A hollow, conducting sphere with an outer radius of 0.250

Problem 1E A flat sheet of paper of area 0.250 m2 is oriented so that the normal to the sheet is at an angle of 60o to a uniform electric field of magnitude 14 N/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 ? 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.

CP CALC 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 \(\rho(r)=3 \alpha r /(2 R) \quad\) for \(r \leq R / 2\) \(\rho(r)=\alpha\left[1-(r / R)^2\right] \quad\) for \(R / 2 \leq r \leq R\) \(\rho(r)=0 \quad\) for \(r \geq R\) 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 \(R / 2 \leq r \leq R\)? (d) What is the magnitude of \(\overrightarrow{\boldsymbol{E}}\) at \(r=R / 2\)? (e) If an electron with charge \(q^{\prime}=-e\) 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.)

Problem 1DQ A rubber balloon has a single point charge in its interior. Does the electric flux through the balloon depend on whether or not it is fully inflated? Explain your reasoning.

Problem 2DQ Suppose that in Fig. 22.15 both charges were positive. What would be the fluxes through each of the four surfaces in the example?

Problem 3DQ In Fig. 22.15, suppose a third point charge were placed outside the purple Gaussian surface ?C?. Would this affect the electric flux through any of the surfaces ?A?, ?B?, ?C?, or D ?in the figure? Why or why not?

Problem 2E A flat sheet is in the shape of a rectangle with sides of lengths 0.400 m and 0.600 m. The sheet is immersed in a uniform electric field of magnitude 75.0 N/C that is directed at 20° from the plane of the sheet (Fig.). Find the magnitude of the electric flux through the sheet. Figure:

Problem 3E You measure an electric field of 1.25 X 106 N/C at a distance of 0.150 m from a point charge. There is no other source of electric field in the region other than this point charge. (a) What is the electric flux through the surface of a sphere that has this charge at its center and that has radius 0.150 m? (b) What is the magnitude of this charge?

A certain region of space bounded by an imaginary closed surface contains no charge. Is the electric field always zero everywhere on the surface? If not, under what circumstances is it zero on the surface?

Problem 5E A hemispherical surface with radius r in a region of uniform electric field has its axis aligned parallel to the direction of the field. Calculate the flux through the surface.

Problem 5DQ A spherical Gaussian surface encloses a point charge q. If the point charge is moved from the center of the sphere to a point away from the center, does the electric field at a point on the surface change? Does the total flux through the Gaussian surface change? Explain.

Problem 6DQ You find a sealed box on your doorstep. You suspect that the box contains several charged metal spheres packed in insulating material. How can you determine the total net charge inside the box without opening the box? Or isn’t this possible?

Problem 6E The cube in ?Fig. E22.6 has sides of length L = 10.0 cm. The electric field is uniform, has magnitude E = 4.00 X 103 N/C, and is parallel to the xy-plane at an angle of 53.1o measured from the +x-axis toward the +y-axis. (a) What is the electric flux through each of the six cube faces S1, S2, S3, S4, S5, and S6? (b) What is the total electric flux through all faces of the cube?

A solid copper sphere has a net positive charge. The charge is distributed uniformly over the surface of the sphere, and the electric field inside the sphere is zero. Then a negative point charge outside the sphere is brought close to the surface of the sphere. Is all the net charge on the sphere still on its surface? If so, is this charge still distributed uniformly over the surface? If it is not uniform, how is it distributed? Is the electric field inside the sphere still zero? In each case justify your answers.

Problem 4E It was shown in Example 21.11 (Section 21.5) that the electric field due to an infinite line of charge is perpendicular to the

Problem 7E BIO As discussed in Section 22.5, human nerve cells have a net negative charge and the material in the interior of the cell is a good conductor. If a cell has a net charge of - 8.65 pC, what are the magnitude and direction (inward or outward) of the net flux through the cell boundary?

Problem 8DQ If the electric field of a point charge were proportional to 1/r3 instead of 1/r2, would Gauss’s law still be valid? Explain your reasoning. (?Hint: Consider a spherical Gaussian surface centered on a single point charge.)

Problem 9DQ In a conductor, one or more electrons from each atom are free to roam throughout the volume of the conductor. Does this contradict the statement that any excess charge on a solid conductor must reside on its surface? Why or why not?

