| Find the electric fluxes ${\text{Φ}}_{\text{A}}$ to ${\text{Φ}}_{\text{E}}$ through surfaces A to E in FIGURE P24.29.

FIGURE P24.29

| FIGURE P24.30 shows four sides of a $3.0\text{cm}\times 3.0\text{cm }\times 3.0\text{cm}$ cube.

FIGURE P24.30

1. What are the electric fluxes ${\text{Φ}}_{\text{A}}$ to ${\text{Φ}}_{\text{D}}$ through sides A to D?

2. What is the net flux through these four sides?

| A 10 nC charge is at the center of a $2.0\text{m}\times 2.0\text{m }\times 2.0\text{m}$ cube. What is the electric flux through the top surface of the cube?

| Charges $q_1=-4Q$ and $q_2=+2Q$ are located at $x=-a$ and $x=+a$, respectively. What is the net electric flux through a sphere of radius 2a centered (a) at the origin and (b) at $x=2a?$

|| A neutral conductor contains a hollow cavity in which there is a $+100\text{nC}$ point charge. A charged rod then transfers $-50\text{nC}$ to the conductor. Afterward, what is the charge (a) on the inner wall of the cavity, and (b) on the exterior surface of the conductor?

|| A spherically symmetric charge distribution produces the electric field $\vec{E}=\left(5000r^2\right)\widehat{r}\text{N}\text{/}\text{C}$, where r is in m.

1. What is the electric field strength at $r=20\text{cm}?$

2. What is the electric flux through a 40-cm-diameter spherical surface that is concentric with the charge distribution?

3. How much charge is inside this 40-cm-diameter spherical surface?

|| A hollow metal sphere has inner radius a and outer radius b. The hollow sphere has charge $+2Q$. A point charge $+Q$ sits at the center of the hollow sphere.

1. Determine the electric fields in the three regions $r\le a$, $a<r<b$, and $r\ge b$.

2. How much charge is on the inside surface of the hollow sphere? On the exterior surface?

|| A 20-cm-radius ball is uniformly charged to 80 nC.

1. What is the ball’s volume charge density $(\text{C}\text{/}{\text{m}}^3)?$

2. How much charge is enclosed by spheres of radii 5, 10, and 20 cm?

3. What is the electric field strength at points 5, 10, and 20 cm from the center?

|| FIGURE P24.37 shows a solid metal sphere at the center of a hollow metal sphere. What is the total charge on (a) the exterior of the inner sphere, (b) the inside surface of the hollow sphere, and (c) the exterior surface of the hollow sphere?

FIGURE P24.37

|| The earth has a vertical electric field at the surface, pointing down, that averages 100 N/C. This field is maintained by various atmospheric processes, including lightning. What is the excess charge on the surface of the earth?

|| Figure 24.32b showed a conducting box inside a parallel-plate capacitor. The electric field inside the box is $\vec{E}=\vec{0}$. Suppose the surface charge on the exterior of the box could be frozen. Draw a picture of the electric field inside the box after the box, with its frozen charge, is removed from the capacitor.

Hint: Superposition.

|| A hollow metal sphere has 6 cm and 10 cm inner and outer radii, respectively. The surface charge density on the inside surface is $-100\text{nC}\text{/}{\text{m}}^2$. The surface charge density on the exterior surface is $+100\text{nC}\text{/}{\text{m}}^2$. What are the strength and direction of the electric field at points 4, 8, and 12 cm from the center?

|| An infinitely wide, horizontal metal plate lies above a horizontal infinite sheet of charge with surface charge density $800{\text{nC/m}}^2$. The bottom surface of the plate has surface charge density $-100{\text{nC/m}}^2$. What is the surface charge density on the top surface of the plate?

|| A uniformly charged ball of radius a and charge $-Q$ is at the center of a hollow metal shell with inner radius b and outer radius c. The hollow sphere has net charge $+2Q$. Determine the electric field strength in the four regions $r\le a\text{,}a<r<b\text{,}b\le r\le c$, and $r>c$.

|| Find the electric field inside and outside a hollow plastic ball of radius R that has charge Q uniformly distributed on its outer surface.

| The three parallel planes of charge shown in FIGURE P24.44 have surface charge densities $-\frac{1}{2}\eta \text{,}\eta$, and $-\frac{1}{2}\eta$. Find the electric fields $\vec{E}{}_{\text{A}}\text{to}\vec{E}{}_{\text{D}}$ in regions A to D. The upward direction is the $+y$-direction.

FIGURE P24.44

|| An infinite slab of charge of thickness $2z_0$ lies in the xy-plane between $z=-z_0$ and $z=+z_0$. The volume charge density $\rho (\text{C}\text{/}{\text{m}}^3)$ is a constant.

