Q: Which equation represents Gauss's law in differential form?

$\displaystyle \nabla \cdot \vec{E} = \frac{\rho}{\varepsilon_0}$

Question 1

Which of the following best describes the physical meaning of Gauss's law in its integral form?

Accepted Answer

The net electric flux through any closed surface is equal to the total charge enclosed divided by the permittivity of free space.

Question 2

A point charge $q$ is placed at the center of a spherical surface of radius $r$. What is the total electric flux through the surface?

Accepted Answer

$\displaystyle \frac{q}{\varepsilon_0}$

Question 3

Which equation represents Gauss's law in differential form?

Accepted Answer

$\displaystyle 
abla \cdot \vec{E} = \frac{ho}{\varepsilon_0}$

Question 4

A long, straight wire carries a uniform line charge density $\lambda$. Using Gauss's law, what is the magnitude of the electric field at a distance $r$ from the wire?

Accepted Answer

$\displaystyle \frac{\lambda}{2\pi \varepsilon_0 r}$

Question 5

Which of the following statements about electric flux $\phi_e$ is correct?

Accepted Answer

Electric flux through a surface is given by $\displaystyle \phi_e = \vec{E} \cdot \vec{n} A$, where $\vec{n}$ is the unit vector perpendicular to the surface.

Question 6

Suppose you have a cube in a region of space with a uniform electric field $\vec{E}$ pointing in the $x$-direction. What is the net electric flux through the cube?

Accepted Answer

Zero, because the flux entering one face is equal to the flux leaving the opposite face.

Question 7

A spherical shell of radius $R$ encloses no charge. What is the value of $\displaystyle \oint_S \vec{E} \cdot d\vec{A}$ over the shell?

Accepted Answer

Zero, because there is no enclosed charge.

Question 8

Which of the following is a correct application of the law of superposition for electric fields?

Accepted Answer

The total electric field at a point is the vector sum of the electric fields produced by each individual charge.

Question 9

A surface charge distribution $\sigma$ is spread uniformly over a flat, infinite plane. What is the magnitude of the electric field just above the surface?

Accepted Answer

$\displaystyle \frac{\sigma}{2\varepsilon_0}$

Question 10

Which of the following is true about the shape of the Gaussian surface used in Gauss's law?

Accepted Answer

The shape of the Gaussian surface is arbitrary; the net flux depends only on the enclosed charge.

Question 11

A non-conducting solid sphere of radius $R$ has a uniform volume charge density $\rho$. What is the electric field at a distance $r < R$ from the center?

Accepted Answer

$\displaystyle \frac{\rho r}{3\varepsilon_0}$

Question 12

Which of the following is a consequence of the divergence theorem as applied to Gauss's law?

Accepted Answer

The surface integral of the electric field over a closed surface equals the volume integral of the charge density divided by $\varepsilon_0$.

Question 13

A lightning bolt delivers a charge $Q$ to the ground. If the ground is modeled as a large conducting plane, what is the electric field just above the ground due to this charge?

Accepted Answer

$\displaystyle \frac{Q}{2\varepsilon_0 A}$, where $A$ is the area over which the charge is distributed.

Question 14

If a charge $q$ is located outside a closed surface $S$, what is the value of $\displaystyle \oint_S \vec{E} \cdot d\vec{A}$?

Accepted Answer

Zero, because the charge is not enclosed by the surface.

Question 15

Which of the following is the correct expression for the electric field produced by a point charge $q$ at a distance $r$?

Accepted Answer

$\displaystyle \vec{E} = \frac{1}{4\pi \varepsilon_0} \frac{q}{r^2} \hat{r}$

Question 16

A cylindrical Gaussian surface of radius $r$ and length $L$ surrounds an infinitely long line of charge with linear charge density $\lambda$. What is the total charge enclosed?

Accepted Answer

$\displaystyle Q_{enclosed} = \lambda L$

Question 17

Which of the following is NOT a valid assumption when using Gauss's law to calculate the electric field?

Accepted Answer

The electric field must be uniform everywhere on the Gaussian surface, regardless of symmetry.

Question 18

A thin spherical shell of radius $R$ carries a total charge $Q$ uniformly distributed over its surface. What is the electric field at a point outside the shell at distance $r > R$?

Accepted Answer

$\displaystyle \vec{E} = \frac{1}{4\pi \varepsilon_0} \frac{Q}{r^2} \hat{r}$

Question 19

Which of the following best explains why the $z$-components of the electric field cancel when calculating the field from an infinite line charge?

Accepted Answer

By symmetry, for every charge element above the observation point, there is an equal element below, producing equal and opposite $z$-components.

LEC 2: Electric Field and Gauss's Law – Physics 2101A Lecture 24 Study Notes

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