Problem 8E The three small spheres shown in ?Fig. E22.8 carry charges q1 = 4.00 nC, q2 = - 7.80 nC, and q3 = 2.40 nC. Find the net electric flux through each of the following closed surfaces shown in cross section in the figure: (a) S1; (b) S2; (c) S3; (d) S4; (e) S5. (f) Do your answers to parts (a)–(e) depend on how the charge is distributed over each small sphere? Why or why not?

Problem 9E A charged paint is spread in a very thin uniform layer over the surface of a plastic sphere of diameter 12.0 cm, giving it a charge of –35.0 ???C. Find the electric field (a) just inside the paint layer; (b) just outside the paint layer; (c) 5.00 cm outside the surface of the paint layer.

Problem 10DQ You charge up the van de Graaff generator shown in Fig. 22.26, and then bring an identical but uncharged hollow conducting sphere near it, without letting the two spheres touch. Sketch the distribution of charges on the second sphere. What is the net flux through the second sphere? What is the electric field inside the second sphere?

Problem 10E A point charge q1 = 4.00 nC is located on the x-axis at x = 2.00 m, and a second point charge q2 = -6.00 nC is on the y -axis at y = 1.00 m. What is the total electric flux due to these two point charges through a spherical surface centered at the origin and with radius (a) 0.500 m, (b) 1.50 m, (c) 2.50 m?

Problem 11DQ A lightning rod is a rounded copper rod mounted on top of a building and welded to a heavy copper cable running down into the ground. Lightning rods are used to protect houses and barns from lightning; the lightning current runs through the copper rather than through the building. Why? Why should the end of the rod be rounded?

Problem 11E A 6.20-µC point charge is at the center of a cube with sides of length 0.500 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 m long? Explain.

Problem 12DQ A solid conductor has a cavity in its interior. Would the presence of a point charge inside the cavity affect the electric field outside the conductor? Why or why not? Would the presence of a point charge outside the conductor affect the electric field inside the cavity? Again, why or why not?

Problem 12E 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 X 10-15 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 X 10-10 m? (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?

Problem 13DQ Explain this statement: “In a static situation, the electric field at the surface of a conductor can have no component parallel to the surface because this would violate the condition that the charges on the surface are at rest.” Would this statement be valid for the electric field at the surface of an ?insulator?? Explain your answer and the reason for any differences between the cases of a conductor and an insulator.

Problem 13E A point charge of +5.00 ???C is located on the ?x?-axis at ?x = 4.00 m, next to a spherical surface of radius 3.00 m centered of the origin. (a) Calculate the magnitude of the electric field at ?x = 3.00 m. (b) Calculate the magnitude of the electric held at ?x = 3.00 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 m) than on the far side (at ?x = – 3.00 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.

Problem 14DQ In a certain region of space, the electric field is uniform. (a) Use Gauss’s law to prove that this region of space must be electrically neutral; that is, the volume charge density ? must be zero. (b) Is the converse true? That is, in a region of space where there is no charge, must be uniform? Explain.

Problem 14E A solid metal sphere with radius 0.450 m carries a net charge of 0.250 nC. Find the magnitude of the electric field (a) at a point 0.100 m outside the surface of the sphere and (b) at a point inside the sphere, 0.100 m below the surface.

Problem 15DQ (a) In a certain region of space, the volume charge density ? has a uniform positive value. Can be uniform in this region? Explain. (b) Suppose that in this region of uniform positive ? there is a “bubble” within which ? = 0. Can be uniform within this bubble? Explain.

Two very long uniform lines of charge are parallel and are separated by 0.300 m. Each line of charge has charge per unit length +5.20 \(\mu \mathrm{C} / \mathrm{m}\). What magnitude of force does one line of charge exert on a 0.0500-m section of the other line of charge?

Poblem 16E Some planetary scientists have suggested that the planet Mars has an electric field somewhat similar to that of the earth, producing a net electric flux of -3.63 X 1016 N ? m2/C at the planet’s surface. Calculate: (a) the total electric charge on the planet; (b) the electric field at the planet’s surface (refer to the astronomical data inside the back cover); (c) the charge density on Mars, assuming all the charge is uniformly distributed over the planet’s surface.

