Problem 61
Coaxial Cylinders. A long metal cylinder with radius a is supported on an insulating stand on the axis of a long, hollow, metal tube with radius \(b\). The positive charge per unit length on the inner cylinder is \(\lambda\), and there is an equal negative charge per unit length on the outer cylinder. (a) Calculate the potential \(V(r)\) for (i) \(r < a\); (ii) \(a < r < b\); (iii) \(r > b\). (\(Hint:\) The net potential is the sum of the potentials due to the individual conductors.) Take \(V = 0\) at \(r = b\). (b) Show that the potential of the inner cylinder with respect to the outer is $$V^{ab} = \frac{\lambda} {2\pi\epsilon_0} ln \frac{b} {a}$$ (c) Use Eq. (23.23) and the result from part (a) to show that the electric field at any point between the cylinders has magnitude $$E(r) = \frac{V_{ab}} {ln(b/a)} \frac{1} {r}$$ (d) What is the potential difference between the two cylinders if the outer cylinder has no net charge?
Problem 64
The vertical deflecting plates of a typical classroom oscilloscope are a pair of parallel square metal plates carrying equal but opposite charges. Typical dimensions are about 3.0 cm on a side, with a separation of about 5.0 mm. The potential difference between the plates is 25.0 V. The plates are close enough that we can ignore fringing at the ends. Under these conditions: (a) how much charge is on each plate, and (b) how strong is the electric field between the plates? (c) If an electron is ejected at rest from the negative plate, how fast is it moving when it reaches the positive plate?
Problem 67
A solid sphere of radius \(R\) contains a total charge \(Q\) distributed uniformly throughout its volume. Find the energy needed to assemble this charge by bringing infinitesimal charges from far away. This energy is called the "self- energy" of the charge distribution. (\(\textit{Hint:}\) After you have assembled a charge q in a sphere of radius \(r\), how much energy would it take to add a spherical shell of thickness \(dr\) having charge \(dq\)? Then integrate to get the total energy.)
Problem 68
A thin insulating rod is bent into a semicircular arc of radius \(a\), and a total electric charge \(Q\) is distributed uniformly along the rod. Calculate the potential at the center of curvature of the arc if the potential is assumed to be zero at infinity.
Problem 69
Charge \(Q = +\)4.00 \(\mu\)C is distributed uniformly over the volume of an insulating sphere that has radius \(R =\) 5.00 cm. What is the potential difference between the center of the sphere and the surface of the sphere?
Problem 70
An insulating spherical shell with inner radius 25.0 cm and outer radius 60.0 cm carries a charge of \(+150.0\) \(\mu\)C uniformly distributed over its outer surface. Point \(a\) is at the center of the shell, point \(b\) is on the inner surface, and point \(c\) is on the outer surface. (a) What will a voltmeter read if it is connected between the following points: (i) \(a\) and \(b\); (ii) \(b\) and \(c\); (iii) \(c\) and infinity; (iv) \(a\) and \(c\)? (b) Which is at higher potential: (i) \(a\) or \(b\); (ii) \(b\) or \(c\); (iii) \(a\) or \(c\)? (c) Which, if any, of the answers would change sign if the charge were \(-\)150 \(\mu\)C?
Problem 71
Two plastic spheres, each carrying charge uniformly distributed throughout its interior, are initially placed in contact and then released. One sphere is 60.0 cm in diameter, has mass 50.0 g, and contains \(-\)10.0 \(\mu\)C of charge. The other sphere is 40.0 cm in diameter, has mass 150.0 g, and contains \(-\)30.0 \(\mu\)C of charge. Find the maximum acceleration and the maximum speed achieved by each sphere (relative to the fixed point of their initial location in space), assuming that no other forces are acting on them. (\(Hint:\) The uniformly distributed charges behave as though they were concentrated at the centers of the two spheres.)
Problem 72
(a) If a spherical raindrop of radius 0.650 mm carries a charge of \(-\)3.60 pC uniformly distributed over its volume, what is the potential at its surface? (Take the potential to be zero at an infinite distance from the raindrop.) (b) Two identical raindrops, each with radius and charge specified in part (a), collide and merge into one larger raindrop. What is the radius of this larger drop, and what is the potential at its surface, if its charge is uniformly distributed over its volume?
Problem 74
An alpha particle with kinetic energy 9.50 MeV (when far away) collides head- on with a lead nucleus at rest. What is the distance of closest approach of the two particles? (Assume that the lead nucleus remains stationary and may be treated as a point charge. The atomic number of lead is 82. The alpha particle is a helium nucleus, with atomic number 2.)
Problem 75
Two metal spheres of different sizes are charged such that the electric potential is the same at the surface of each. Sphere \(A\) has a radius three times that of sphere \(B\). Let \(Q_A\) and \(Q_B\) be the charges on the two spheres, and let \(E_A\) and \(E_B\) be the electric-field magnitudes at the surfaces of the two spheres. What are (a) the ratio \(Q_B/Q_A\) and (b) the ratio \(E_B/E_A\)?
