Question 1

A solid sphere of radius $$R$$ has a uniform charge density $$\rho$$ ($$\mathrm{C/m^3$}). $$What is the electric field at a distance $$r$$ from the center, where $$r \u003c R$$? Use Gauss's Law to derive your answer.

Accepted Answer

$$E = \dfrac{\rho r}{3\$$varepsilon_0$$}$$

Question 2

For a uniformly charged solid sphere of radius $$R$$ and total charge $$Q$$, what is the electric field at a point outside the sphere at distance $$r$$ ($$r \u003e R$$)?

Accepted Answer

$$E = \dfrac{Q}{4\pi\$$varepsilon_0$$ r^2$}$$

Question 3

A solid sphere of radius $$R$$ has a uniform charge density $$\rho. $$What is the total charge $$Q$$ contained in the sphere?

Accepted Answer

$$Q = \dfrac{4}{3}\pi R^3$ \rho$$

Question 4

Consider a spherical conducting shell with inner radius $$b$$ and outer radius $$c$$, concentric with a solid sphere of radius $$a at $$the center. The sphere has charge $$+q$$, and the shell has net charge $$-q. $$What is the electric field at a point $$r$$ where $$a \u003c r \u003c b$$?

Accepted Answer

$$E = \dfrac{q}{4\pi\$$varepsilon_0$$ r^2$}$$

Question 5

A solid sphere of radius $$a is $$surrounded by a conducting shell with inner radius $$b = 2a$$ and outer radius $$c = 2.5a. $$The sphere has charge $$+q$$, and the shell has net charge $$-q. $$What is the electric field at $$r = 3.5a$$?

Accepted Answer

$$E = 0$$

Question 6

In the scenario above, what is the net charge on the inner surface of the conducting shell?

Accepted Answer

$$-q$$

Question 7

In the same scenario, what is the net charge on the outer surface of the conducting shell?

Accepted Answer

$$0$$

Question 8

Two solid spheres, each of radius $$R$$, have uniformly distributed charges $$q_1$$ and $$q_2. $$Point $$P$$ lies on the line connecting their centers, at a distance $$R/2$$ from the center of sphere 1. If the net electric field at $$P is $$zero, what is the ratio $$q_2$$/$$q_1$$?

Accepted Answer

$$q_2$$/$$q_1 = 1/8$

Question 9

Which of the following statements about the electric field inside a uniformly charged solid sphere is correct?

Accepted Answer

The electric field increases linearly with distance from the center.

Question 10

A student claims that the electric field just outside a uniformly charged solid sphere is the same as if all the charge were concentrated at the center. Is this correct? Why?

Accepted Answer

Yes, because outside the sphere, the field is identical to that of a point charge at the center.

Question 11

A solid sphere of radius $$R$$ has a uniform charge density $$\rho. $$What is the electric field at the center of the sphere?

Accepted Answer

$$E = 0$$

Question 12

Which Gaussian surface is most appropriate for calculating the electric field inside a uniformly charged solid sphere?

Accepted Answer

A sphere of radius $$r \u003c R$$ centered at the same point as the charged sphere.

Question 13

A solid sphere of radius $$R$$ has a uniform charge density $$\rho. $$What is the electric field at a point $$r = R (on $$the surface)?

Accepted Answer

$$E = \dfrac{\rho R}{3\$$varepsilon_0$$}$$

Question 14

A spherical shell has a net charge $$-q$$ and surrounds a solid sphere of charge $$+q. $$What is the electric field at a point inside the shell but outside the sphere?

Accepted Answer

$$E = \dfrac{q}{4\pi\$$varepsilon_0$$ r^2$}$$

Question 15

If a solid sphere of radius $$R$$ has a non-uniform charge density that increases with radius, which of the following statements is true about the electric field inside the sphere?

Accepted Answer

The electric field increases faster than linearly with $$r.$$

Question 16

A solid sphere of radius $$R$$ has a uniform charge density $$\rho. If $$the sphere is placed in a vacuum, what is the direction of the electric field at a point outside the sphere?

Accepted Answer

Radially outward from the center of the sphere.

Question 17

A solid sphere of radius $$R$$ has a uniform charge density $$\rho. $$What is the electric field at a point $$r \u003e R$$ from the center?

Accepted Answer

$$E = \dfrac{\rho R^3$}{3\$$varepsilon_0$$ r^2$}$$

Question 18

Which of the following best explains why the electric field inside a conductor is zero?

Accepted Answer

Free charges move until they cancel any internal field.

Electric Fields of Spherical Charge Distributions: Gauss's Law Applications

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