Problem 17E How many excess electrons must be added to an isolated spherical conductor 32.0 cm in diameter to produce an electric field of 1150 N/C just outside the surface?

Problem 18E The electric field 0.400 m from a very long uniform line of charge is 840 N/C. How much charge is contained in a 2.00-cm section of the line?

Problem 19E A very long uniform line of charge has charge per unit length 4.80 µC/m and lies along the x-axis. A second long uniform line of charge has charge per unit length -2.40 µC/m and is parallel to the x-axis at y = 0.400 m. What is the net electric field (magnitude and direction) at the following points on the y-axis: (a) y = 0.200 m and (b) y = 0.600 m?

Problem 20E (a) At a distance of 0.200 cm from the center of a charged conducting sphere with radius 0.100 cm, the electric field is 480 N/C. What is the electric field 0.600 cm from the center of the sphere? (b) At a distance of 0.200 cm from the axis of a very long charged conducting cylinder with radius 0.100 cm, the electric field is 480 N/C. What is the electric field 0.600 cm from the axis of the cylinder? (c) At a distance of 0.200 cm from a large uniform sheet of charge, the electric field is 480 N/C. What is the electric field 1.20 cm from the sheet?

Problem 21E A hollow, conducting sphere with an outer radius of 0.250 m and an inner radius of 0.200 m has a uniform surface charge density of +6.37 X 10-6 C/m2. A charge of -0.500 µC is now introduced at the center of 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 flux through a spherical surface just inside the inner surface of the sphere?

Problem 22E A point charge of –2.00 ?C is located in the center of a spherical cavity of radius 6.50 cm inside an insulating charged solid. The charge density in the solid is ?? = 7.35 × 10–4 C/m3. Calculate the electric field inside the solid at a distance of 9.50 cm from the center of the cavity.

Problem 23E The electric field at a distance of 0.145 m from the sur-face of a solid insulating sphere with radius 0.355 m is 1750 N/C. (a) Assuming the sphere’s charge is uniformly distributed, what is the charge density inside it? (b) Calculate the electric field inside the sphere at a distance of 0.200 m from the center.

CP A very small object with mass \(8.20 \times 10^{-9}\) kg and positive charge \(6.50 \times 10^{-9}\) C is projected directly toward a very large insulating sheet of positive charge that has uniform surface charge density \(5.90 \times 10^{-8} \mathrm{C} / \mathrm{m}^{2}\). The object is initially 0.400 m from the sheet. What initial speed must the object have in order for its closest distance of approach to the sheet to be 0.100 m?

Problem 25E CP At time t = 0 a proton is a distance of 0.360 m from a very large insulating sheet of charge and is moving parallel to the sheet with speed 9.70 X 102 m/s. The sheet has uniform surface charge density 2.34 X 10-9 C/m2. What is the speed of the proton at t = 5.00 X 10-8 s?

Problem 26E CP An electron is released from rest at a distance of 0.300 m from a large insulating sheet of charge that has uniform surface charge density +2.90 X 10-12 C/m2. (a) How much work is done on the electron by the electric field of the sheet as the electron moves from its initial position to a point 0.050 m from the sheet? (b) What is the speed of the electron when it is 0.050 m from the sheet?

Problem 27E An insulating sphere of radius R = 0.160 m has uniform charge density ?? = +7.20 × 10?9 C/m3. A small object that can be treated as a point charge is released from rest just outside the surface of the sphere. The small object has positive charge ?q = 3.40 × 10–6 C. How much work does the electric field of the sphere do oil the object as the object moves to a point very far from the sphere?

Problem 28E A conductor with an inner cavity, like that shown in Fig.

Problem 30E A square insulating sheet 80.0 cm on a side is held horizontally. The sheet has 7.50 nC of charge spread uniformly over its area. (a) Calculate the electric field at a point 0.100 mm above the center of the sheet. (b) Estimate the electric field at a point 100 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?