1. Use Gauss’s law to find an expression for the electric field strength inside the slab $(-z_0\le z\le z_0)$.

2. Find an expression for the electric field strength above the slab $(z\ge z_0)$.

3. Draw a graph of E from $z=0$ to $z=3z_0$.

|| FIGURE P24.46 shows an infinitely wide conductor parallel to and distance d from an infinitely wide plane of charge with surface charge density $\eta$. What are the electric fields $\vec{E}{}_{\text{A}}$ to $\vec{E}{}_{\text{D}}$ in regions A to D?

FIGURE P24.46

|| FIGURE P24.47 shows two very large slabs of metal that are parallel and distance l apart. The top and bottom of each slab has surface area A. The thickness of each slab is so small in comparison to its lateral dimensions that the surface area around the sides is negligible. Metal 1 has total charge $Q_1=Q$ and metal 2 has total charge $Q_2=2Q$. Assume Q is positive. In terms of Q and A, determine

1. The electric field strengths $E_{\text{A}}$ to $E_{\text{E}}$ in regions A to E.

2. The surface charge densities ${\eta }_{\text{a}}$ to ${\eta }_{\text{d}}$ on the four surfaces a to d.

FIGURE P24.47

|| CALC An infinite slab of charge is centered in the xy-plane. It has charge density $\rho ={\rho }_0{e}^{-\left|z\right|/z_0}$, where ${\rho }_0$ and $z_0$ are constants. This is a charge density that decreases exponentially as you move away from $z=0$ in either the positive or negative direction. Find the electric field strength at distance z from the center of the slab.

|| A very long, uniformly charged cylinder has radius R and linear charge density $\text{λ}$. Find the cylinder’s electric field strength (a) outside the cylinder, $r\ge R$, and (b) inside the cylinder, $r\le R$. (c) Show that your answers to parts a and b match at the boundary, $r=R$.

|| A long, thin straight wire with linear charge density $\text{λ}$ runs down the center of a thin, hollow metal cylinder of radius R. The cylinder has a net linear charge density $2\text{λ}$. Assume $\text{λ}$ is positive. Find expressions for the electric field strength (a) inside the cylinder, $r<R$, and (b) outside the cylinder, $r>R$. In what direction does the electric field point in each of the cases?

|| The electric field must be zero inside a conductor in electrostatic equilibrium, but not inside an insulator. It turns out that we can still apply Gauss’s law to a Gaussian surface that is entirely within an insulator by replacing the right-hand side of Gauss’s law, $Q_{\text{in}}/{ϵ}_0$, with $Q_{\text{in}}/ϵ$, where $ϵ$ is the permittivity of the material. (Technically, ${ϵ}_0$ is called the vacuum permittivity.) Suppose that a 50 nC point charge is surrounded by a thin, 32-cm-diameter spherical rubber shell and that the electric field strength inside the rubber shell is $2500\text{N/C}$. What is the permittivity of rubber?

|| Three long, concentric cylindrical shells of radii R, 2R, and 3R have linear charge densities $-{\text{λ}}_0$, $+2{\text{λ}}_0$, and $-{\text{λ}}_0$, respectively. The thickness of each shell is negligible. What are the electric field strength and direction in the four regions (a) $r<R$, (b) $R<r<2R$, (c) $2R<r<3R$, and (4) $r>3R?$

||| A long cylinder with radius b and volume charge density $\rho$ has a spherical hole with radius $a<b$ centered on the axis of the cylinder. What is the electric field strength inside the hole at radial distance $r<a$ in a plane that is perpendicular to the cylinder through the center of the hole?

Hint: Can you create this charge distribution as a superposition of charge distributions for which you can use Gauss’s law to find the electric field?

|| A spherical shell has inner radius $R_{\text{in}}$ and outer radius $R_{\text{out}}$. The shell contains total charge Q, uniformly distributed. The interior of the shell is empty of charge and matter.

1. Find the electric field strength outside the shell, $r\ge R_{\text{out}}$.

2. Find the electric field strength in the interior of the shell, $r\le R_{\text{in}}$.

3. Find the electric field strength within the shell, $R_{\text{in}}\le r\le R_{\text{out}}$.

4. Show that your solutions match at both the inner and outer boundaries.

||| An early model of the atom, proposed by Rutherford after his discovery of the atomic nucleus, had a positive point charge $+Ze$ (the nucleus) at the center of a sphere of radius R with uniformly distributed negative charge $-Ze$. Z is the atomic number, the number of protons in the nucleus and the number of electrons in the negative sphere.

1. Show that the electric field strength inside this atom is

   $$E_{\text{in}}={\displaystyle \frac{Ze}{4\pi {ϵ}_0}}({\displaystyle \frac{1}{r^2}}-{\displaystyle \frac{r}{R^3}})$$

2. What is E at the surface of the atom? Is this the expected value? Explain.

3. A uranium atom has $Z=92$ and $R=0.10\text{nm}$. What is the electric field strength at $r=\frac{1}{2}R?$

|| CALC Newton’s law of gravity and Coulomb’s law are both inverse-square laws. Consequently, there should be a “Gauss’s law for gravity.”

