You may have seen a “coaxial cable” connected to a television set. As shown in Figure 22.107, a coaxial cable consists of a
central copper wire of radius r1 surrounded by a hollow copper tube (typically made of braided copper wire) of inner radius
r2 and outer radius r3. Normally the space between the central wire and the outer tube is filled with an insulator, but in this
problem assume for simplicity that this space is filled with air. Assume that no current runs in the cable.
Suppose that a coaxial cable is straight and has a very long length L, and that the central wire carries a charge +Q uniformly
distributed along the wire (so that the charge per unit length is +Q/L everywhere along the wire). Also suppose that the
outer tube carries a charge −Q uniformly distributed along its length L. The cylindrical symmetry of the situation indicates
that the electric field must point radially outward or radially inward. The electric field cannot have any component parallel
to the cable. In this problem, draw mathematical Gaussian cylinders of length d (with d much less than the cable length L)
and appropriate radius r, centered on the central wire.
(a) Use a mathematical Gaussian cylinder located inside the central wire (r <>
radius in the interior of the outer tube (r2 <><>
inner and outer conductors. (Hint: What do you know about the electric field in the interior of the two conductors?
What do you know about the flux on the ends of your Gaussian cylinders?)
(b) Use a mathematical Gaussian cylinder whose radius is in the air gap (r1 <><>
the gap as a function of r. (Don't forget to consider the flux on the ends of your Gaussian cylinder.)
(c) Use a mathematical Gaussian cylinder whose radius is outside the cable (r > r3) to determine the electric field
outside the cable. (Don't forget to consider the flux on the ends of your Gaussian cylinder.)