.3

Find the sign of the charge

What is the sign of the charge that produces an electric field that points toward the charge? ANSWER:

positive negative

Since the electric field produced by A at P points south toward A, the charge of A must be negative. ANSWER:

1.11×10−9 −1.11×10−9 2.78×10−10 −2.78×10−10 5.75×1012 −5.75×1012

Part B If a second object B with the same charge as A is placed at 0.250 Hint B.1

south of A (so that objects A and B and point P follow a straight line), what is the magnitude of the total electric field produced by the two objects at P?

How to approach the problem

Since the electric field is a vector quantity, you need to apply the principle of superposition to find the total field at P. The principle of superposition in of electric fields says that the total electric field at any point due to two or more charges is the vector sum of the fields that would be produced at that point by the individual charges. Hint B.2

Find the vector sum of the electric fields

Which of the following diagrams, where

ANSWER:

and

are the electric fields produced by A and B, respectively, correctly represents the situation described in this problem?

C

Now find the magnitude of the vector sum. Hint B.3

Find the electric field produced by B at P

What is the magnitude of the electric field Hint B.3.1

produced by the second object B at point P?

Magnitude of the electric field of a point charge

Given a point charge , the magnitude of the electric field

at a distance

from the charge is given by

,

where the constant of proportionality is Hint B.3.2 How far (

= 8.99×109

.

Find the distance from P to the second object ) is P from B? Recall that P is located 0.250

north of A and B is located 0.250

south of A.

Express your answer in meters. ANSWER:

=

Express your answer in newtons per coulomb. ANSWER:

ANSWER:

=

40.0 50.0 30.0 10.0

Electric Field Conceptual Question Description: Simple conceptual question about identifying where on the x axis the electric field would be zero, given two charges. Part A For the charge distribution provided, indicate the region (A to E) along the horizontal axis where a point exists at which the net electric field is zero.

Hint A.1

Zeros of the electric field

The net electric field can only be zero if the electric fields due to the two charges point in opposite directions and have equal magnitudes. Therefore, first determine the region(s) where the two constituent electric fields point in opposite directions. Then, in each region determine whether a point exists where the fields have equal magnitude. If there is such a point, then select that region. If no such region exists on the horizontal axis choose the last option (nowhere). ANSWER:

A B C D E nowhere

Part B For the charge distribution provided, indicate the region (A to E) along the horizontal axis where a point exists at which the net electric field is zero.

Hint B.1

Zeros of the electric field

The net electric field can only be zero if the electric fields due to the two charges point in opposite directions and have equal magnitudes. Therefore, first determine the region(s) where the two constituent electric fields point in opposite directions. Then, in each region determine whether a point exists where the fields have equal magnitude. If there is such a point, then select that region. Hint B.2

Determine the regions where the electric fields could cancel

In which region(s) do the electric fields from the two source charges point in opposite directions? List all the correct answers in alphabetical order. ANSWER:

BCD

Since the two charges produce fields that point in opposite directions in these regions, if the magnitude of the fields are equal, the net electric field will be zero. Hint B.3

Consider the magnitude of the electric field

For each of the three regions found in the previous hint, determine whether it is possible for the magnitudes to be equal. As an example, consider the point directly between the two charges. Which charge produces the largest magnitude field directly between the two charges? ANSWER:

the charge on the right the charge on the left neither, because they have the same magnitude

Therefore, the point directly between the two charges is not the correct answer since the right charge dominates at this point. Check the other two possible regions. If no such region exists on the horizontal axis choose the last option (nowhere). ANSWER:

A B C D E nowhere

Part C For the charge distribution provided, indicate the region (A to E) along the horizontal axis where a point exists at which the net electric field is zero.

Hint C.1

Zeros of the electric field

The net electric field can only be zero if the electric fields due to the two charges point in opposite directions and have equal magnitudes. Therefore, first determine the region(s) where the two constituent electric fields point in opposite directions. Then, in each region determine whether a point exists where the fields have equal magnitude. If there is such a point, then select that region. If no such region exists on the horizontal axis choose the last option (nowhere). ANSWER:

A B C D E nowhere

Part D For the charge distribution provided, indicate the region (A to E) along the horizontal axis where a point exists at which the net electric field is zero.

Hint D.1

Zeros of the electric field

The net electric field can only be zero if the electric fields due to the two charges point in opposite directions and have equal magnitudes. Therefore, first determine the region(s) where the two constituent electric fields point in opposite directions. Then, in each region determine whether a point exists where the fields have equal magnitude. If there is such a point, then select that region. ANSWER:

A B C D E Nowhere along the finite x axis

The Electric Field at a Point Due to Two Point Charges Description: Calculate the magnitude and direction of the total electric field at the origin from two point charges at given locations. A point charge

is at the point

meters,

meters, and a second point charge

is at the point

meters,

.

Part A Calculate the magnitude Hint A.1

of the net electric field at the origin due to these two point charges. How to approach the problem

First, draw a diagram of the charge system with both of the charges at their correct locations. Next calculate the x and y components of the electric field at the origin from each charge separately. Use vector addition to obtain the components of the combined field in each direction. Finally, use the x and y components to find the magnitude of the combined electric field using the Pythagorean theorem.

Calculate the x component of the field created by

Hint A.2 Calculate

, the x component of

, the electric field at the origin created by

, including its sign.

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

Calculate the y component of

Hint A.3 Calculate

, the y component of

, the electric field at the origin created by

, including its sign.

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

=

Since the second charge is on the x axis, there will be no y component of

Calculate the magnitude of the field created by the first charge

Hint A.4 Calculate

, the magnitude of the electric field at the origin created by charge

Hint A.4.1 Calculate

.

. Recall that magnitude is always nonnegative.

Calculate the distance from the first charge to the origin , the distance from the first charge,

, to the origin.

ANSWER: =

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

Calculate the x component of

Hint A.5 Calculate

, the x component of

Hint A.5.1

, the electric field at the origin created by

, including its sign.

Splitting a vector into its components

Use the x and y components of the location of charge

, along with the distance from the charge to the origin, to construct a right triangle. Use this triangle to find the sine or cosine of the angle that

you calculate, and use the direction in which the field points to find the appropriate sign for the component. Express your answer in newtons per coulomb to three significant figures. ANSWER: =

Hint A.6

Calculate the y component of

Calculate the y component of Hint A.6.1

, the electric field created at the origin by

Splitting a vector into its components

, including its sign.

makes with the x axis. To find the x component, use the cosine that

Use the x and y components of the location of charge

, along with the distance from the charge to the origin, to construct a right triangle. Use this triangle to find the sine or cosine of the angle that

makes with the x axis. To find the y component, use the sine that you

calculate, and use the direction in which the field points to find the appropriate sign for the component. Express your answer in newtons per coulomb to three significant figures. ANSWER: =

Hint A.7

Putting it all together

Add the components from each charge to obtain the components of the total electric field (e.g.,

). Use the Pythagorean theorem to calculate the magnitude of the field:

.

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

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

Using the components of the electric field from Part A, calculate the tangent of the angle as the ratio of the y component to the x component, then take the arctangent to find the angle. Since the angle is taken with respect to the negative x axis, use the x component. Express your answer in degrees to three significant figures. ANSWER: =

Score Summary: Your score on this assignment is 0%. You received 0 out of a possible total of 25 points.

up from the negative x axis

instead of

for