Chapter 22

22.1. A flat sheet of paper of area 0.250 \(\mathrm{m}^{2}\) is oriented so that the normal to the sheet is at an angle of \(60^{\circ}\) to a uniform electric field of magnitude 14 \(\mathrm{N} / \mathrm{C}\) (a) Find the magnitude of the electric flux through the sheet. (b) Does the answer to part (a) depend on the shape of the sheet? Why or why not?(c) For what angle \(\phi\) between the normal to the sheet and the electric field is the magnitude of the flux through the sheet (i) largest and (ii) smallest? Explain your answers.

22.3 You measure an electric field of \(1.25 \times 10^{6} \mathrm{N} / \mathrm{C}\) at a distance of 0.150 \(\mathrm{m}\) from a point charge. (a) What is the electric flux through a sphere at that distance from the charge? (b) What is the magnitude of the charge?

22.5. A hemispherical surface with radius \(r\) in a region of uniform electric field \(\vec{E}\) has its axis aligned parallel to the direction of the field. Calculate the flux through the surface.

22.9. A charged paint is spread in a very thin uniform layer over the surface of a plastic sphere of diameter \(12.0 \mathrm{cm},\) giving it a charge of \(-15.0 \mu \mathrm{C}\) . Find the electric field (a) just inside the paint layer; (b) just outside the paint layer: (c) 5.00 \(\mathrm{cm}\) outside the surface of the paint layer.

22.10. A point charge \(q_{1}=4.00 \mathrm{nC}\) is located on the \(x\) -axis at \(x=2.00 \mathrm{m},\) and a second point charge \(q_{2}=-6.00 \mathrm{nC}\) is on the \(y\) -axis at \(y=1.00 \mathrm{m}\) . What is the total electric flux due to these two point charges through a spherical surface centered at the origin and with radins (a) \(0.500 \mathrm{m},(\mathrm{b}) 1.50 \mathrm{m},(\mathrm{c}) 2.50 \mathrm{m} ?\)

22.11. In a certain region of space, the electric field \(\overrightarrow{\boldsymbol{E}}\) is uniform. (a) Use Gauss's law to prove that this region of space must be electrically neutral; that is, the volume charge density \(\rho\) must be zero. (b) Is the converse true? That is, in a region of space where there is no charge, must \(\overrightarrow{\boldsymbol{E}}\) be uniform? Explain.

22.12. (a) In a certain region of space, the volume charge density \(\rho\) has a uniform positive value. Can \(\overrightarrow{\boldsymbol{E}}\) be uniform in this region? Explain. (b) Suppose that in this region of uniform positive \(\rho\) there is a "bubble" within which \(\rho=0 .\) Can \(\vec{E}\) be uniform within this bubble? Explain.

22\. 13. A \(9.60-\mu\) C point charge is at the center of a cube with sides of length 0.500 \(\mathrm{m}\) (a) What is the electric flux through one of the six faces of the cube? (b) How would your answer to part (a) change if the sides were 0.250 \(\mathrm{m}\) long? Explain.

22\. 14. Electric Fields in an Atom. The nuclei of large atoms, such as uranium, with 92 protons, can be modeled as spherically symmetric spheres of charge. The radius of the uranium nucleus is approximately \(7.4 \times 10^{-15} \mathrm{m}\) (a) What is the electric field this nucleus produces just outside its surface? (b) What magnitude of electric field does it produce at the distance of the electrons, which is about \(1.0 \times 10^{-10} \mathrm{m} ?(\mathrm{c})\) The electrons can be modeled as forming a uniform shell of negative charge. What net electric field do they produce at the location of the nucleus?

22.15. A point charge of \(+5.00 \mu C\) is located on the \(x\) -axis at \(x=4.00 \mathrm{m},\) next to a spherical surface of radius 3.00 \(\mathrm{m}\) centered at the origin. (a) Calculate the magnitude of the electric field at \(x=3.00 \mathrm{m} .\) (b) Calculate the magnitude of the electric field at \(x=-3.00 \mathrm{m} .\) (c) According to Gauss's law, the net flux through the sphere is zero because it contains no charge. Yet the field due to the external charge is much stronger on the near side of the sphere (i.e., at \(x=3.00 \mathrm{m}\) ) than on the far side (at \(x=-3.00 \mathrm{m}\) ). How, then, can the flux into the sphere (on the near side) equal the flux out of it (on the far side)? Explain. A sketch will help.