Problem 29E Apply Gauss’s law to the Gaussian surfaces and

Two very large, nonconducting plastic sheets, each 10.0 cm thick, carry uniform charge densities \(\sigma_{1}\), \(\sigma_{2}\), \(\sigma_{3}\), and \(\sigma_{4}\) on their surfaces (?Fig. E22.30 ). These surface charge densities have the values \(\sigma_{1}\) = \(-6.00 \mu \mathrm{C} / \mathrm{m}^{2}\), \(\sigma_{2}=+5.00 \mu \mathrm{C} / \mathrm{m}^{2}\), \(\sigma_{3}=+2.00 \mu \mathrm{C} / \mathrm{m}^{2}\), and \(\sigma_{4}=+4.00\) \(\mu \mathrm{C} / \mathrm{m}^{2}\). Use Gauss’s law to find the magnitude and direction of the electric field at the following points, far from the edges of these sheets: (a) point A, 5.00 cm from the left face of the left-hand sheet; (b) point B, 1.25 cm from the inner surface of the right-hand sheet; (c) point C, in the middle of the right-hand sheet.

Problem 31E An infinitely long cylindrical conductor has radius? ?and 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).

Problem 35P The electric field in ?Fig. P22.35 is everywhere parallel to the x -axis, so the components Ey and Ez are zero. The x -component of the field Ex depends on x but not on y or z. At points in the yz-plane (where x = 0), Ex = 125 N/C. (a) What is the electric flux through surface I in Fig. P22.35? (b) What is the electric flux through surface II? (c) The volume shown is a small section of a very large insulating slab 1.0 m thick. If there is a total charge of -24.0 nC within the volume shown, what are the magnitude and direction of at the face opposite surface I? (d) Is the electric field produced by charges only within the slab, or is the field also due to charges outside the slab? How can you tell?

Problem 33E A negative charge –?Q is placed inside the cavity of a hollow metal solid. The outside or 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 effects of the charge ??Q?? In principle, could the same thing be done for gravity? Why or why not?

Problem 34P A cube has sides of length ?L ?= 0.300 m. It is placed with one corner at the origin as shown in Fig. E22.6. The electric

Problem 37P The electric field at one face of a parallelepiped is uniform over the entire face and is directed out of the face. At the opposite face, the electric field is also uniform over the entire face and is directed into that face (?Fig. P22.37?). The two faces in question are inclined at 30.0o from the horizontal, while both are horizontal; has a magnitude of 2.50 X 104 N/C, and has a magnitude of 7.00 X 104 N/C. (a) Assuming that no other electric field lines cross the surfaces of the parallelepiped, determine the net charge contained within. (b) Is the electric field produced by the charges only within the parallelepiped, or is the field also due to charges outside the parallelepiped? How can you tell?

Problem 36P CALC In a region of space there is an electric field that is in the z-direction and that has magnitude E = [964 N/(C ? m)]x . Find the flux for this field through a square in the xy-plane at z = 0 and with side length 0.350 m. One side of the square is along the +x-axis and another side is along the +y-axis.

Problem 38P A long line carrying a uniform linear charge density +50.0 µC/m runs parallel to and 10.0 cm from the surface of a large, flat plastic sheet that has a uniform surface charge density of -100 µC/m2 on one side. Find the location of all points where an ? particle would feel no force due to this arrangement of charged objects.

Problem 39P The Coaxial 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 ?. 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 = 2c. (d) Find the charge per unit length on the inner surface and on the outer surface of the outer cylinder.

Problem 40P A very long conducting tube (hollow cylinder) has inner radius a and outer radius b. It carries charge per unit length +?, where a 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 +?. (a) Calculate the electric field in terms of ? and the distance r from the axis of the tube for (i) r < a ; (ii) a < r < b; (iii) r > b. 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?

Problem 41P Repeat Problem, but now let the conducting tube have charge per unit length ? ??. As in the line of charge has charge per unit length +? ??.

Problem 42P A very long, solid cylinder with radius R has positive charge uniformly distributed throughout it, with charge per unit volume ?. (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 ?. (b) What is the electric field at a point outside the volume in terms of the charge per unit length ? 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 = 3R.

Problem 43P CP A small sphere with mass 4.00 X 10-6 kg and charge 5.00 X 10-8 C hangs from a thread near a very large, charged insulating sheet (?Fig. P22.33?). The charge density on the surface of the sheet is uniform and equal to -2.50 X 10-9 C/m2. Find the angle of the thread.

Problem 44P A Sphere in a Sphere. A solid conducting sphere carrying 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 r < a, a < r < b, b < r < c, and r > c. (b) Graph the magnitude of the electric field as a function of r from r = 0 to r = 2c. (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 2c.

