Here is a fourth way of computing the energy of a

Problem 58P A point charge q is at the center of an uncharged spherical conducting shell, of inner radius a and outer radius b. Question: How much work would it take to move the charge out to infinity (through a tiny hole drilled in the shell)? [Answer:

Problem 1P (a) Twelve equal charges, q, are situated at the corners of a regular 12-sided polygon (for instance, one on each numeral of a clock face). What is the net force on a test charge Q at the center? (b) Suppose one of the 12 q’s is removed (the one at “6 o’clock”). What is the force on Q? Explain your reasoning carefully. (c) Now 13 equal charges, q, are placed at the corners of a regular 13-sided polygon. What is the force on a test charge Q at the center? (d) If one of the 13 q’s is removed, what is the force on Q? Explain your reasoning.

Find the electric field a distance z above one end of a straight line segment of length L (Fig. 2.7) that carries a uniform line charge \(\lambda\). Check that your formula is consistent with what you would expect for the case z ? L.

Problem 2P Find the electric field (magnitude and direction) a distance z above the midpoint between equal and opposite charges (±q), a distance d apart (same as Example 2.1, except that the charge at x = +d/2 is ?q). Reference Example 2.1

Find the electric field a distance z above the center of a square loop (side a) carrying uniform line charge ? (Fig. 2.8). [Hint: Use the result of Ex. 2.2.]

Problem 6P Find the electric field a distance z above the center of a flat circular disk of radius R (Fig. 2.10) that carries a uniform surface charge ?.What does your formula give in the limit R??? Also check the case z ?R. Reference figure 2.10

Problem 5P Find the electric field a distance z above the center of a circular loop of radius r (Fig. 2.9) that carries a uniform line charge ?. Reference figure 2.9

Problem 8P Use your result in Prob. 2.7 to find the field inside and outside a solid sphere of radius R that carries a uniform volume charge density ?. Express your answers in terms of the total charge of the sphere, q. Draw a graph of |E| as a function of the distance from the center. Reference: Prob. 2.7. Find the electric field a distance z from the center of a spherical surface of radius R (Fig. 2.11) that carries a uniform charge density ?. Treat the case z < R (inside) as well as z > R (outside). Express your answers in terms of the total charge q on the sphere. [Hint: Use the law of cosines to write r in terms of R and ?. Be sure to take the positive square root: if R > z, but it’s (z ? R) if R < z.] Reference: Fig.2.11.

Problem 7P Find the electric field a distance z from the center of a spherical surface of radius R (Fig. 2.11) that carries a uniform charge density ?. Treat the case z < R (inside) as well as z > R (outside). Express your answers in terms of the total charge q on the sphere. [Hint: Use the law of cosines to write r in terms of R and ?. Be sure to take the positive square root: but it’s (z ? R) if R < z.] Reference figure 2.11

Problem 10P A charge q sits at the back corner of a cube, as shown in Fig. 2.17. What is the flux of E through the shaded side? Reference figure 2.17

Problem 11P Use Gauss’s law to find the electric field inside and outside a spherical shell of radius R that carries a uniform surface charge density ?. Compare your answer to Prob. 2.7. Reference:Prob.2.7 Find the electric field a distance z from the center of a spherical surface of radius R (Fig. 2.11) that carries a uniform charge density ?. Treat the case z < R (inside) as well as z > R (outside). Express your answers in terms of the total charge q on the sphere. [Hint: Use the law of cosines to write r in terms of R and ?. Be sure to take the positive square root:

Problem 9P Suppose the electric field in some region is found to be in spherical coordinates (k is some constant). (a) Find the charge density ?. (b) Find the total charge contained in a sphere of radius R, centered at the origin. (Do it two different ways.)

Problem 12P Use Gauss’s law to find the electric field inside a uniformly charged solid sphere (charge density ?). Compare your answer to Prob. 2.8. Reference: Prob. 2.8. Use your result in Prob. 2.7 to find the field inside and outside a solid sphere of radius R that carries a uniform volume charge density ?. Express your answers in terms of the total charge of the sphere, q. Draw a graph of |E| as a Function of the distance from the center. Reference: Prob.2.7. Find the electric field a distance z from the center of a spherical surface of radius R (Fig. 2.11) that carries a uniform charge density ?. Treat the case z < R (inside) as well as z > R (outside). Express your answers in terms of the total charge q on the sphere. [Hint: Use the law of cosines to write r in terms of R and ?. Be sure to take the positive square root: Reference: Fig.2.11.

