Q: Two point charges, $$+3.30\ \mu\text{C}$$ and $$-7.20\ \mu\text{C}$$, are separated by $$2.90\ \text{m}. $$What is the electric potential at the midpoint between them?

$$V = -2.42 \times 10^4$\ \text{V}$$

Q: Two protons are moving directly toward one another, each with an initial speed of $$2.00 \times 10^5$\ \text{m/s}$$ when far apart. What is the distance of closest approach?

$$r_B = 3.45 \times 10$$^{-12}$$\ \text{m}$$

Question 1

A thin rod of length $$L$$ lies along the $$x-$$axis, as shown in the diagram, and has a non-uniform linear charge density $$\lambda = c x$$, where $$c is a $$constant and $$x is $$the distance from the left end. What is the expression for the electric potential at point $$P_1$$, located a distance $$d$$ from the left end of the rod along the $$x-$$axis, assuming $$V = 0 at $$infinity?

Accepted Answer

$$V = \frac{c}{4\pi\$$epsilon_0$$} \left[ L - d \ln \left( 1 + \frac{L}{d} \right) \right]$$

Question 2

A solid metal sphere of diameter $$1\ 	ext{cm}$$ carries a charge of $$-3.00\ 	ext{nC}. $$What is the electric potential $$5.00\ 	ext{cm}$$ away from its center?

Accepted Answer

$$V = -539\ 	ext{V}$$

Question 3

Two point charges, $$+3.30\ \mu	ext{C}$$ and $$-7.20\ \mu	ext{C}$$, are separated by $$2.90\ 	ext{m}. $$What is the electric potential at the midpoint between them?

Accepted Answer

$$V = -2.42 	imes 10^4$\ 	ext{V}$$

Question 4

Two protons are moving directly toward one another, each with an initial speed of $$2.00 	imes 10^5$\ 	ext{m/s}$$ when far apart. What is the distance of closest approach?

Accepted Answer

$$r_B = 3.45 	imes 10$$^{-12}$$\ 	ext{m}$$

Question 5

Two conducting spheres, A and B, are initially given equal amounts of positive charge and are isolated. Sphere A has five times the radius of sphere B. After connecting them with a wire, what is the ratio of the charge on sphere A to that on sphere B ($$q_A/q_B$$)?

Accepted Answer

$$q_A/q_B = 5$$

Question 6

What is the electrostatic energy required to assemble a uniformly charged sphere of radius $$R$$ and total charge $$Q$$ from infinitesimal charges located infinitely far away?

Accepted Answer

$$U = \frac{3kQ^2$}{5R}$$

Question 7

Four charges $$q_1 = +2.0$$\ \mu	ext{C}$$, q_2$ = +3.0\ \mu	ext{C}$$, $$q_3 = +4.0$$\ \mu	ext{C}$$, q_4$ = +5.0\ \mu	ext{C}$$ are placed at the vertices of a square of side $$s = 1.0\ 	ext{cm}. $$What is the total work required to assemble this configuration?

Accepted Answer

$$W = 57.8\ 	ext{J}$$

Question 8

A uniformly charged wire of length $$L$$ and total charge $$Q$$ lies along the $$x-$$axis. What is the electric potential at a point $$P$$ located a perpendicular distance $$d$$ from the center of the wire?

Accepted Answer

$$V = \frac{\lambda}{4\pi\$$epsilon_0$$} \ln \left( \frac{\sqrt{L^2$/4 + d^2$} + L/2}{\sqrt{L^2$/4 + d^2$} - L/2} \right)$$

Question 9

Using the result for the electric potential from the previous question, what is the electric field at point $$P$$ located a perpendicular distance $$d$$ from the center of the wire?

Accepted Answer

$$E = \frac{\lambda}{2\pi\$$epsilon_0$$} \cdot \frac{1}{\sqrt{d^2$ + (L/2)^2$}}$$

Question 10

For an infinitely long uniformly charged wire, what is the expression for the electric field at a perpendicular distance $$d$$ from the wire?

Accepted Answer

$$E = \frac{\lambda}{2\pi\$$epsilon_0$$ d}$$

Question 11

A thin rod of length $$L$$ lies along the $$x-$$axis, as shown in the diagram, and has a non-uniform linear charge density $$\lambda = c x. $$What substitution is used to solve the integral for the electric potential at point $$P_1$$?

Accepted Answer

$$u = d + x$$

Question 12

Which of the following statements about electric potential is correct?

Accepted Answer

Electric potential is a scalar quantity and can be summed directly for multiple charges.

Question 13

If a positive charge is placed near a negatively charged sphere, what happens to its electric potential energy as it moves closer to the sphere?

Accepted Answer

The electric potential energy decreases (becomes more negative).

Question 14

What is the work required to bring a charge $$q$$ from infinity to a point at distance $$r$$ from another charge $$Q$$?

Accepted Answer

$$W = \frac{kQq}{r}$$

Question 15

A system consists of two protons separated by a distance $$r. $$What is the expression for their electric potential energy?

Accepted Answer

$$EPE = \frac{k e^2$}{r}$$

Question 16

When assembling a system of point charges, which formula gives the total work required?

Accepted Answer

$$W = k \sum_{i\u003cj} \frac{q_i q_j}{r_{ij}}$$

Electric Potential Due to a Nonuniformly Charged Rod

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