Simple Nature - Light and Matter

Problem 29.

together so that their interiors coincide. A capacitor with the plates

close together has a nearly uniform electric field between the plates,

and almost zero field outside; these capacitors don’t have their plates

very close together compared to the dimensions of the plates, but

for the purposes of this problem, assume that they still have ap-

proximately the kind of idealized field pattern shown in the figure.

Each capacitor has an interior volume of 1.00 m^3 , and is charged up

to the point where its internal field is 1.00 V/m.

(a) Calculate the energy stored in the electric field of each capacitor

when they are separate.

√

(b) Calculate the magnitude of the interior field when the two ca-

pacitors are put together in the manner shown. Ignore effects arising

from the redistribution of each capacitor’s charge under the influ-

ence of the other capacitor.

√

(c) Calculate the energy of the put-together configuration. Does as-

sembling them like this release energy, consume energy, or neither?√

28 Find the capacitance of the surface of the earth, assuming

there is an outer spherical “plate” at infinity. (In reality, this outer

plate would just represent some distant part of the universe to which

we carried away some of the earth’s charge in order to charge up the

earth.)

√

29 (a) Show that the field found in example 13 on page 596

reduces toE= 2kλ/Rin the limit ofL→∞.

(b) An infinite strip of widthbhas a surface charge densityσ. Find

the field at a point at a distancezfrom the strip, lying in the plane

perpendicularly bisecting the strip.

√

(c) Show that this expression has the correct behavior in the limit

wherezapproaches zero, and also in the limit ofzb. For the

latter, you’ll need the result of problem 22a, which is given on page

1065.

30 A solid cylinder of radiusband length`is uniformly charged

with a total chargeQ. Find the electric field at a point at the center

of one of the flat ends.

31 Find the potential at the edge of a uniformly charged disk.

(DefineV = 0 to be infinitely far from the disk.)√

.Hint, p. 1032

32 Find the energy stored in a capacitor in terms of its capaci-

tance and the voltage difference across it.

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33 (a) Find the capacitance of two identical capacitors in series.

(b) Based on this, how would you expect the capacitance of a

parallel-plate capacitor to depend on the distance between the plates?

34 (a) Use complex number techniques to rewrite the function

f(t) = 4 sinωt+ 3 cosωtin the formAsin(ωt+δ).

√

662 Chapter 10 Fields