Problem 45P 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.

Problem 46P A conducting spherical shell with inner radius a and outer radius b has a positive point charge Q located at its center. The total charge on the shell is - 3Q, and it is insulated from its surroundings (?Fig. P22.44?). (a) Derive expressions for the electric-field magnitude E in terms of the distance r from the center for the regions r < a, a < r < b, and r > b. What is the surface charge density (b) on the inner surface of the conducting shell; (c) on the outer surface of the conducting shell? (d) Sketch the electric field lines and the location of all charges. (e) Graph E as a function of r.

Problem 47P Concentric Spherical Shells. A small conducting spherical shell with inner radius a and outer radius b is concentric with a larger conducting spherical shell with inner radius c and outer radius d (?Fig. P22.45?). The inner shell has total charge +2q, and the outer shell has charge +4q. (a) Calculate the electric field (magnitude and direction) in terms of q and the distance r from the common center of the two shells for (i) r < a; (ii) a < r < b; (iii) b < r < c; (iv) c < r < d; (v) r > d. Graph the radial component of as a function of r . (b) What is the total charge on the (i) inner surface of the small shell; (ii) outer surface of the small shell; (iii) inner surface of the large shell; (iv) outer surface of the large shell?

Problem 48P Repeat Problem 22.45, but now let the outer shell have charge -2q. The inner shell still has charge +2q. 22.45 . ?Concentric Spherical Shells. A small conducting spherical shell with inner radius a and outer radius b is concentric with a larger conducting spherical shell with inner radius c and outer radius d (?Fig. P22.45?). The inner shell has total charge +2q, and the outer shell has charge +4q. (a) Calculate the electric field (magnitude and direction) in terms of q and the distance r from the common center of the two shells for (i) r < a; (ii) a < r < b; (iii) b < r < c; (iv) c < r < d; (v) r > d. Graph the radial component of as a function of r . (b) What is the total charge on the (i) inner surface of the small shell; (ii) outer surface of the small shell; (iii) inner surface of the large shell; (iv) outer surface of the large shell?

Problem 49P Repeal Problem, but now let the outer shell have charge ?4?q?. As in Problem, the inner shell has charge +2?q?. Problem: Concentric Spherical Shells. A small conducting spherical shell with inner radius a and outer radius ?b is concentric with a larger conducting spherical shell with inner radius ?c and outer radius ?d (Fig.) The inner shell has total charge +2?q?, and the outer shell has charge +4?q?. (a) Calculate the electric field (magnitude and direction) in terms of ?q and the distance ?r from the common center of the two shells for (i) ?r < ?a; (ii) ?a < ?r < ?b?; (iii) ?b < ?r < ?c?; (iv) ?c < ?r < ?d?; (v) ?r > ?d. Show your results in a graph of the radial component of as a function of ?r?. (b) What is the total charge on the (i) inner surface of the small shell; (ii) outer surface of the small shell; (iii) inner surface of the large shell; (iv) outer surface of the large shell? Figure:

Problem 50P 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 2R. The insulating shell has a uniform charge density ?. (a) Find the value of ? so that the net charge of the entire system is zero. (b) If ? has the value found in part (a), find the electric field (magnitude and direction) in each of the regions 0 < r < R, R < r < 2R, and r > 2R. Graph the radial component of 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.

Problem 51P 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).

Problem 52P (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?

CALC An insulating hollow sphere has inner radius \(a\) and outer radius \(b\). Within the insulating material the volume charge density is given by \(\rho(r)=\frac{\alpha}{r}\), 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?

Problem 54P CP Thomson’s Model of the Atom. 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 X 1014 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.)

Problem 55P Thomson’s Model of the Atom, Continued. 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 (?Fig. P22.51?). Find the distance d in terms of the other properties of the atom. 22.50 .. ?CP Thomson’s Model of the Atom. 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 X 1014 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.)

Problem 56P A Uniformly Charged Slab. 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 ?. (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.

Problem 57P CALC A Nonuniformly Charged Slab. Repeat Problem 22.54, but now let the charge density of the slab be given by ?(x) = ?0(x/d)2, where ?0 is a positive constant. 22.54 . ?A Uniformly Charged Slab. 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 ?. (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.

