Unit 8

Electric Charges, Fields, and Gauss's Law

8 practice questions · Free response

Notes

$\displaystyle F = \frac{1}{4\pi\varepsilon_0}\frac{|q_1q_2|}{r^2}$, where $\displaystyle \frac{1}{4\pi\varepsilon_0} = k = 9.0 \times 10^9$ N·m²/C².
$\displaystyle \vec E = \frac{\vec F}{q_0}$. Point charge: $\displaystyle E = \frac{kq}{r^2}$.
Continuous distributions: $\displaystyle \vec E = \int\frac{k\,dq}{r^2}\hat r$, with $dq = \lambda\,dx$, $\sigma\,dA$, or $\rho\,dV$. Use symmetry to cancel components first.
On the axis of a ring of charge $Q$, radius $a$: $\displaystyle E = \frac{kQx}{(x^2 + a^2)^{3/2}}$.

Flux: $\Phi_E = \oint\vec E \cdot d\vec A$.
Gauss's law: $\displaystyle \oint\vec E \cdot d\vec A = \frac{Q_{enc}}{\varepsilon_0}$.
Choose a Gaussian surface that matches the symmetry so $E$ is constant on it:

Charge distribution	Gaussian surface	Result
Point charge or sphere (outside)	Sphere	$\displaystyle E = \frac{kQ}{r^2}$
Long line of charge $\lambda$	Cylinder	$\displaystyle E = \frac{\lambda}{2\pi\varepsilon_0r}$
Infinite plane $\sigma$	Pillbox	$\displaystyle E = \frac{\sigma}{2\varepsilon_0}$
Uniform solid sphere, inside ($r < R$)	Sphere	$\displaystyle E = \frac{kQr}{R^3}$

Flux through a closed surface depends only on enclosed charge, not on the surface's shape or charges outside.

The electric field 2.0 m from a 4.0 μC point charge is about:

A closed surface encloses $+3q$ and $-q$, and a charge $+5q$ sits outside. The net flux through the surface is:

Which Gaussian surface is best for an infinitely long line of charge?

The field at a distance $r$ from a long line with charge density $\lambda$ varies as:

What is the electric field at the center of a uniformly charged ring?

Inside a uniformly charged solid insulating sphere, the field:

The field near a large flat sheet of uniform charge density $\sigma$:

A cube has a uniform field passing straight through two opposite faces. The net flux through the cube is:

Write your answer first, then open the scoring guide.

Free-response question

A solid insulating sphere of radius $R$ has a volume charge density $\rho(r) = \beta r$, where $\beta$ is a positive constant.

(a) Find the total charge $Q$ of the sphere.

Show the answer and scoring guide

$\displaystyle Q = \int_0^R \beta r(4\pi r^2)\,dr = 4\pi\beta\frac{R^4}{4} = \pi\beta R^4$.

Check off the points you earned:

(b) Use Gauss's law to find $E$ inside the sphere ($r < R$).

Show the answer and scoring guide

$Q_{enc} = \pi\beta r^4$. $\displaystyle E(4\pi r^2) = \frac{\pi\beta r^4}{\varepsilon_0} \Rightarrow E = \frac{\beta r^2}{4\varepsilon_0}$.

Check off the points you earned:

(c) Find $E$ outside the sphere ($r > R$).

Show the answer and scoring guide

$\displaystyle E(4\pi r^2) = \frac{\pi\beta R^4}{\varepsilon_0} \Rightarrow E = \frac{\beta R^4}{4\varepsilon_0r^2}$.

Check off the points you earned: