Problem Set 38.022 ESG

DourmashkinOctober 1, 2010

Problem 1Chapter 2, Exercise 27. Use the result stated in Eq. 24:

φ =

∫ π/2

−π/22σacosθ dθ = 4σa

to calculate the energy stored in the electric field of the charged disk described in Section 2.6. (Hint:Consider the work done in building the disk of charge out from zero radius to radius a by adding successiverings of width dr. Express the total energy in terms of radius a and total charge Q = σπa2.)Ans. 8Q2/3πa

Problem 2Chapter 2, Exercise 31. A flat nonconducting sheet lies in the xy plane. The only charges in the systemare on this sheet. In the half-space above the sheet, z > 0, the potential is φ = φ0e

−kz cos(kx), where φ0and k are constants.

(a) Verify that φ satisfies Laplace’s equation in the space above the sheet.

(b) What do the electric field lines look like?

(c) Describe the charge distribution on the sheet.

Problem 3Chapter 3, Exercise 1. A spherical conductor A contains two spherical cavities. The total charge on theconductor itself is zero. However, there is a point charge qb at the center of one cavity and qc at thecenter of the other. A considerable distance r away is another charge qd. What force acts on each of thefour objects A, qb, qc, qd? Which answers, if any, are only approximate, and depend on r being relativelylarge?

Problem 4Chapter 3, Exercise 3. In the field of the point charge over the plane (Fig. 3.9), if you follow a field linethat starts out from the point charge in a horizontal direction, that is, parallel to the plane, where doesit meet the surface of the conductor? (You’ll need Gauss’s Law and a simple integration.)

Problem 5Chapter 3, Exercise 5. A charge Q is located h cm above a conducting plane, just as in Fig. 3.8a. Askedto predict the amount of work that would have to be done to move this charge out to infinite distancefrom the plane, one student says that it is the same as the work required to separate to infinite distancestwo charges Q and −Q which are initially 2h cm apart, hence W = Q2/2h. Another student calculatesthe force that acts on the charge as it is being moved and integrates F dx, but gets a different answer.What did the second student get, and who is right?

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Problem Set 38.022 ESG

DourmashkinOctober 1, 2010

Problem 6Chapter 3, Exercise 8. Three conducting plates are placed parallel to one another as shown. (The outerplates are 5 cm and 8 cm from the inner plate.) The outer plates are connected by a wire. The inner plateis isolated and carries a charge amounting to 10 esu per square centimeter of plate. In what proportionmust this charge divide itself into a surface charge σ1 on one face of the inner plate and a surface chargeσ2 on the other side of the same plate?

Problem 7Chapter 3, Exercise 10. What is the capacitance C of a capacitor that consists of two concentric sphericalmetal shells? The inner radius of the outer shell is a; the outer radius of the inner shell is b. Check yourresult by considering the limiting case with the gap between the conductors, a− b, much smaller than b.In that limit the formula for capacitance of the flat parallel-plate capacitor ought to be applicable.

Problem 8Chapter 3, Exercise 19. In the apparatus shown, ions are accelerated through a potential difference V0and then enter the space between the semicylindrical electrodes A and B. Show that an ion will followthe semicircular path of radius r0 if the potentials of the outer and inner electrodes are maintained,respectively, at 2V0 ln(b/r0) and 2V0 ln(a/r0). (The cylindrical electrodes A and B are assumed to belong, in the direction perpendicular to the diagram, compared with the space between them.)

Problem 9Chapter 3, Exercise 23.

(a) Find the capacitance of a capacitor that consists of two coaxial cylinders, of radii a and b, and lengthL. Assume L >> b − a, so that end corrections may be neglected. Check your results by showingthat, if the gap between the cylinders, b − a, is very small compared with the radius, your formulareduces to one that could have been obtained by using the formula for the parallel-plate capacitor.

(b) A cylinder of 2.00-inch outer diameter hangs, with its axis vertical, from one arm of a beam balance.The lower portion of the hanging cylinder is surrounded by a stationary cylinder, coaxial, with innerdiameter 3.00 inches. Calculate the magnitude of the force tending to pull the hanging cylinderfurther down when the potential difference between the two cylinders is 5 kilovolts.

Problem 10Charged Slab and P-N junction. Consider a slab of insulating material which is infinitely large in two ofits three dimensions, and has a thickness d in the third dimension. An edge view of the slab is shown inthe figure. The slab has a uniform positive charge density ρ.

(a) Calculate the electric field everywhere, both inside and outside the slab.

(b) When two slabs of N-type and P-type semiconductors are put in contact, the relative affinities of thematerials cause electrons to migrate out of the N-type material across the junction to the P- typematerial. This leaves behind a volume in the N-type material that is positively charged and creates

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Problem Set 38.022 ESG

DourmashkinOctober 1, 2010

a negatively charged volume in the P-type material.Let us model this as two infinite slabs of charge, both of thickness d with the junction lying on theplane x = 0. The N-type material lies in the range 0 < x < d and has uniform charge density +ρ0.The adjacent P-type material lies in the range −d < x < 0 and has uniform charge density −ρ0.Thus:

ρ(x) =

+ρ0 0 < x < d

−ρ0 −d < x < 0

0 |x| > d

(i) Find the electric field everywhere.

(ii) Find the potential difference between the points P1 and P2. The point P1 is located on a planeparallel to the slab a distance x1 > d from the center of the slab. The point P2 is located on aplane parallel to the slab a distance x2 < −d from the center of the slab.

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