22.16. A solid metal sphere with radius 0.450 \(\mathrm{m}\) carries a net charge of 0.250 \(\mathrm{nC}\) . Find the magnitude of the electric field (a) at a point 0.100 \(\mathrm{m}\) outside the surface of the sphere and \((\mathrm{b})\) at a point inside the sphere, 0.100 \(\mathrm{m}\) below the surface.

22.19. How many excess electrons must be added to an isolated spherical conductor 32.0 \(\mathrm{cm}\) in diameter to produce an electric field of 1150 \(\mathrm{N} / \mathrm{C}\) just outside the surface?

22.20. The electric field 0.400 \(\mathrm{m}\) from a very long uniform line of charge is 840 \(\mathrm{N} / \mathrm{C}\) . How much charge is contained in a \(2.00-\mathrm{cm}\) section of the line?

22.21. A very long uniform line of charge has charge per unit length 4.80\(\mu \mathrm{C} / \mathrm{m}\) and lies along the \(x\) -axis. A second long uniform line of charge has charge per unit length \(-2.40 \mu \mathrm{C} / \mathrm{m}\) and is parallel to the \(x\) -axis at \(y=0.400 \mathrm{m}\) . What is the net clectric field (magnitude and direction) at the following points on the \(y\) -axis: (a) \(y=0.200 \mathrm{m}\) and \((\mathrm{b}) y=0.600 \mathrm{m} ?\)

22.22. (a) At a distance of 0.200 \(\mathrm{cm}\) from the center of a charged conducting sphere with radius \(0.100 \mathrm{cm},\) the electric field is 480 \(\mathrm{N} / \mathrm{C}\) . What is the electric field 0.600 \(\mathrm{cm}\) from the center of the sphere? (b) At a distance of 0.200 \(\mathrm{cm}\) from the axis of a very long charged conducting cylinder with radius \(0.100 \mathrm{cm},\) the electric field is 480 \(\mathrm{N} / \mathrm{C}\) . What is the electric field 0.600 \(\mathrm{cm}\) from the axis of the cylinder? (c) At a distance of 0.200 \(\mathrm{cm}\) from a large uniform sheet of charge, the electric field is 480 \(\mathrm{N} / \mathrm{C}\) . What is the electric field 1.20 \(\mathrm{cm}\) from the sheet?

22.23. A hollow, conducting sphere with an outer radius of 0.250 \(\mathrm{m}\) and an inner radius of 0.200 \(\mathrm{m}\) has a uniform surface charge density of \(+6.37 \times 10^{-6} \mathrm{C} / \mathrm{m}^{2} .\) A charge of \(-0.500 \mu \mathrm{C}\) is now introduced into the cavity inside the sphere. (a) What is the new charge density on the outside of the sphere? (b) Calculate the strength of the electric field just outside the sphere. (c) What is the electric fiux through a spherical surface just inside the inner surface of the sphere?

22.24. A point charge of \(-2.00 \mu \mathrm{C}\) is located in the center of a spherical cavity of radius 6.50 \(\mathrm{cm}\) inside an insulating charged solid. The charge density in the solid is \(\rho=7.35 \times 10^{-4} \mathrm{C} / \mathrm{m}^{3} .\) Calculate the electric field inside the solid at a distance of 9.50 \(\mathrm{cm}\) from the center of the cavity.

22.25. The electric field at a distance of 0.145 \(\mathrm{m}\) from the surface of a solid insulating sphere with radius 0.355 \(\mathrm{m}\) is 1750 \(\mathrm{N} / \mathrm{C}\) . (a) Assuming the sphere's charge is uniformly distributed, what is the charge density inside it? ( 6 ) Calculate the electric field inside the sphere at a distance of 0.200 \(\mathrm{m}\) from the center.

22\. 28 . A square insulating sheet 80.0 \(\mathrm{cm}\) on a side is held horizontally. The sheet has 7.50 \(\mathrm{nC}\) of charge spread uniformly over its area. (a) Calculate the electric field at a point 0.100 \(\mathrm{mm}\) above the center of the sheet. (b) Estimate the electric field at a point 100 \(\mathrm{m}\) above the center of the sheet. (c) Would the answers to parts (a) and (b) be different if the sheet were made of a conducting material? Why or why not?