Problem 14P Find the electric field inside a sphere that carries a charge density proportional to the distance from the origin, ? = kr, for some constant k. [Hint: This charge density is not uniform, and you must integrate to get the enclosed charge.]

Problem 13P Find the electric field a distance s from an infinitely long straight wire that carries a uniform line charge ?. Compare Eq. 2.9. Reference equation 2.9

Problem 15P A thick spherical shell carries charge density (Fig. 2.25). Find the electric field in the three regions: (i) r < a, (ii) a < r < b, (iii) r > b. Plot |E| as a function of r , for the case b = 2a. Reference figure 2.25

Problem 20P One of these is an impossible electrostatic field. Which one? Here k is a constant with the appropriate units. For the possible one, find the potential, using the origin as your reference point. Check your answer by computing ?V. [Hint: You must select a specific path to integrate along. It doesn’t matter what path you choose, since the answer is path-independent, but you simply cannot integrate unless you have a definite path in mind.]

Problem 16P long coaxial cable (Fig. 2.26) carries a uniform volume charge density ? on the inner cylinder (radius a), and a uniform surface charge density on the outer cylindrical shell (radius b). This surface charge is negative and is of just the right magnitude that the cable as a whole is electrically neutral. Find the electric field in each of the three regions: (i) inside the inner cylinder (s < a), (ii) between the cylinders (a < s < b), (iii) outside the cable (s > b). Plot |E| as a function of s. Reference figure 2.26

Problem 18P Two spheres, each of radius R and carrying uniform volume charge densities +? and ??, respectively, are placed so that they partially overlap (Fig. 2.28). Call the vector from the positive center to the negative center d. Show that the field in the region of overlap is constant, and find its value. [Hint: Use the answer to Prob. 2.12.] Reference: Fig. 2.28. Reference: Prob. 2.12. Use Gauss’s law to find the electric field inside a uniformly charged solid sphere (charge density ?). Compare your answer to Prob. 2.8. Reference: Prob.2.8. Use your result in Prob. 2.7 to find the field inside and outside a solid sphere of radius R that carries a uniform volume charge density ?. Express your answers in terms of the total charge of the sphere, q. Draw a graph of |E| as a function of the distance from the center. Reference: Prob.2.7 Find the electric field a distance z from the center of a spherical surface of radius R (Fig. 2.11) that carries a uniform charge density ?. Treat the case z < R (inside) as well as z > R (outside). Express your answers in terms of the total charge q on the sphere. [Hint: Use the law of cosines to write r in terms of R and ?. Be sure to take the positive square root: if R > z, but it’s (z ? R) if R < z.] Reference: Fig.2.11.

Problem 17P An infinite plane slab, of thickness 2d, carries a uniform volume charge density ? (Fig. 2.27). Find the electric field, as a function of y, where y = 0 at the center. Plot E versus y, calling E positive when it points in the +y direction and negative when it points in the ?y direction. Reference figure 2.27

Problem 21P Find the potential inside and outside a uniformly charged solid sphere whose radius is R and whose total charge is q. Use infinity as your reference point. Compute the gradient of V in each region, and check that it yields the correct field. Sketch V(r ).

Problem 22P Find the potential a distance s from an infinitely long straight wire that carries a uniform line charge ?. Compute the gradient of your potential, and check that it yields the correct field.

Problem 24P For the configuration of Prob. 2.16, find the potential difference between a point on the axis and a point on the outer cylinder. Note that it is not necessary to commit yourself to a particular reference point, if you use Eq. 2.22. Reference equation 2.22 Reference prob 2.16 long coaxial cable (Fig. 2.26) carries a uniform volume charge density ñ on the inner cylinder (radius a), and a uniform surface charge density on the outer cylindrical shell (radius b). This surface charge is negative and is of just the right magnitude that the cable as a whole is electrically neutral. Find the electric field in each of the three regions: (i) inside the inner cylinder (s < a), (ii) between the cylinders (a < s < b), (iii) outside the cable (s > b). Plot |E| as a function of s. Reference figure 2.26

