Chapter 22

A solid conducting sphere with radius \(R\) that carries positive charge \(Q\) is concentric with a very thin insulating shell of radius \(2R\) that also carries charge \(Q\). The charge \(Q\) is distributed uniformly over the insulating shell. (a) Find the electric field (magnitude and direction) in each of the regions \(0 < r < R, R < r < 2R\), and \(r > 2R\). (b) Graph the electric-field magnitude as a function of \(r\).

Negative charge \(-Q\) is distributed uniformly over the surface of a thin spherical insulating shell with radius R. Calculate the force (magnitude and direction) that the shell exerts on a positive point charge \(q\) located a distance (a) \(r > R\) from the center of the shell (outside the shell); (b) \(r < R\) from the center of the shell (inside the shell).

A solid conducting sphere with radius \(R\) carries a positive total charge \(Q\). The sphere is surrounded by an insulating shell with inner radius \(R\) and outer radius 2\(R\). The insulating shell has a uniform charge density \(\rho\). (a) Find the value of \(\rho\) so that the net charge of the entire system is zero. (b) If \(\rho\) has the value found in part (a), find the electric field \(\overrightarrow{E}\) (magnitude and direction) in each of the regions 0 \(< r < R, R < r < 2R\), and \(r > 2R\). Graph the radial component of \(\overrightarrow{E}\) as a function of r. (c) As a general rule, the electric field is discontinuous only at locations where there is a thin sheet of charge. Explain how your results in part (b) agree with this rule.

An insulating hollow sphere has inner radius \(a\) and outer radius \(b\). Within the insulating material the volume charge density is given by \(\rho\) (\(r\)) \(= \alpha/r\), where \(\alpha\) is a positive constant. (a) In terms of \(\alpha\) and \(a\), what is the magnitude of the electric field at a distance \(r\) from the center of the shell, where \(a < r < b\)? (b) A point charge \(q\) is placed at the center of the hollow space, at \(r =\) 0. In terms of \(\alpha\) and \(a\), what value must \(q\) have (sign and magnitude) in order for the electric field to be constant in the region \(a < r < b\), and what then is the value of the constant field in this region?

Early in the 20th century, a leading model of the structure of the atom was that of English physicist J. J. Thomson (the discoverer of the electron). In Thomson's model, an atom consisted of a sphere of positively charged material in which were embedded negatively charged electrons, like chocolate chips in a ball of cookie dough. Consider such an atom consisting of one electron with mass \(m\) and charge \(-e\), which may be regarded as a point charge, and a uniformly charged sphere of charge \(+e\) and radius \(R\). (a) Explain why the electron's equilibrium position is at the center of the nucleus. (b) In Thomson's model, it was assumed that the positive material provided little or no resistance to the electron's motion. If the electron is displaced from equilibrium by a distance less than \(R\), show that the resulting motion of the electron will be simple harmonic, and calculate the frequency of oscillation. (\(Hint:\) Review the definition of SHM in Section 14.2. If it can be shown that the net force on the electron is of this form, then it follows that the motion is simple harmonic. Conversely, if the net force on the electron does not follow this form, the motion is not simple harmonic.) (c) By Thomson's time, it was known that excited atoms emit light waves of only certain frequencies. In his model, the frequency of emitted light is the same as the oscillation frequency of the electron(s) in the atom. What radius would a Thomson-model atom need for it to produce red light of frequency 4.57 \(\times\) 10\(^{14}\) Hz? Compare your answer to the radii of real atoms, which are of the order of 10\(^{-10}\) m (see Appendix F). (d) If the electron were displaced from equilibrium by a distance greater than \(R\), would the electron oscillate? Would its motion be simple harmonic? Explain your reasoning. (\(Historical\) \(note:\) In 1910, the atomic nucleus was discovered, proving the Thomson model to be incorrect. An atom's positive charge is not spread over its volume, as Thomson supposed, but is concentrated in the tiny nucleus of radius 10\(^{-14}\) to 10\(^{-15}\) m.)

Using Thomson's (outdated) model of the atom described in Problem 22.50, consider an atom consisting of two electrons, each of charge \(-e\), embedded in a sphere of charge \(+2e\) and radius \(R\). In equilibrium, each electron is a distance \(d\) from the center of the atom (\(\textbf{Fig. P22.51}\)). Find the distance \(d\) in terms of the other properties of the atom.

(a) How many excess electrons must be distributed uniformly within the volume of an isolated plastic sphere 30.0 cm in diameter to produce an electric field of magnitude 1390 N/C just outside the surface of the sphere? (b) What is the electric field at a point 10.0 cm outside the surface of the sphere?

A nonuniform, but spherically symmetric, distribution of charge has a charge density \(\rho\)(\(r\)) given as follows: $$\rho(r) = \rho_0 \bigg(1 - \frac{r}{R}\bigg) \space \space \space \mathrm{for} \space r \leq R$$ $$\rho(r) = 0 \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \mathrm{for} \space r \leq R$$ where \(\rho_0 = 3Q/{\pi}R^3\) is a positive constant. (a) Show that the total charge contained in the charge distribution is \(Q\). (b) Show that the electric field in the region \(r \geq R\) is identical to that produced by a point charge \(Q\) at \(r =\) 0. (c) Obtain an expression for the electric field in the region \(r \leq R\). (d) Graph the electric-field magnitude \(E\) as a function of \(r\). (e) Find the value of \(r\) at which the electric field is maximum, and find the value of that maximum field.

A slab of insulating material has thickness 2d and is oriented so that its faces are parallel to the yz-plane and given by the planes \(x = d\) and \(x = -d\). The \(y\)- and \(z\)-dimensions of the slab are very large compared to \(d\); treat them as essentially infinite. The slab has a uniform positive charge density \(\rho\). (a) Explain why the electric field due to the slab is zero at the center of the slab (\(x =\) 0). (b) Using Gauss's law, find the electric field due to the slab (magnitude and direction) at all points in space.

A nonuniform, but spherically symmetric, distribution of charge has a charge density \(\rho(r)\) given as follows: $$\rho(r) = \rho_0 \bigg(1 - \frac{4r}{3R}\bigg) \space \space \space \mathrm{for} \space r \leq R$$ $$\rho(r) = 0 \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \mathrm{for} \space r \leq R$$ where \(\rho_0\) is a positive constant. (a) Find the total charge contained in the charge distribution. Obtain an expression for the electric field in the region (b) \(r \geq R; (c) r \leq R\). (d) Graph the electricfield magnitude \(E\) as a function of \(r\). (e) Find the value of \(r\) at which the electric field is maximum, and find the value of that maximum field.