22.29. An infinitely long cylindrical conductor has radius \(R\) and uniform surface charge density \(\sigma\) . (a) In terms of \(\sigma\) and \(R\) , what is the charge per unit length \(\lambda\) for the cylinder? (b) 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? (c) Express the result of part (b) 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. Compare your result to the result for a line of charge in Example 22.6 (Section \(22.4 ) .\)

22.31. A negative charge \(-Q\) is placed inside the cavity of a hol- low metal solid. The outside of the solid is grounded by connecting a conducting wire between it and the earth. (a) Is there any excess charge induced on the inner surface of the piece of metal? If so, find its sign and magnitude. (b) Is there any excess charge on the outside of the piece of metal? Why or why not?(c) Is there an electric field in the cavity? Explain. (d) Is there an electric field within the metal? Why or why not? Is there an electric field outside the piece of metal? Explain why or why not. (e) Would someone outside the solid measure an electric field due to the charge \(-Q ?\) Is it reasonable to say that the grounded conductor has shielded the region from the ciffects of the charge \(-Q ?\) In principle, could the same thing be done for gravity? Why or why not?

12.32. A cube has sides of length \(L .\) It is placed with one corner at the origin as shown in Fig. 22.32 . The electric field is uniform and given by \(\overrightarrow{\boldsymbol{E}}=-B \hat{\boldsymbol{i}}+\boldsymbol{C} \hat{\boldsymbol{j}}-\boldsymbol{D} \hat{\boldsymbol{k}},\) where \(\boldsymbol{B}, \boldsymbol{C},\) and \(\boldsymbol{D}\) are positive constants. (a) Find the electric flux through each of the six cube faces \(S_{1}, S_{2}, S_{3}, S_{4}, S_{5},\) and \(S_{6}\) . ( \(b )\) Find the electric flux through the entire cube.

What Aflat, square surface with sides of length \(L\) is described by the equations $$ x-L \quad(0 \leq y \leq L, 0 \leq z \leq L) $$ (a) Draw this square and show the \(x\) - \(y\) - and \(z\) -axes. (b) Find the electric flux through the square due to a positive point charge \(q\) located at the origin \((x=0, y=0, z=0)\) . (Hint: Think of the square as part of a cube centered on the origin.)

22.36. A long line carying a uniform linear charge density \(+50.0 \mu C / m\) runs parallel to and 10.0 \(\mathrm{cm}\) from the surface of a large, flat plastic sheet that has a uniform surface charge density of \(-100 \mu \mathrm{C} / \mathrm{m}^{2}\) on one side. Find the location of all points where an \(\alpha\) particle would feel no force due to this arrangement of charged objects.

22.37. The Coasial Cable. A long coaxial cable consists of an inner cylindrical conductor with radius \(a\) and an outer coaxial cylinder with inner radius \(b\) and outer radius \(c\) . The outer cylinder is mounted on insulating supports and has no net charge. The inner cylinder has a uniform positive charge per unit length \(\lambda\) . Calculate the electric field (a) at any point between the cylinders a distance \(r\) from the axis and \((b)\) at any point outside the outer cylinder. (c) Graph the magnitude of the electric field as a function of the distance \(r\) from the axis of the cable, from \(r=0\) to \(r=2 c\) (d) Find the charge per unit length on the inner surface and on the outer surface of the outer cylinder.

22\. 38. 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 \(\mathrm{C} / \mathrm{m}\) . A line of charge lies along the axis of the tube. The line of charge has charge per unit length \(+\alpha\) (a) Calculate the electric field in terms of \(\alpha\) and the distance \(r\) from the axis of the tube for \((i) rb\) . Show your results in a graph of \(E\) as a function of \(r\) . (b) What is the charge per unit length on (i) the inner surface of the tube and (ii) the outer surface of the tube?

22.40. A very long, solid cylinder with radius \(R\) has positive charge uniformly distributed throughout it, with charge per unit volume \(\rho\) (a) Derive the expression for the electric field inside the volume at a distance \(r\) from the axis of the cylinder in terms of the charge density \(\rho .\) (b) What is the electric field at a point outside the volume in terms of the charge per unit length \(\lambda\) in the cylinder? (c) Compare the answers to parts (a) and (b) for \(r=R\) (d) Graph the electric-field magnitude as a function of \(r\) from \(r=0\) to \(r=3 R\) .