Problem 19P Calculate ? × E directly from Eq. 2.8, by the method of Sect. 2.2.2. Refer to Prob. 1.63 if you get stuck. Reference figure 2.8 Reference Prob. 1.63 (a) Find the divergence of the function First compute it directly, as in Eq. 1.84. Test your result using the divergence theorem, as in Eq. 1.85. Is there a delta function at the origin, as there was for ?What is the general formula for the divergence of unless n = ?2, in which case it is 4??3(r); for n < ? , the divergence is ill-defined at the origin.] (b) Find the curl of Test your conclusion using Prob. 1.61b. [Answer: equation 1.85

Problem 23P For the charge configuration of Prob. 2.15, find the potential at the center, using infinity as your reference point. Reference prob 2.15 A thick spherical shell carries charge density (Fig. 2.25). Find the electric field in the three regions: (i) r < a, (ii) a < r < b, (iii) r > b. Plot |E| as a function of r , for the case b = 2a. Reference figure 2.25

Problem 26P A conical surface (an empty ice-cream cone) carries a uniform surface charge ?. The height of the cone is h, as is the radius of the top. Find the potential difference between points a (the vertex) and b (the center of the top).

Problem 27P Find the potential on the axis of a uniformly charged solid cylinder, a distance z from the center. The length of the cylinder is L, its radius is R, and the charge density is ?. Use your result to calculate the electric field at this point. (Assume that z > L/2.)

Problem 29P Check that Eq. 2.29 satisfies Poisson’s equation, by applying the Laplacian and using Eq. 1.102. Reference equation Eq. 2.29 Reference equation Eq. 1.102.

Problem 31P (a) Three charges are situated at the corners of a square (side a), as shown in Fig. 2.41. How much work does it take to bring in another charge, +q, from far away and place it in the fourth corner? (b) How much work does it take to assemble the whole configuration of four charges? Figure 2.41

Problem 25P Using Eqs. 2.27 and 2.30, find the potential at a distance z above the center of the charge distributions in Fig. 2.34. In each case, compute E = ??V, and compare your answers with Ex. 2.1, Ex. 2.2, and Prob. 2.6, respectively. Suppose that we changed the right-hand charge in Fig. 2.34a to ?q; what then is the potential at P? What field does that suggest? Compare your answer to Prob. 2.2, and explain carefully any discrepancy. Eqs. 2.27 Eqs. 2.30 Reference example 2.1 Find the electric field a distance z above the midpoint between two equal charges (q), a distance d apart (Fig. 2.4a). Reference example 2.1 Find the electric field a distance z above the midpoint of a straight line segment of length 2L that carries a uniform line charge ? (Fig. 2.6). Reference prob.2.6 Find the electric field a distance z above the center of a flat circular disk of radius R (Fig. 2.10) that carries a uniform surface charge ?.What does your formula give in the limit R??? Also check the case z ?R. Reference figure 2.10 Reference prob.2.2 Find the electric field (magnitude and direction) a distance z above the midpoint between equal and opposite charges (±q), a distance d apart (same as Example 2.1, except that the charge at x = +d/2 is ?q). Reference example 2.1 Find the electric field a distance z above the midpoint between two equal charges (q), a distance d apart (Fig. 2.4a).

Problem 28P Use Eq. 2.29 to calculate the potential inside a uniformly charged solid sphere of radius R and total charge q. Compare your answer to Prob. 2.21. Reference equation 2.29 Reference prob 2.21 Find the potential inside and outside a uniformly charged solid sphere whose radius is R and whose total charge is q. Use infinity as your reference point. Compute the gradient of V in each region, and check that it yields the correct field. Sketch V(r ).

Consider an infinite chain of point charges, ±q (with alternating signs), strung out along the x axis, each a distance a from its nearest neighbors. Find the work per particle required to assemble this system. [Partial Answer: \(-a q^{2} /\left(4 \pi_{0} a\right)\), for some dimensionless number ?; your problem is to determine ?. It is known as the Madelung constant. Calculating the Madelung constant for 2- and 3-dimensional arrays is much more subtle and difficult.]

