Q: A large, thin circular sheet of radius $5.00\ \mathrm{m}$ contains $2.50 \times 10^{12}$ protons spread evenly over its surface. What is the total charge on the sheet in Coulombs?

$4.0 \times 10^{-7}\ \mathrm{C}$

Q: Given a surface charge density $\sigma = 0.4\ \mu\mathrm{C}/\mathrm{m}^2$ on a circular sheet of area $A = 78.5\ \mathrm{m}^2$, what is the total charge $q$ on the sheet?

$q = \sigma A = 0.4 \times 10^{-6}\ \mathrm{C}/\mathrm{m}^2 \times 78.5\ \mathrm{m}^2 = 3.14 \times 10^{-5}\ \mathrm{C}$

Q: In the same square charge configuration, what is the net electric field at the center due to all three charges if $q_1 = -q_2 = -q_3$?

$E_{\text{net}} = 2\sqrt{2} \dfrac{k q_1}{L^2}$

Q: A square surface of side $L = 2\ \mathrm{m}$ is oriented in the $x$-$y$ plane. The electric field at the surface is $\vec{E} = (220\ \hat{i} - 70\ \hat{j} - 10\ \hat{k})\ \mathrm{N}/\mathrm{C}$. What is the absolute value of the flux through the surface?

$|\Phi| = |E_z| A = 10 \times (2)^2 = 40\ \mathrm{N} \cdot \mathrm{m}^2/\mathrm{C}$

Q: An infinitely-long insulating cylinder has radius $R$ and uniform volume charge density $\rho$. What is the electric field at a point $r$ inside the cylinder ($r < R$)?

$E = \dfrac{\rho r}{2\varepsilon_0}$

Question 1

A large, thin circular sheet of radius $5.00\ \mathrm{m}$ contains $2.50 	imes 10^{12}$ protons spread evenly over its surface. What is the total charge on the sheet in Coulombs?

Accepted Answer

$4.0 	imes 10^{-7}\ \mathrm{C}$

Question 2

Given a surface charge density $\sigma = 0.4\ \mu\mathrm{C}/\mathrm{m}^2$ on a circular sheet of area $A = 78.5\ \mathrm{m}^2$, what is the total charge $q$ on the sheet?

Accepted Answer

$q = \sigma A = 0.4 	imes 10^{-6}\ \mathrm{C}/\mathrm{m}^2 	imes 78.5\ \mathrm{m}^2 = 3.14 	imes 10^{-5}\ \mathrm{C}$

Question 3

What is the magnitude of the electric field very close to the center of a large charged sheet with surface charge density $\sigma$?

Accepted Answer

$E = \dfrac{\sigma}{2\varepsilon_0}$

Question 4

A point charge $Q = -15\ \mu\mathrm{C}$ is placed in an electric field and experiences a force $\vec{F} = (-4.0\ \hat{i} + 8.0\ \hat{j})\ \mathrm{N}$. What is the electric field at this location?

Accepted Answer

$\vec{E} = \dfrac{\vec{F}}{Q} = (0.27\ \hat{i} - 0.53\ \hat{j})\ \mathrm{kV}/\mathrm{C}$

Question 5

Three point charges are at the corners of a square of side $L$. $q_1$ is positive, $q_2$ and $q_3$ are negative. Point $P$ is at the center. What is the $y$-component of the electric field at $P$ due to $q_1$?

Accepted Answer

$E_{y, q_1} = \dfrac{k q_1}{L^2} \dfrac{1}{\sqrt{2}}$

Question 6

In the same square charge configuration, what is the net electric field at the center due to all three charges if $q_1 = -q_2 = -q_3$?

Accepted Answer

$E_{	ext{net}} = 2\sqrt{2} \dfrac{k q_1}{L^2}$

Question 7

A uniformly charged rod of length $L$ lies along the $x$-axis with its left end at the origin. The rod has a total charge $q$. Point $P$ is located on the $x$-axis a distance $d$ to the left of the origin. What is the correct integral to find the electric field at $P$?

Accepted Answer

$\vec{E} = \int_0^L \dfrac{k q}{d + x)^2} ( -\hat{i} )$

Question 8

A square surface of side $L = 2\ \mathrm{m}$ is oriented in the $x$-$y$ plane. The electric field at the surface is $\vec{E} = (220\ \hat{i} - 70\ \hat{j} - 10\ \hat{k})\ \mathrm{N}/\mathrm{C}$. What is the absolute value of the flux through the surface?

Accepted Answer

$|\Phi| = |E_z| A = 10 	imes (2)^2 = 40\ \mathrm{N} \cdot \mathrm{m}^2/\mathrm{C}$

Question 9

A charged sheet with surface charge density $\sigma = +0.15\ \mu\mathrm{C}/\mathrm{m}^2$ produces an electric field $\vec{E}_s = + (2,500\ \hat{z})\ \mathrm{N}/\mathrm{C}$. Using Gauss's law, what is the magnitude of the electric field just above the sheet?

Accepted Answer

$E = \dfrac{\sigma}{2\varepsilon_0}$

Question 10

An infinitely-long insulating cylinder has radius $R$ and uniform volume charge density $\rho$. What is the electric field at a point $r$ inside the cylinder ($r < R$)?

Accepted Answer

$E = \dfrac{\rho r}{2\varepsilon_0}$

Question 11

Suppose the same cylinder is replaced by a conducting cylinder with the same total charge. What happens to the electric field inside the cylinder?

Accepted Answer

The field inside becomes zero.

Question 12

An electron is moved from point $A$ to $B$ above an infinite charged sheet. The field is $E = 2550\ \mathrm{N}/\mathrm{C}$, and $\Delta V = -102\ \mathrm{V}$. What is the distance $d$ between $A$ and $B$?

Accepted Answer

$d = \dfrac{\Delta V}{-E} = \dfrac{-102}{-2550} = 0.04\ \mathrm{m}$

Question 13

What is the change in potential energy $\Delta U_e$ for the electron's movement in the previous question?

Accepted Answer

$\Delta U_e = q \Delta V = (-1.6 	imes 10^{-19}\ \mathrm{C})(-102\ \mathrm{V}) = 1.63 	imes 10^{-17}\ \mathrm{J}$

Question 14

A proton is moved instead of an electron in the previous scenario. What is $\Delta V_{AB}$ in this case?

Accepted Answer

$\Delta V_{AB} = -102\ \mathrm{V}$

Question 15

Three point charges $q_1$, $q_2$, $q_3$ are arranged on a rectangle with $d_1 = 4\ \mathrm{cm}$ and $d_2 = 3\ \mathrm{cm}$. $q_1 = 2\ \mu\mathrm{C}$, $q_2 = -1.5\ \mu\mathrm{C}$, $q_3 = -2\ \mu\mathrm{C}$. What is the potential energy of this configuration?

Accepted Answer

$U = k_e \left( \dfrac{q_1 q_2}{d_1} + \dfrac{q_1 q_3}{d_2} + \dfrac{q_2 q_3}{\sqrt{d_1^2 + d_2^2}} \right )$

Question 16

A charged sheet with surface charge density $\sigma$ produces an electric field $E$. If a test charge is placed just above the sheet, what is the direction of the electric field?

Accepted Answer

Away from the sheet if $\sigma > 0$, toward the sheet if $\sigma < 0$

Question 17

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

Accepted Answer

$\Phi = \dfrac{Q}{\varepsilon_0}$

Electrostatics: Charge Distributions, Electric Fields, and Gauss's Law – Study Notes

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