Q: How long does it take the capacitor to have charge $$\frac{2}{5} of $$the maximum value? (Answer in terms of time constant $$\tau.)$$

$$t = -\tau \ln(1 - \frac{2}{5})$$

Question 1

Two point charges $$q_1 = +e$ at x=0$ and q_2$ = -2e at$$ $$x=3a$$ are placed along the x-axis. What is the magnitude and direction of the electric field at $$x=a$$?

Accepted Answer

The field is $$E = \frac{7ke}{a^2$} to $$the left.

Question 2

For the same two charges as above, what is the electric potential at $$x=a$$?

Accepted Answer

$$V = \frac{ke}{a} - \frac{2ke}{2a}$$

Question 3

Where is the net electric field zero (other than at infinity) for the two charges $$q_1 = +e$ at x=0$ and q_2$ = -2e at$$ $$x=3a$$?

Accepted Answer

At $$x = 2a$$

Question 4

A third charge $$q_3 = -2e$ is placed at x=a$. What is the magnitude and direction of the electric force on q_3$?

Accepted Answer

Force is $$F = 14\frac{ke^2$}{a^2$} to $$the left.

Question 5

A positive charge $$Q is $$distributed uniformly along the x-axis. What is the electric potential at the origin?

Accepted Answer

$$V = k \int \frac{dq}{r}$$

Question 6

A charge $$q = -2.0\ \mathrm{C}$$ moves 3.0 m in the opposite direction of a uniform electric field of $$100\ \mathrm{V/m}$$ pointing in the positive x direction. What is the change in potential energy of the charge?

Accepted Answer

$$\Delta U = -600\ \mathrm{J}$$

Question 7

What is the work done on the charge by the electric field in the scenario above?

Accepted Answer

$$W = +600\ \mathrm{J}$$

Question 8

A uniform electric field $$\vec{E} = (2.0\ \mathrm{N/C})\hat{i} - (7.0\ \mathrm{N/C})\hat{k}$$ passes through a rectangular plane of area $$A = 30\ \mathrm{cm}^2$$ lying in the $$yz$$ plane. What is the electric flux through the plane?

Accepted Answer

$$\Phi = -2.1\ \mathrm{N\cdot m^2$/C}$$

Question 9

Two capacitors, $$C_A = 2.0\ \mu\mathrm{F}$$ and $$C_B = 4.0\ \mu\mathrm{F}$$, are connected in series to a $$6.0\ \mathrm{V}$$ battery. Which capacitor has more charge?

Accepted Answer

Both have the same charge

Question 10

What is the potential difference across capacitor A in the above scenario?

Accepted Answer

$$4.0\ \mathrm{V}$$

Question 11

The separation of the two plates of a charged capacitor connected to a battery is halved. How does the capacitance change?

Accepted Answer

Capacitance doubles

Question 12

How does the charge on the capacitor change when the plate separation is halved (with battery connected)?

Accepted Answer

Charge doubles

Question 13

Wire A and wire B are connected in series. If the diameter of wire A is three times that of wire B, how does the electron drift speed $$v_{dA} in $$wire A compare with $$v_{dB} in $$wire B?

Accepted Answer

$$v_{dA} = \frac{1}{9} v_{dB}$$

Question 14

The potential at the surface of a solid metal sphere of radius $$a is$$ $$V_0. $$What is the potential at a point of distance $$b$$ ($$b \u003c a) $$from the center of the sphere?

Accepted Answer

$$V_0$$

Question 15

An uncharged capacitor of capacitance $$C$$ and a resistor of resistance $$R$$ are connected in series to a battery of voltage $$V. If$$ $$C = 2.00\ \mu\mathrm{F}$$ and $$R = 5.00\ \mathrm{k}\Omega$$, what is the time constant?

Accepted Answer

$$	au = 10\ \mathrm{ms}$$

Question 16

How long does it take the capacitor to have charge $$\frac{2}{5} of $$the maximum value? (Answer in terms of time constant $$	au.)$$

Accepted Answer

$$t = -	au \ln(1 - \frac{2}{5})$$

Question 17

In a region of space, the electric potential is given by $$V(x, y, z) = Ax^3$y + By^2$z. $$What is the electric field in vector form?

Accepted Answer

$$\vec{E} = -\$$left(3Ax^2$$y\hat{i} + (Ax^3$ + 2Byz)\hat{j} + By^2$\hat{k}\right)$$

Question 18

A small insulating solid sphere with radius $$a is $$concentric with a larger conducting spherical shell with inner radius $$b$$ and outer radius $$c. $$The solid sphere has total charge $$-3q$$ distributed uniformly over the entire volume, and the shell has charge $$-4q. $$Using Gauss’s law, what is the electric field for $$r \u003c a$$?

Accepted Answer

$$E = -\frac{3q r}{4\pi \$$epsilon_0$$ a^3$}$$

Question 19

For the same system as above, what is the charge on the inner surface of the conducting shell?

Accepted Answer

$$+3q$$

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