Problem 30P (a) Check that the results of Exs. 2.5 and 2.6, and Prob. 2.11, are consistent with Eq. 2.33. (b) Use Gauss’s law to find the field inside and outside a long hollow cylindrical tube, which carries a uniform surface charge ?. Check that your result is consistent with Eq. 2.33. (c) Check that the result of Ex. 2.8 is consistent with boundary conditions 2.34 and 2.36. Reference example 2.5 An infinite plane carries a uniform surface charge ?. Find its electric field. Reference example 2.6 Two infinite parallel planes carry equal but opposite uniform charge densities ±? (Fig. 2.23). Find the field in each of the three regions: (i) to the left of both, (ii) between them, (iii) to the right of both. Reference prob 2.11 Use Gauss’s law to find the electric field inside and outside a spherical shell of radius R that carries a uniform surface charge density ?. Compare your answer to Prob. 2.7. Reference Prob. 2.7 Reference example 2.8 Find the potential of a uniformly charged spherical shell of radius R (Fig. 2.33).

Problem 34P Find the energy stored in a uniformly charged solid sphere of radius R and charge q. Do it three different ways: (a) Use Eq. 2.43. You found the potential in Prob. 2.21. (b) Use Eq. 2.45. Don’t forget to integrate over all space. (c) Use Eq. 2.44. Take a spherical volume of radius a. What happens as a??? Reference Eq. 2.43 Reference Eq. 2.45. Reference Eq. 2.44 Reference Prob. 2.21 Find the potential inside and outside a uniformly charged solid sphere whose radius is R and whose total charge is q. Use infinity as your reference point. Compute the gradient of V in each region, and check that it yields the correct field. Sketch V(r ).

Problem 39P Two spherical cavities, of radii a and b, are hollowed out from the interior of a (neutral) conducting sphere of radius R (Fig. 2.49). At the center of each cavity a point charge is placed—call these charges qa and qb. (a) Find the surface charge densities ?a , ?b, and ?R. (b) What is the field outside the conductor? (c) What is the field within each cavity? (d) What is the force on qa and qb? (e) Which of these answers would change if a third charge, qc, were brought near the conductor? Reference figure 2.49

Problem 35P Here is a fourth way of computing the energy of a uniformly charged solid sphere: Assemble it like a snowball, layer by layer, each time bringing in an infinitesimal charge dq from far away and smearing it uniformly over the surface, thereby increasing the radius. How much work dW does it take to build up the radius by an amount dr? Integrate this to find the work necessary to create the entire sphere of radius R and total charge q.

Problem 36P Consider two concentric spherical shells, of radii a and b. Suppose the inner one carries a charge q, and the outer one a charge ?q (both of them uniformly distributed over the surface). Calculate the energy of this configuration, (a) using Eq. 2.45, and (b) using Eq. 2.47 and the results of Ex. 2.9. Reference equation 2.45 Reference equation 2.47 Reference example 2.9 Find the energy of a uniformly charged spherical shell of total charge q and radius R.

Problem 38P A metal sphere of radius R, carrying charge q, is surrounded by a thick concentric metal shell (inner radius a, outer radius b, as in Fig. 2.48). The shell carries no net charge. (a) Find the surface charge density ? at R, at a, and at b. (b) Find the potential at the center, using infinity as the reference point. (c) Now the outer surface is touched to a grounding wire, which drains off charge and lowers its potential to zero (same as at infinity). How do your answers to (a) and (b) change? Reference figure 2.48

Problem 40P (a) A point charge q is inside a cavity in an uncharged conductor (Fig. 2.45). Is the force on q necessarily zero?11 (b) Is the force between a point charge and a nearby uncharged conductor always attractive?12 Reference figure 2.45

Problem 41P Two large metal plates (each of area A) are held a small distance d apart. Suppose we put a charge Q on each plate; what is the electrostatic pressure on the plates?

Problem 42P A metal sphere of radius R carries a total charge Q.What is the force of repulsion between the “northern” hemisphere and the “southern” hemisphere?

Problem 43P Find the capacitance per unit length of two coaxial metal cylindrical tubes, of radii a and b (Fig. 2.53).

Problem 44P Suppose the plates of a parallel-plate capacitor move closer together by an infinitesimal distance ?, as a result of their mutual attraction. (a) Use Eq. 2.52 to express the work done by electrostatic forces, in terms of the field E, and the area of the plates, A. (b) Use Eq. 2.46 to express the energy lost by the field in this process. (This problem is supposed to be easy, but it contains the embryo of an alternative derivation of Eq. 2.52, using conservation of energy.) Reference equation 2.52 Reference equation 2.46 Reference equation

Problem 46P If the electric field in some region is given (in spherical coordinates) by the expression for some constant k, what is the charge density? [Answer:

Problem 45P Find the electric field at a height z above the center of a square sheet (side a) carrying a uniform surface charge ?. Check your result for the limiting cases a??and z ?a.