Problem 58P CALC A nonuniform, but spherically symmetric, distribution of charge has a charge density ?(r) given as follows: where ?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 ? R; (c) r ? 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.

Problem 59P Gauss’s Law for Gravitation. The gravitational force between two point masses separated by a distance ?r is proportional to 1/?r?2, just like the electric force between two point charges. Because of this similarity between gravitational and electric interactions, there is also a Gauss’s law for gravitation. (a) Let be the acceleration due to gravity caused by a point mass ?m at the origin, so that = ?( ?Gm?/?r?2) . Consider a spherical Gaussian surface with radius ?r centered on this point mass, and show that the flux of through this surface is given by (b) By following the same logical steps used in Section 22.3 to obtain Gauss’s law for the electric field, show that the flux of through ?any closed surface is given by where ?M? ncl is the total mass enclosed within the closed surface.

Problem 61P (a) An insulating sphere with radius a has a uniform charge density ?. The sphere is not centered at the origin but at Show that the electric field inside the sphere is given by (b) An insulating sphere of radius R has a spherical hole of radius a located within its volume and centered a distance b from the center of the sphere, where a < b < R (a cross section of the sphere is shown in ?Fig. P22.57?). The solid part of the sphere has a uniform volume charge density ?. Find the magnitude and direction of the electric field inside the hole, and show that is uniform over the entire hole. [?Hint: Use the principle of superposition and the result of part (a).]

Problem 62P A very long, solid insulating cylinder has radius R; bored along its entire length is a cylindrical hole with radius a. The axis of the hole is a distance b from the axis of the cylinder, where a < b < R (?Fig. P22.58?). The solid material of the cylinder has a uniform volume charge density ?. Find the magnitude and direction of the electric field inside the hole, and show that is uniform over the entire hole. (?Hint:? See Problem 22.57.) 22.57 . (a) An insulating sphere with radius a has a uniform charge density ?. The sphere is not centered at the origin but at Show that the electric field inside the sphere is given by (b) An insulating sphere of radius R has a spherical hole of radius a located within its volume and centered a distance b from the center of the sphere, where a < b < R (a cross section of the sphere is shown in ?Fig. P22.57?). The solid part of the sphere has a uniform volume charge density ?. Find the magnitude and direction of the electric field inside the hole, and show that is uniform over the entire hole. [?Hint: Use the principle of superposition and the result of part (a).]

Problem 63P 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). Find the magnitude and direction of the net electric field due to these two distributions of charge al the following points on the ?x?-axis: (a) ?x = 0; (b) ?x = ?R?/2; (c) ?x? = ?R?; ? ? = 3?R?. Figure:

Problem 64P Repeat Problem, but now let the left-hand sphere have positive charge ?Q and let the right-hand sphere have negative charge ??Q?. Problem: 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.). Find the magnitude and direction of the net electric field due to these two distributions of charge al the following points on the ?x?-axis: (a) ?x = 0; (b) ?x = ?R?/2; (c) ?x? = ?R?;? ? = 3?R?. Figure:

Problem 65P CALC A nonuniform, but spherically symmetric, distribution of charge has a charge density ?(r) given as follows: where ?0 = 3Q/?R3 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 ? 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 ? 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.

Problem 66CP A region in space contains a total positive charge ?Q that is distributed spherically such that the volume charge density ? ?? ?r?) is given by Here ? is a positive constant having units of C/m3. (a) Determine ? in terms of ?Q and ?R?. (b) Using Gauss’s law, derive an expression for the magnitude of 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 ? ?R?/2? (d) If an electron with charge ?q?’ = –?e is 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 14.2. If, and only if, the net force on the electron is proportional to its displacement from equilibrium, then the motion is simple harmonic.) (e) What is the period of the motion in part (d)? (f) If the amplitude of the motion described in part (e) is greater than ?R?/2, is the motion still simple harmonic? Why or why not?

Problem 60P Applying Gauss’s Law for Gravitation. ?Using Gauss’s law for gravitation (derived in part (b) of Problem 22.59), show that the following statements are true: (a) For any spherically symmetric mass distribution with total mass ?M?, the acceleration due to gravity outside the distribution is the same as though all the mass