Question 1

A point charge $$q = 1\ \mathrm{C} is $$located at $$(1,2)\ \mathrm{m}$$ and a test charge $$q_0 is at$$ $$(2,-2)\ \mathrm{m}. $$What is the magnitude of the electric field $$\vec{E} at $$the location of $$q_0$$ due to $$q$$?

Accepted Answer

$$\vec{E} = \dfrac{1}{4\pi\$$epsilon_0$$} \dfrac{q}{r^2$}$$, where $$r = \sqrt{(2-1)^2$ + (-2-2)^2$}$$

Question 2

Two charges $$2q$$ and $$-q$$ are placed on the $$x-$$axis at $$x=0$$ and $$x=a. At $$what point $$x$$ along the axis is the electric field zero?

Accepted Answer

$$x = 2a$$

Question 3

Two positive point charges are placed on opposite sides of the $$x-$$axis, each a distance $$x$$ from the center. What is the direction of the electric field $$\vec{E}$$ along the $$y-$$axis at some height $$y$$?

Accepted Answer

The field points along the $$y-$$axis, away from the origin if $$y\u003e0$$

Question 4

Four charges $$q, -q, q, -q$$ are equally spaced on the $$x-$$axis at positions $$-3a, -a, a, 3a. At $$what $$y$$ value along the $$y-$$axis is the $$x-$$component of the electric field zero?

Accepted Answer

$$y = \sqrt{3}a$$

Question 5

Two spheres of mass $$m$$ and charge $$q$$ are suspended from threads of length $$L$$ and settle at an equilibrium separation $$d. $$Using the small angle approximation $$\sin	heta \approx 	an	heta$$, what is the electric force $$F_e$$ between the spheres?

Accepted Answer

$$F_e = mg 	an	heta$$

Question 6

A half-ring of charge $$q$$ and radius $$a$$ has a uniform charge density $$\lambda = q/(\pi a). $$What is the vertical component $$E_y of $$the electric field at the center due to the half-ring?

Accepted Answer

$$E_y = \dfrac{1}{4\pi\$$epsilon_0$$} \dfrac{2q}{a^2$}$$

Question 7

A non-conducting sphere of radius $$a$$ and charge $$q$$ has a constant interior charge density $$\rho. $$Using Gauss's Law, what is the electric field $$E$$ inside the sphere ($$r \u003c a$$)?

Accepted Answer

$$E = \dfrac{1}{4\pi\$$epsilon_0$$} \dfrac{qr}{a^3$}$$

Question 8

Using Gauss's Law, what is the electric field strength $$E of an $$infinite plane with uniform charge density $$\sigma$$?

Accepted Answer

$$E = \dfrac{\sigma}{2\$$epsilon_0$$}$$

Question 9

A solid sphere of charge $$q$$ and radius $$a is $$surrounded by a cloud of charges with density $$\rho = 1/r. $$What is the net external electric field at $$r \u003e a$$?

Accepted Answer

$$E = \dfrac{1}{4\pi\$$epsilon_0$$} \dfrac{q}{r^2$}$$

Question 10

Two metal plates, each of area $$A$$, separated by distance $$d$$, have charge densities $$+\sigma$$ and $$-\sigma. $$What is the electric field strength $$E$$ between the plates using Gauss's Law?

Accepted Answer

$$E = \dfrac{\sigma}{\$$epsilon_0$$}$$

Question 11

What is the capacitance $$C of $$two concentric spherical metal shells with radii $$a$$ and $$b$$ ($$b \u003e a$$)?

Accepted Answer

$$C = 4\pi\$$epsilon_0$$ ab/(b-a)$$

Question 12

Consider two parallel wires of constant density $$\lambda$$ separated by $$3\ \mathrm{m}$$, with opposite charge. The electric field halfway between them is $$E = 1.5 	imes 10^4$\ \mathrm{V/m}. $$How much charge rests on a $$1\ \mathrm{km}$$ stretch of one wire?

Accepted Answer

$$\lambda = \dfrac{E \cdot 2\pi\$$epsilon_0$$ d}{1}$$

Question 13

Calculate the potential energy $$U of a $$sphere of radius $$a$$ with charge $$Q$$ uniformly distributed throughout the interior. Show that the internal potential energy is a fifth of the external potential energy.

Accepted Answer

$$U_{int} = \dfrac{1}{5} U_{ext}$$

Question 14

In the circuit shown, what is the equivalent resistance $$R_{eq} if$$ $$R_1$$ and $$R_2$$ are in parallel?

Accepted Answer

$$R_{eq} = \dfrac{R_1$ R_2$}{R_1$ + R_2$}$$

Question 15

A capacitor of $$C = 100\ \mu\mathrm{F} is $$connected in series to $$R_1 = 1$$\ \mathrm{k}\Omega$$ and R_2$ = 5\ \mathrm{k}\Omega. $$What is the time constant $$	au$$ for the discharge?

Accepted Answer

$$	au = C(R_1$ + R_2$)$$

Question 16

A metal block partially inserted into a capacitor can be modeled as three capacitors. If the block has width $$w$$ and is inserted a distance $$y$$, which of the following best describes the equivalent capacitance?

Accepted Answer

$$C_{eq} = C_1$ + C_2$ + C_3$

Question 17

A wire of length $$L$$ and cross-sectional area $$A$$ has constant resistivity $$\rho. $$What is its resistance $$R$$?

Accepted Answer

$$R = \dfrac{\rho L}{A}$$

Question 18

A wire segment has resistivity $$\rho(x) = \$$rho_0$$ (L-x)/L$$ from $$x=0 to$$ $$x=L. $$What is the equivalent resistance between $$x=0$$ and $$x=L$$?

Accepted Answer

$$R = \dfrac{\$$rho_0$$ L}{2A}$$

Question 19

In the diamond-shaped circuit with resistors $$R$$ and $$2R as $$shown, what is the equivalent resistance between the two terminals?

Accepted Answer

$$R_{eq} = \dfrac{5}{6}R$$

Electrostatics and Circuits: Practice Problems and Study Guide

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