Problem 49P A sphere of radius R carries a charge density ?(r ) = kr (where k is a constant). Find the energy of the configuration. Check your answer by calculating it in at least two different ways. [Answer:

An inverted hemispherical bowl of radius R carries a uniform surface charge density \(\sigma\). Find the potential difference between the “north pole” and the center. [Answer: \(\left(R \sigma / 2 \epsilon_{0}\right)(\sqrt{2}-1)\)]

Problem 47P Find the net force that the southern hemisphere of a uniformly charged solid sphere exerts on the northern hemisphere. Express your answer in terms of the radius R and the total charge Q. [Answer:

Problem 50P The electric potential of some configuration is given by the expression where A and ? are constants. Find the electric field E(r), the charge density ?(r ), and the total charge Q. [Answer :

Problem 52P Two infinitely long wires running parallel to the x axis carry uniform charge densities +? and ??(Fig. 2.54). (a) Find the potential at any point (x, y, z), using the origin as your reference. (b) Show that the equipotential surfaces are circular cylinders, and locate the axis and radius of the cylinder corresponding to a given potential V0.

Problem 51P Find the potential on the rim of a uniformly charged disk (radius R, charge density ?). [Hint: First show that V = k(? R/??0), for some dimensionless number k, which you can express as an integral. Then evaluate k analytically, if you can, or by computer.]

Problem 56P All of electrostatics follows from the 1/r 2 character of Coulomb’s law, together with the principle of superposition. An analogous theory can therefore be constructed for Newton’s law of universal gravitation. What is the gravitational energy of a sphere, of mass M and radius R, assuming the density is uniform? Use your result to estimate the gravitational energy of the sun (look up the relevant numbers). Note that the energy is negative—masses attract, whereas (like) electric charges repel. As the matter “falls in,” to create the sun, its energy is converted into other forms (typically thermal), and it is subsequently released in the form of radiation. The sun radiates at a rate of 3.86 × 1026 W; if all this came from gravitational energy, how long would the sun last? [The sun is in fact much older than that, so evidently this is not the source of its power.14]

Problem 53P In a vacuum diode, electrons are “boiled” off a hot cathode, at potential zero, and accelerated across a gap to the anode, which is held at positive potential V0. The cloud of moving electrons within the gap (called space charge) quickly builds up to the point where it reduces the field at the surface of the cathode to zero. From then on, a steady current I flows between the plates. Suppose the plates are large relative to the separation (A ? d2 in Fig. 2.55), so that edge effects can be neglected. Then V, ?, and v (the speed of the electrons) are all functions of x alone. (a) Write Poisson’s equation for the region between the plates. (b) Assuming the electrons start from rest at the cathode, what is their speed at point x, where the potential is V(x)? (c) In the steady state, I is independent of x. What, then, is the relation between ? and v? (d) Use these three results to obtain a differential equation for V, by eliminating ? and v. (e) Solve this equation for V as a function of x, V0, and d. Plot V(x), and compare it to the potential without space-charge. Also, find ? and v as functions of x. (f) Show that and find the constant K. (Equation 2.56 is called the Child-Langmuir law. It holds for other geometries as well, whenever space-charge limits the current. Notice that the space-charge limited diode is nonlinear—it does not obey Ohm’s law.)

Problem 55P Suppose an electric field E(x, y, z) has the form E x = ax, Ey = 0, Ez = 0 where a is a constant. What is the charge density? How do you account for the fact that the field points in a particular direction, when the charge density is uniform? [This is a more subtle problem than it looks, and worthy of careful thought.]

Problem 57P We know ! that the charge on a conductor goes to the surface, but just how it distributes itself there is not easy to determine. One famous example in which the surface charge density can be calculated explicitly is the ellipsoid: In this case 15 where Q is the total charge. By choosing appropriate values for a, b, and c, obtain (from Eq. 2.57): (a) the net (both sides) surface charge density ?(r ) on a circular disk of radius R; (b) the net surface charge density ?(x) on an infinite conducting “ribbon” in the xy plane, which straddles the y axis from x = ?a to x = a (let be the total charge per unit length of ribbon); (c) the net charge per unit length ?(x) on a conducting “needle,” running from x = ?a to x = a. In each case, sketch the graph of your result.