You’ve been assigned the task of determining the magnitude and direction of the electric field at a point in space. Give a step-by-step procedure of how you will do so. List any objects you will use, any measurements you will make, and any calculations you will need to perform. Make sure that your measurements do not disturb the charges that are creating the field.

Reproduce FIGURE Q23.2 on your paper. For each part, draw a dot or dots on the figure to show any position or positions (other than infinity) where $\vec{E}=\vec{0}$.

FIGURE Q23.2

Rank in order, from largest to smallest, the electric field strengths $E_1$ to $E_4$ at points 1 to 4 in FIGURE Q23.3. Explain.

FIGURE Q23.3

A small segment of wire in FIGURE Q23.4 contains 10 nC of charge.

FIGURE Q23.4

1. The segment is shrunk to one-third of its original length. What is the ratio ${\text{λ}}_{\text{f}}\text{/}{\text{λ}}_{\text{i}}$, where ${\text{λ}}_{\text{i}}$ and ${\text{λ}}_{\text{f}}$ are the initial and final linear charge densities?

2. A proton is very far from the wire. What is the ratio $F_{\text{f}}\text{/}{F}_{\text{i}}$ of the electric force on the proton after the segment is shrunk to the force before the segment was shrunk?

3. Suppose the original segment of wire is stretched to 10 times its original length. How much charge must be added to the wire to keep the linear charge density unchanged?

An electron experiences a force of magnitude F when it is 1 cm from a very long, charged wire with linear charge density $\text{λ}$. If the charge density is doubled, at what distance from the wire will a proton experience a force of the same magnitude F?

FIGURE Q23.6 shows a hollow soda straw that has been uniformly charged with positive charge. What is the electric field at the center (inside) of the straw? Explain.

FIGURE Q23.6

The irregularly shaped area of charge in FIGURE Q23.7 has surface charge density ${\eta }_{\text{i}}$. Each dimension (x and y) of the area is reduced by a factor of 3.163.

FIGURE Q23.7

1. What is the ratio ${\eta }_{\text{f}}\text{/}{\eta }_{\text{i}}$, where ${\eta }_{\text{f}}$ is the final surface charge density?

2. An electron is very far from the area. What is the ratio $F_{\text{f}}\text{/}{F}_{\text{i}}$ of the electric force on the electron after the area is reduced to the force before the area was reduced?

A circular disk has surface charge density $8{\text{nC/cm}}^2$. What will the surface charge density be if the radius of the disk is doubled?

A sphere of radius R has charge Q. The electric field strength at distance $r>R$ is $E_{\text{i}}$. What is the ratio $E_{\text{f}}\text{/}{E}_{\text{i}}$ of the final to initial electric field strengths if (a) Q is halved, (b) R is halved, and (c) r is halved (but is still$>R$)? Each part changes only one quantity; the other quantities have their initial values.

The ball in FIGURE Q23.10 is suspended from a large, uniformly charged positive plate. It swings with period T. If the ball is discharged, will the period increase, decrease, or stay the same? Explain.

FIGURE Q23.10

Rank in order, from largest to smallest, the electric field strengths $E_1$ to $E_5$ at the five points in FIGURE Q23.11. Explain.

FIGURE Q23.11

A parallel-plate capacitor consists of two square plates, size $L\times L$, separated by distance d. The plates are given charge $\pm Q$. What is the ratio $E_{\text{f}}\text{/}{E}_{\text{i}}$ of the final to initial electric field strengths if (a) Q is doubled, (b) L is doubled, and (c) d is doubled? Each part changes only one quantity; the other quantities have their initial values.

A small object is released at point 3 in the center of the capacitor in Figure Q23.11. For each situation, does the object move to the right, to the left, or remain in place? If it moves, does it accelerate or move at constant speed?

1. A positive object is released from rest.

2. A neutral but polarizable object is released from rest.

3. A negative object is released from rest.

A proton and an electron are released from rest in the center of a capacitor.

1. Is the force ratio $F_{\text{p}}\text{/}{F}_{\text{e}}$ greater than 1, less than 1, or equal to 1? Explain.

2. Is the acceleration ratio $a_{\text{p}}\text{/}{a}_{\text{e}}$ greater than 1, less than 1, or equal to 1? Explain.

Three charges are placed at the corners of the triangle in FIGURE Q23.15. The $++$ charge has twice the quantity of charge of the two $-$ charges; the net charge is zero. Is the triangle in equilibrium? If so, explain why. If not, draw the equilibrium orientation.

FIGURE Q23.15