Q: According to Gauss’ law, what is the electric flux through a closed surface that contains multiple charges inside?

$$\Phi_E = \frac{q_{\text{in}}}{\$$epsilon_0$$}$$

Q: A charge $$-1 nC is $$inside a toroidal surface, and a charge $$+100 nC is at $$the center of the torus. What is the total electric flux through the toroidal surface?

$$\Phi_E = \frac{-1 \times 10$$^{-9}$$}{\$$epsilon_0$$}$$

Question 1

Which of the following correctly defines the electric flux $$\Phi_E$$ through a flat surface of area $$A$$ when a uniform electric field $$\vec{E} is $$perpendicular to the surface?

Accepted Answer

$$\Phi_E = E \cdot A$$

Question 2

For a non-uniform electric field $$\vec{E}(\vec{r})$$ passing through a curved surface, which expression gives the total electric flux $$\Phi_E$$?

Accepted Answer

$$\Phi_E = \int_S \vec{E}(\vec{r}) \cdot d\vec{A}(\vec{r})$$

Question 3

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

Accepted Answer

$$\Phi_E = \frac{q}{\$$epsilon_0$$}$$

Question 4

According to Gauss’ law, what is the electric flux through a closed surface that contains multiple charges inside?

Accepted Answer

$$\Phi_E = \frac{q_{	ext{in}}}{\$$epsilon_0$$}$$

Question 5

A charge $$-1 nC is $$inside a toroidal surface, and a charge $$+100 nC is at $$the center of the torus. What is the total electric flux through the toroidal surface?

Accepted Answer

$$\Phi_E = \frac{-1 	imes 10$$^{-9}$$}{\$$epsilon_0$$}$$

Question 6

Which symmetry is most appropriate for applying Gauss’ law to a uniformly charged infinite plane?

Accepted Answer

Planar symmetry

Question 7

A uniformly charged ball of radius $$R$$ has charge density $$\rho. $$What is the electric field $$\vec{E}(r)$$ inside the ball ($$r \u003c R$$)?

Accepted Answer

$$\vec{E}(r) = \frac{\rho r}{3\$$epsilon_0$$} \hat{r}$$

Question 8

A uniformly charged thin spherical shell of radius $$R$$ has surface charge density $$\sigma. $$What is the electric field $$\vec{E}(r)$$ outside the shell ($$r \u003e R$$)?

Accepted Answer

$$\vec{E}(r) = \frac{\sigma R^2$}{\$$epsilon_0$$ r^2$} \hat{r}$$

Question 9

What is the electric field produced by a surface charge density $$\sigma$$ uniformly spread in the $$(yz)-$$plane at a point a distance $$x$$ from the plane?

Accepted Answer

$$\vec{E} = \frac{\sigma}{2\$$epsilon_0$$} \hat{x}$$

Question 10

A linear charge density $$\lambda is $$uniformly distributed along the $$x-$$axis. What is the electric field at a point a distance $$r$$ from the axis?

Accepted Answer

$$\vec{E} = \frac{\lambda}{2\pi \$$epsilon_0$$ r} \hat{r}$$

Question 11

If the electric field $$\vec{E} is $$tangential to a surface $$A$$, what is the electric flux $$\Phi_E$$ through the surface?

Accepted Answer

$$\Phi_E = 0$$

Question 12

A charge $$1 nC is $$located at a distance $$a/2$$ directly above the center of a square of side length $$a. $$What is the electric flux through the square?

Accepted Answer

$$\Phi_E = \frac{q}{2\$$epsilon_0$$}$$

Question 13

Which step is NOT part of the general strategy for applying Gauss’ law to find the electric field?

Accepted Answer

Calculate the magnetic field inside the surface

Question 14

If a charge is located outside a closed surface, what is its contribution to the electric flux through the surface?

Accepted Answer

It does not contribute to the flux

Question 15

Which of the following is the correct vector form for the electric flux $$\Phi_E$$ through a surface?

Accepted Answer

$$\Phi_E = \vec{E} \cdot \vec{A}$$

Question 16

For a spherical charge distribution $$\rho(r)$$, which symmetry should be used to apply Gauss’ law?

Accepted Answer

Spherical symmetry

Question 17

If the charge distribution does not have any particular symmetry, which expression should be used to find the electric field $$\vec{E}(\vec{r})$$?

Accepted Answer

$$\vec{E}(\vec{r}) = k_e \int \frac{dQ(\vec{r}')}{|\vec{r} - \vec{r}'|^3} (\vec{r} - \vec{r}')$$

Gauss' Law and Electric Field Flux: Study Notes

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