Chapter 23

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?

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?

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.)

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.

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?

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?

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.)

(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?

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.)

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 \(E_B/E_A\)?

A metal sphere with radius \(R_1\) has a charge \(Q_1\) . Take the electric potential to be zero at an infinite distance from the sphere. (a) What are the electric field and electric potential at the surface of the sphere? This sphere is now connected by a long, thin conducting wire to another sphere of radius \(R_2\) that is several meters from the first sphere. Before the connection is made, this second sphere is uncharged. After electrostatic equilibrium has been reached, what are (b) the total charge on each sphere; (c) the electric potential at the surface of each sphere; (d) the electric field at the surface of each sphere? Assume that the amount of charge on the wire is much less than the charge on each sphere.

The electric potential V in a region of space is given by $$V(x, y, z) = A(x^2 - 3y^2 + z^2)$$ where \(A\) is a constant. (a) Derive an expression for the electric field \(\overrightarrow{E}\) at any point in this region. (b) The work done by the field when a 1.50-\(\mu\)C test charge moves from the point \((x, y, z) = (0, 0, 0.250 m)\) to the origin is measured to be 6.00 \(\times 10^{-5}\) J. Determine A. (c) Determine the electric field at the point \((0, 0, 0.250 m)\). (d) Show that in every plane parallel to the \(xz\)-plane the equipotential contours are circles. (e) What is the radius of the equipotential contour corresponding to \(V = 1280\) \(V\) and \(y = 2.00\) m?

A hollow, thin-walled insulating cylinder of radius \(R\) and length \(L\) (like the cardboard tube in a roll of toilet paper) has charge \(Q\) uniformly distributed over its surface. (a) Calculate the electric potential at all points along the axis of the tube. Take the origin to be at the center of the tube, and take the potential to be zero at infinity. (b) Show that if \(L \ll R\), the result of part (a) reduces to the potential on the axis of a ring of charge of radius \(R\). (See Example 23.11 in Section 23.3.) (c) Use the result of part (a) to find the electric field at all points along the axis of the tube.

For a particular experiment, helium ions are to be given a kinetic energy of 3.0 MeV. What should the voltage at the center of the accelerator be, assuming that the ions start essentially at rest? (a) -3.0 MV; (b) +3.0 MV; (c) +1.5 MV; (d) +1.0 MV.

A helium ion (He\(^{++}\)) that comes within about 10 fm of the center of the nucleus of an atom in the sample may induce a nuclear reaction instead of simply scattering. Imagine a helium ion with a kinetic energy of 3.0 MeV heading straight toward an atom at rest in the sample. Assume that the atom stays fixed. What minimum charge can the nucleus of the atom have such that the helium ion gets no closer than 10 fm from the center of the atomic nucleus? (1 fm = 1 \(\times\) 10\(^{-15}\) m, and \(e\) is the magnitude of the charge of an electron or a proton.) (a) 2\(e\); (b) 11\(e\); (c) 20\(e\); (d) 22\(e\).

The maximum voltage at the center of a typical tandem electrostatic accelerator is 6.0 MV. If the distance from one end of the acceleration tube to the midpoint is 12 m, what is the magnitude of the average electric field in the tube under these conditions? (a) 41,000 V/m; (b) 250,000 V/m; (c) 500,000 V/m; (d) 6,000,000 V/m.