22.42. A Sphere in a Sphere. A solid conducting sphere carry- ing charge \(q\) has radius a. It is inside a concentric hollow conducting sphere with inner radius \(b\) and outer radius \(c\) . The hollow sphere has no net charge. (a) Derive expressions for the electric-field magnitude in terms of the distance \(r\) from the center for the regions \(rc .\) (b) Graph the magnitude of the electric field as a function of \(r\) from \(r=0\) to \(r=2 c\) (c) What is the charge on the inner surface of the hollow sphere? (d) On the outer surface? (e) Represent the charge of the small sphere by four plus signs. Sketch the field lines of the system within a spherical volume of radius 2 c.

22.43. A solid conducting sphere with radius \(R\) that carries poositive charge \(Q\) is concentric with a very thin insulating shell of radius 2\(R\) 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 \(02 R\) . (b) Graph the electric- field magnitude as a function of \(r .\)

22.48. 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 (magnitude and direction) in each of the regions \(02 R\) . Show your results in a graph of the radial component of \(\overrightarrow{\boldsymbol{E}}\) as a function of \(\boldsymbol{r}\) . (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.

22.48. Negative charge \(-Q\) is distributed uniformly over the sur- face 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) a distance \(r>R\) from the center of the shell (outside the shell) and (b) a distance \(r

22.50. (a) How many excess electrons must be distributed uni- formly within the volume of an isolated plastic sphere 30.0 \(\mathrm{cm}\) in diameter to produce an electric field of 1150 \(\mathrm{N} / \mathrm{C}\) just outside the surface of the sphere? (b) What is the electric field at a point 10.0 \(\mathrm{cm}\) outside the surface of the sphere?

22.52. Thomson's Model of the Atom. In the early years of the 20 th century, a leading model of the structure of the atom was that of the English physicist. I. 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 equilibrium position of the electron 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 motion of the electron. 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 simple harmonic motion in Section 13.2 . 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 or electrons in the atom. What would the radius of a Thomson-model atom have to be for it to produce red light of frequency \(4.57 \times 10^{14} \mathrm{Hz}\) . Compare your answer to the radii of real atoms, which are of the order of \(10^{-10} \mathrm{m}\) (see Appendix F for data about the electron). (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 sup- posed, but is concentrated in the tiny nucleus of radius \(10^{-14}\) to \(10^{-13} \mathrm{m} .\)

22.54. A Uniformiy Charged Stab. A slab of insulating material has thickness 2\(d\) and is oriented so that its faces are parallel to the \(y z\) -plane and given by the planes \(x=d\) and \(x=-d\) . The \(y\) - and \(z\) -dimensions of the slab are very large compared to \(d\) and may be treated 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.

22.56. Can Electric Forces Alone Give Stable Rquilibrium? In Chapter \(21,\) several examples were given of calculating the force exerted on a point charge by other point charges in its sur- roundings. (a) Consider a positive point charge \(+q .\) Give an example of how you would place two other point charges of your choosing so that the net force on charge \(+q\) will be zero. \((b)\) If the net force on charge \(+q\) is zero, then that charge is in equilibrium. The equilibrium will be stable if, when the charge \(+q\) is displaced slightly in any direction from its position of equilibrium, the net force on the charge pushes it back toward the equilibrium position. For this to be the case, what must the direction of the electric field \(\overrightarrow{\boldsymbol{E}}\) be due to the other charges at points surrounding the equilibrium position of \(+q ?(c)\) Imagine that the charge \(+q\) is moved very far away, and imagine a small Gaussian surface centered on the position where \(+q\) was in equilibrium. By applying Gauss's law to this surface, show that it is impossible to satisfy the condition for stability described in part (b). In other words, a charge \(+q\) cannot be held in stable equilibrium by electrostatic forces alone. This result is known as Earnshaw's theorem. (d) Parts (a)-(c) referred to the equilibrium of a positive point charge \(+q\) . Prove that Earnshaw's theorem also applies to a negative point charge \(-q\) .

