Problem 23.
locationsx=±b,y=±bof the four points on the surfaces that
are closest to the central axis.) There is no obvious, pedestrian way
to determine the field or potential in the central vacuum region,
but there’s a trick that works: with a little mathematical insight,
we see that the potentialV =Vob−^2 xyis consistent with all the
given information. (Mathematicians could prove that this solution
was unique, but a physicist knows it on physical grounds: if there
were two different solutions, there would be no physical way for the
system to decide which one to do!)
(a) Use the techniques of subsection 10.2.2 to find the field in the
vacuum region.
(b) Sketch the field as a “sea of arrows.”√21 (a) A certain region of three-dimensional space has a potential
that varies asV =br^2 , whereris the distance from the origin. Use
the techniques of subsection 10.2.2 to find the field.√(b) Write down another potential that gives exactly the same field.22 (a) Example 13 on page 596 gives the field of a charged rod in
its midplane. Starting from this result, take the limit as the length
of the rod approaches infinity. Note thatλis not changing, so asL
gets bigger, the total chargeQincreases. .Answer, p. 1065
(b) In the text, I have shown (by several different methods) that the
field of an infinite, uniformly charged plane is 2πkσ. Now you’re
going to rederive the same result by a different method. Suppose
that it is thex−yplane that is charged, and we want to find the
field at the point (0, 0,z). (Since the plane is infinite, there is no
loss of generality in assumingx= 0 andy= 0.) Imagine that we
slice the plane into an infinite number of straight strips parallel to
theyaxis. Each strip has infinitesimal width dx, and extends from
xtox+ dx. The contribution any one of these strips to the field
at our point has a magnitude which can be found from part a. By
vector addition, prove the desired result for the field of the plane of
charge.
23 Consider the electric field created by a uniformly charged
cylindrical surface that extends to infinity in one direction.
(a) Show that the field at the center of the cylinder’s mouth is 2πkσ,
which happens to be the same as the field of an infiniteflatsheet of
charge!
(b) This expression is independent of the radius of the cylinder.
Explain why this should be so. For example, what would happen if
you doubled the cylinder’s radius?
24 In an electrical storm, the cloud and the ground act like a
parallel-plate capacitor, which typically charges up due to frictional
electricity in collisions of ice particles in the cold upper atmosphere.
Lightning occurs when the magnitude of the electric field builds up
to a critical value,Ec, at which air is ionized.660 Chapter 10 Fields