24. Electric Force & Field; Gauss' Law

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.

A rod of length $L$ lies along the $y$-axis with its center at the origin. The rod has a nonuniform linear charge density $λ=a|y|$, where a is a constant with the units C/m². Determine the constant a in terms of $L$ and the rod's total charge $Q$.

A thin rod of length L and total charge Q has the nonuniform linear charge distribution $\lambda \left(x\right)={\lambda }_{0}x/L$, where x is measured from the rod's left end. What is ${\lambda }_0$ in terms of Q and L?

FIGURE EX24.2 shows a cross section of two concentric spheres. The inner sphere has a negative charge. The outer sphere has a positive charge larger in magnitude than the charge on the inner sphere. Draw this figure on your paper, then draw electric field vectors showing the shape of the electric field.

The electric field is constant over each face of the tetrahedron shown in FIGURE EX24.4. Does the box contain positive charge, negative charge, or no charge? Explain.

The three parallel planes of charge shown in FIGURE P24.44 have surface charge densities ─ ½ η , η , and ─ ½ η. Find the electric fields $\overrightarrow{E_A}$ to $\overrightarrow{E_D}$ in regions A to D. The upward direction is the + y-direction.

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