24. Electric Force & Field; Gauss' Law

What is the net electric flux through the cylinder of FIGURE EX24.21?

A conductor with an inner cavity, like that shown in Fig. $22.23$c, carries a total charge of $+5.00$ nC. The charge within the cavity, insulated from the conductor, is $-6.00$ nC. How much charge is on (a) the inner surface of the conductor and (b) the outer surface of the conductor?

A very large, horizontal, nonconducting sheet of charge has uniform charge per unit area $\sigma=5.00\times10^{-6}$ C/m². A small sphere of mass $m=8.00\times10^{-6}$ kg and charge $q$ is placed $3.00$ cm above the sheet of charge and then released from rest. What is $q$ if the sphere is released $1.50$ cm above the sheet?

A very large, horizontal, nonconducting sheet of charge has uniform charge per unit area $\sigma =5.00\times {10}^{-6}$ C/m². A small sphere of mass $m=8.00\times {10}^{-6}$ kg and charge $q$ is placed $3.00$ cm above the sheet of charge and then released from rest. If the sphere is to remain motionless when it is released, what must be the value of $q$?

An infinitely long cylindrical conductor has radius $r$ and uniform surface charge density $\sigma$. In terms of $\sigma$, what is the magnitude of the electric field produced by the charged cylinder at a distance $r>R$ from its axis? Then, express the result in terms of $\lambda$ and show that the electric field outside the cylinder is the same as if all the charge were on the axis.

An infinitely long cylindrical conductor has radius $r$ and uniform surface charge density $\sigma$. In terms of $\sigma$ and $R$, what is the charge per unit length $\lambda$ for the cylinder?

A very long conducting tube (hollow cylinder) has inner radius $A$ and outer radius $b$. It carries charge per unit length $+α$, where $\alpha$ is a positive constant with units of C/m. A line of charge lies along the axis of the tube. The line of charge has charge per unit length. Calculate the electric field in terms of $\alpha$ and the distance $r$ from the axis of the tube for (i) ; (ii) ; (iii) $r>b$. Show your results in a graph of $E$ as a function of $R$.

A very long conducting tube (hollow cylinder) has inner radius $A$ and outer radius $b$. It carries charge per unit length $+\alpha$, where $\alpha$ is a positive constant with units of C/m. A line of charge lies along the axis of the tube. The line of charge has charge per unit length. What is the charge per unit length on (i) the inner surface of the tube and (ii) the outer surface of the tube?

"(II) Suppose the thick spherical shell of Problem 29 is a conductor. It carries a total net charge Q and at its center there is a point charge +q. What total charge is found on the outer surface of the shell?"

(II) Suppose the thick spherical shell of Problem 29 is a conductor. It carries a total net charge Q and at its center there is a point charge +q. What total charge is found on the inner surface of the shell?

An infinite cylinder of radius R has a linear charge density λ . The volume charge density (C/m³) within the cylinder (r ≤ R ) is p (r) = rp₀ / R, where p₀ is a constant to be determined. The charge within a small volume dV is dq = pdV. The integral of pdV over a cylinder of length L is the total charge Q = λL within the cylinder. Use this fact to show that p₀ = 3λ / 2πR² Hint: Let dV be a cylindrical shell of length L, radius r, and thickness dr. What is the volume of such a shell?

A rod of length $L$ lies along the $y$-axis with its center at the origin. The rod has a nonuniform linear charge density $\lambda =a\mid y\mid$, where a is a constant with the units C/m². Draw a graph of $\lambda$ versus $y$ over the length of the rod.

A 20-cm-radius ball is uniformly charged to 80 nC. How much charge is enclosed by spheres of radii 5, 10, and 20 cm?

A proton orbits a long charged wire, making $1.0\times {10}^6$ revolutions per second. The radius of the orbit is $1.0$ cm. What is the wire's linear charge density?

CALC A 12-cm-long thin rod has the nonuniform charge density $\lambda \left(x\right)=\left(2.0\text{nC/cm}\right)e^{-\mid x\mid /\left(6.0\text{cm}\right)}$, where x is measured from the center of the rod. What is the total charge on the rod? Hint: This exercise requires an integration. Think about how to handle the absolute value sign.