22.58. A nonuniform, but spherically symmetric, distribution of charge has a charge density \(\rho(r)\) given as follows: $$ \begin{array}{ll}{\rho(r)=\rho_{0}(1-4 r / 3 R)} & {\text { for } r \leq R} \\\ {\rho(r)=0} & {\text { for } r \geq R}\end{array} $$ where \(\rho_{0}\) is a positive constant. (a) Find the total charge contained in the charge distribution. (b) Obtain an expression for the electric field in the region \(r \geq R .\) (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. $$ \oint \overrightarrow{\boldsymbol{g}} \cdot d \overrightarrow{\boldsymbol{A}}=-4 \pi G m $$ (b) By following the same logical steps used in Section 22.3 to obtain Gauss's law for the electric field, show the flux of \(\overrightarrow{\boldsymbol{g}}\) through any closed surface is given by $$ \oint \overrightarrow{\boldsymbol{g}} \cdot d \boldsymbol{A}=-4 \pi G M_{\mathrm{encl}} $$

22.63. Positive charge \(Q\) is distributed uniformly over each of two spherical volumes with radius \(R\) . One sphere of charge is centered at the origin and the other at \(x=2 R\) (Fig. 2244 ). Find the magnitude and direction of the net electric field due to these two distributions of charge at the following points on the \(x\) -axis: (a) \(x=0 ;\) (b) \(x=R / 2 ;(c) x=R ;\) (d) \(x=3 R\) .

\(22.66 .\) A region in space contains a total positive charge \(Q\) that is distributed spherically such that the volume charge density \(\rho(r)\) is given by $$ \begin{array}{ll}{\rho(r)=\alpha} & {\text { for } r \leq R / 2} \\\ {\rho(r)=2 \alpha(1-r / R)} & {\text { for } R / 2 \leq r \leq R} \\\ {\rho(r)=0} & {\text { for } r \geq R}\end{array} $$ Here \(\alpha\) is a positive constant having units of \(\mathrm{C} / \mathrm{m}^{3}\) . (a) Determine \(\alpha\) in terms of \(Q\) and \(R .\) (b) Using Gauss's law, derive an expression for the magnitude of \(\vec{E}\) as a function of \(r .\) Do this separately for all three regions. Express your answers in terms of the total charge \(Q\) . Be sure to check that your results agree on the boundaries of the regions. (c) What fraction of the total charge is contained within the region \(r \leq R / 2 ?\left(\text { d) If an electron with charge } q^{\prime}=-e \text { is }\right.\) oscillating back and forth about \(r=0\) (the center of the distribution) with an amplitude less than \(R / 2,\) show that the motion is simple harmonic. (Hint: Review the discussion of simple harmonic motion in Section 13.2. If, and only if, the net force on the electron is proportional to its displacement from equilibrium, then the motion is simple harmonic. \()(\mathrm{e})\) What is the period of the motion in part \((\mathrm{d}) ?(\mathrm{f})\) If the amplitude of the motion described in part (e) is not?

22.67. A region in space contains a total positive charge \(Q\) that is distributed spherically such that the volume charge density \(\rho(r)\) is given by $$ \begin{array}{ll}{\rho(r)=3 \alpha r /(2 R)} & {\text { for } r \leq R / 2} \\\ {\rho(r)=\alpha\left[1-(r / R)^{2}\right]} & {\text { for } R / 2 \leq r \leq R} \\ {\rho(r)=0} & {\text { for } r \geq R}\end{array} $$ Here \(\alpha\) is a positive constant having units of \(\mathrm{C} / \mathrm{m}^{3}\) . (a) Determine \(\alpha\) in terms of \(Q\) and \(R\) . (b) Using Gauss's law, derive an expression for the magnitude of the electric field as a function of \(r .\) Do this separately for all three regions. Express your answers in terms of the total charge \(Q .\) (c) What fraction of the total charge is contained within the region \(R / 2 \leq r \leq R ?(\text { d) What is the magnitude }\) of \(\overrightarrow{\boldsymbol{E}}\) at \(\boldsymbol{r}=\boldsymbol{R} / 2 ?(\mathrm{e})\) If an electron with charge \(\boldsymbol{q}^{\prime}=-e\) is released from rest at any point in any of the three regions, the resulting motion will be oscillatory but not simple harmonic. Why? (See Challenge Problem \(22.66 .\) )