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

A spherical ball of charge has radius R and total charge Q. The electric field strength inside the ball $(r \leq R) \text { is } E(r)=r^{4} E_{\max } / R^{4}$. a. What is $E_{\max }$ in terms of Q and R? b. Find an expression for the volume charge density $\rho(r)$ inside the ball as a function of r. Verify that your charge density gives the total charge Q when integrated over the volume of the ball.

The fundamental frequency of an open-open tube is 1500 Hz when the tube is filled with $0^{\circ} \mathrm{C}$ helium. What is its frequency when filled with $0^{\circ} \mathrm{C}$ air?

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, $Q_{\text {in }} / \epsilon_0$. with $Q_{\text {in }} / \epsilon$, where $\epsilon$ is the permittivity of the material. (Technically. $\boldsymbol{\epsilon}_0$ is called the vacuum permittivity.) Suppose that a $50 \mathrm{nC}$ point charge is surrounded by a thin, $32-\mathrm{cm}$-diameter spherical rubber shell and that the electric field strength inside the rubber shell is $2500 \mathrm{~N} / \mathrm{C}$. What is the permittivity of rubber?

Two 2.0-cm-diameter disks spaced 2.0 mm apart form a parallel-plate capacitor. The electric field between the disks is $5.0 \times 10^{5} \mathrm{V} / \mathrm{m}$. a. What is the voltage across the capacitor? b. An electron is launched from the negative plate. It strikes the positive plate at a speed of $2.0 \times 10^{7} \mathrm{m} / \mathrm{s}$. What was the electron’s speed a s it left the negative plate?

A 20-cm-radius ball is uniformly charged to 80 nC. a. What is the ball’s volume charge density $\left(C / m^{3}\right)$? b. How much charge is enclosed by spheres of radii 5, 10, and 20 cm? c. What is the electric field strength at points 5, 10, and 20 cm from the center?

Two loudspeakers in a $20^{\circ} \mathrm{C}$ room emit 686 Hz sound waves along the x-axis. a. If the speakers are in phase, what is the smallest distance between the speakers for which the interference of the sound waves is maximum destructive? b. If the speakers are out of phase, what is the smallest distance between the speakers for which the interference of the sound waves is maximum constructive?

A room with 3.0-m-high ceilings has a metal plate on the floor with V = 0 V and a separate metal plate on the ceiling. A 1.0 g glass ball charged to +49 nC is shot straight up at 5.0 m/s. How high does the ball go if the ceiling voltage is (a) $+3.0 \times 10^{6} \mathrm{V}$ and (b) $-3.0 \times 10^{6} \mathrm{V} ?$?

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. a. Determine the electric fields in the three regions $r \leq a$, $a<r<b, \text { and } r \geq b$. b. How much charge is on the inside surface of the hollow sphere? On the exterior surface?

A proton travels along the $x$-axis through an electric potential $V=(200 \mathrm{~V} / \mathrm{m}) x$. Its speed is $2.5 \times 10^5 \mathrm{~m} / \mathrm{s}$ as it passes the origin, moving in the $+x$-direction. What is the proton's speed at $x=1.0 \mathrm{~m}$ ?

The electron gun in an old TV picture tube accelerates electrons between two parallel plates 1.2 cm apart with a 25 kV potential difference between them. The electrons enter through a small hole in the negative plate, accelerate, then exit through a small hole in the positive plate. Assume that the holes are small enough not to affect the electric field or potential. a. What is the electric field strength between the plates? b. With what speed does a n electron exit the electron gun if its entry speed is close to zero? NOTE The exit speed is so fast that we really need to use the theory of relativity to compute an accurate value. Your answer to part b is in the right range but a little too big.

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

two very large slabs of metal that are parallel and distance $I$ 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 $Q_2=2 Q$. Assume $Q$ is positive. In terms of $Q$ and $A$, determine a. The electric field strengths $E_{\mathrm{A}}$ to $E_{\mathrm{L}}$ in regions $\mathrm{A}$ to $\mathrm{E}$. b. The surface charge densities $\eta_{\mathrm{a}}$ to $\eta_{\mathrm{d}}$ on the four surfaces a to $\mathrm{d}$.

A $1000 \mathrm{~Hz}$ sound wave traveling through $20^{\circ} \mathrm{C}$ air causes the pressure to oscillate around atmospheric pressure by $\pm 0.050 \%$. What is the maximum speed of an oscillating air molecule? Give your answer in $\mathrm{mm} / \mathrm{s}$.