Q: Which of the following is a correct application of Gauss's Law for a spherical charge distribution?

$$\Phi_E = \oint \vec{E} \cdot d\vec{A} = \dfrac{Q_{\text{enc}}}{\$$epsilon_0$$}$$

Question 1

A solid insulating sphere of radius $$5.00\ \mathrm{cm}$$ carries a charge of $$+3.00\ \mu\mathrm{C}. $$What is the magnitude of the electric field at a distance $$4.00\ \mathrm{cm}$$ from the center of the sphere?

Accepted Answer

$$E = \dfrac{1}{4\pi\$$epsilon_0$$} \dfrac{Q r}{R^3$}$$

Question 2

A conducting shell surrounds a conducting rod. If the rod is positively charged and the shell is grounded, what is the net electric field inside the shell?

Accepted Answer

Zero everywhere inside the shell

Question 3

A point charge $$q = 3.80 \ \mu\mathrm{C} is $$placed $$8.00\ \mathrm{cm}$$ from a thin, flat, infinite sheet with uniform surface charge density $$\sigma. If $$the total electric field is zero at a point halfway between the charge and the sheet, what is the value of $$\sigma$$?

Accepted Answer

Question 4

Which of the following statements about Gauss's Law is correct?

Accepted Answer

Gauss's Law can be used to calculate the electric field only for charge distributions with spherical or cylindrical symmetry.

Question 5

An infinite cylindrical rod has a uniform volume charge density $$\rho. $$What is the magnitude of the electric field at a distance $$r$$ from the axis, where $$r \u003c r_0$ (the rod's radius)?

Accepted Answer

$$E = \dfrac{\rho r}{2\$$epsilon_0$$}$$

Question 6

A long straight wire has a linear charge density $$\lambda. $$What is the magnitude of the electric field at a distance $$r$$ from the wire?

Accepted Answer

$$E = \dfrac{\lambda}{2\pi\$$epsilon_0$$ r}$$

Question 7

A solid conducting sphere of radius $$R$$ carries a total charge $$Q. $$What is the electric field at a point outside the sphere at a distance $$r \u003e R$$?

Accepted Answer

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

Question 8

A thin, infinite sheet has a uniform surface charge density $$\sigma. $$What is the magnitude of the electric field produced by the sheet?

Accepted Answer

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

Question 9

A charged sphere of radius $$R$$ has a total charge $$Q$$ distributed uniformly throughout its volume. What is the electric field at a distance $$r \u003c R$$ from the center?

Accepted Answer

$$E = \dfrac{1}{4\pi\$$epsilon_0$$} \dfrac{Q r}{R^3$}$$

Question 10

You place a point charge $$q at a $$distance $$d$$ from an infinite line of charge with linear charge density $$\lambda. $$What is the magnitude of the electric force on the point charge?

Accepted Answer

$$F = \dfrac{q \lambda}{2\pi\$$epsilon_0$$ d}$$

Question 11

Which of the following is a correct application of Gauss's Law for a spherical charge distribution?

Accepted Answer

$$\Phi_E = \oint \vec{E} \cdot d\vec{A} = \dfrac{Q_{	ext{enc}}}{\$$epsilon_0$$}$$

Question 12

A cylindrical Gaussian surface of radius $$r$$ and length $$L$$ encloses a uniformly charged rod with volume charge density $$\rho. $$What is the total charge enclosed by the Gaussian surface?

Accepted Answer

$$Q_{	ext{enc}} = ho \pi r^2$ L$$

Question 13

For a finite closed cylinder used as a Gaussian surface around a uniformly charged rod, which part of the surface does the electric field point perpendicular to?

Accepted Answer

The curved wall of the cylinder

Question 14

If the electric field outside a uniformly charged sphere is measured at various distances $$r$$ from the center, which graph best represents $$E$$ versus $$r$$?

Accepted Answer

A graph showing $$E$$ decreasing as $$1/r^2$$

Question 15

A student claims that the electric field inside a conductor is always zero. Which scenario would contradict this claim?

Accepted Answer

A conductor with a time-varying magnetic field

Question 16

A charged sphere is placed near a thin, infinite sheet with uniform surface charge density. If the force between them is attractive, what can you conclude about the sign of their charges?

Accepted Answer

They have opposite signs

Question 17

Which of the following is the correct expression for the surface area of the curved wall of a cylinder with radius $$r$$ and length $$L$$?

Accepted Answer

$$A = 2\pi r L$$

Question 18

A student measures the electric field at various distances from a uniformly charged line and plots $$E$$ versus $$r. $$Which relationship should the graph show?

Accepted Answer

$$E$$ decreases as $$1/r$$

Question 19

A spherical shell of radius $$R$$ carries a charge $$Q$$ distributed uniformly over its surface. What is the electric field inside the shell ($$r \u003c R$$)?

Accepted Answer

$$E = 0$$

Gauss's Law and Electric Fields: Problems and Applications

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