1. The electric field was defined as $\vec{E}=\vec{F}{}_{\text{on }q}/q$, and we used this to find the electric field of a point charge. Using analogous reasoning, what is the gravitational field $\vec{g}$ of a point mass? Write your answer using the unit vector $\widehat{r}$, but be careful with signs; the gravitational force between two “like masses” is attractive, not repulsive.

2. What is Gauss’s law for gravity, the gravitational equivalent of Equation 24.18? Use ${\text{Φ}}_{\text{G}}$ for the gravitational flux, $\vec{g}$ for the gravitational field, and $M_{\text{in}}$ for the enclosed mass.

3. A spherical planet is discovered with mass M, radius R, and a mass density that varies with radius as $\rho ={\rho }_0(1-r/2R)$, where ${\rho }_0$ is the density at the center. Determine ${\rho }_0$ in terms of M and R.

   Hint: Divide the planet into infinitesimal shells of thickness dr, then sum (i.e., integrate) their masses.

4. Find an expression for the gravitational field strength inside the planet at distance $r<R$.

||| CALC All examples of Gauss’s law have used highly symmetric surfaces where the flux integral is either zero or EA. Yet we’ve claimed that the net ${\text{Φ}}_{\text{e}}=Q_{\text{in}}\text{/}{ϵ}_0$ is independent of the surface. This is worth checking. FIGURE CP24.57 shows a cube of edge length L centered on a long thin wire with linear charge density $\text{λ}$. The flux through one face of the cube is not simply EA because, in this case, the electric field varies in both strength and direction. But you can calculate the flux by actually doing the flux integral.

FIGURE CP24.57

1. Consider the face parallel to the yz-plane. Define area $d\vec{A}$ as a strip of width dy and height L with the vector pointing in the x-direction. One such strip is located at position y. Use the known electric field of a wire to calculate the electric flux $d\text{Φ}$ through this little area. Your expression should be written in terms of y, which is a variable, and various constants. It should not explicitly contain any angles.

2. Now integrate $d\text{Φ}$ to find the total flux through this face.

3. Finally, show that the net flux through the cube is ${\text{Φ}}_{\text{e}}=Q_{\text{in}}\text{/}{ϵ}_0$.

||| CALC An infinite cylinder of radius R has a linear charge density $\text{λ}$. The volume charge density $(\text{C}\text{/}{\text{m}}^3)$ within the cylinder $(r\le R)$ is $\rho \left(r\right)=r{\rho }_0\text{/}R$, where ${\rho }_0$ is a constant to be determined.

1. Draw a graph of $\rho$ versus x for an x-axis that crosses the cylinder perpendicular to the cylinder axis. Let x range from $-2R$ to 2R.

2. The charge within a small volume dV is $dq=\rho dV$. The integral of $\rho dV$ over a cylinder of length L is the total charge $Q=\text{λ}L$ within the cylinder. Use this fact to show that ${\rho }_0=3\text{λ}\text{/}2\pi R^2$.

   Hint: Let dV be a cylindrical shell of length L, radius r, and thickness dr. What is the volume of such a shell?

3. Use Gauss’s law to find an expression for the electric field strength E inside the cylinder, $r\le R$, in terms of $\text{λ}$ and R.

4. Does your expression have the expected value at the surface, $r=R?$ Explain.

||| CALC A sphere of radius R has total charge Q. The volume charge density $(\text{C}\text{/}{\text{m}}^3)$ within the sphere is $\rho (r)=C\text{/}{r}^2$, where C is a constant to be determined.

1. The charge within a small volume dV is $dq=\rho dV$. The integral of $\rho dV$ over the entire volume of the sphere is the total charge Q. Use this fact to determine the constant C in terms of Q and R.

   Hint: Let dV be a spherical shell of radius r and thickness dr. What is the volume of such a shell?

2. Use Gauss’s law to find an expression for the electric field strength E inside the sphere, $r\le R$, in terms of Q and R.

3. Does your expression have the expected value at the surface, $r=R$? Explain.

||| CALC A sphere of radius R has total charge Q. The volume charge density $(\text{C}\text{/}{\text{m}}^3)$ within the sphere is

This charge density decreases linearly from ${\rho }_0$ at the center to zero at the edge of the sphere.

1. Show that ${\rho }_0=3Q\text{/}\pi R^3$.

2. Show that the electric field inside the sphere points radially outward with magnitude

   $$E={\displaystyle \frac{Qr}{4\pi {ϵ}_0{R}^3}}(4-3{\displaystyle \frac{r}{R}})$$

3. Show that your result of part b has the expected value at $r=R$.

||| CALC A spherical ball of charge has radius R and total charge Q. The electric field strength inside the ball $(r\le R)$ is $E\left(r\right)=r^4{E}_{\text{max}}/R^4$.

1. What is $E_{\text{max}}$ in terms of Q and R?

2. Find an expression for the volume charge density $\rho \left(r\right)$ inside the ball as a function of r.

3. Verify that your charge density gives the total charge Q when integrated over the volume of the ball.