Minimum Thickness of Dielectric

[iblackford]

1. Basically, we have coaxial cable with an inner wire of radius a,
perfectly conducting, straight and infinitely long, with radius a, surrounded by an outer hollow cylinder, also made
from perfectly conducting material, having inner radius b and outer radius c. The hollow cylinder functions as a
return wire, and is coaxial with the inner wire. If the space between the wires is air or vacuum, the electric field
between the inner and return wires is shown to be

E = (Q/L)*(1/2*pi*e*p)

where Q/L is the charge per unit length and  the cylindrical distance between the wires.
Suppose the ~E field cannot be allowed to reach values above the value it reaches in the middle of the arrangement,
where p = (a+b)/2
2 . This can be achieved by shielding the inner cable with a dielectric of permittivity e.
a) What is the minimum value of e and the minimum thickness of dielectric coating that will ensure that the field
is everywhere below its maximal value?
b) What is the capacitance per unit length of the resulting arrangement.

I'm not sure how to do the first part of the problem, I think I would sub in my p = a+b/2 into my first equation, but I'm not sure what to do from there. Any help would be highly appreciated.

 

[Andrew Mason]

1. Basically, we have coaxial cable with an inner wire of radius a,
perfectly conducting, straight and infinitely long, with radius a, surrounded by an outer hollow cylinder, also made
from perfectly conducting material, having inner radius b and outer radius c. The hollow cylinder functions as a
return wire, and is coaxial with the inner wire. If the space between the wires is air or vacuum, the electric field
between the inner and return wires is shown to be
E = (Q/L)*(1/2*pi*e*p)
where Q/L is the charge per unit length and p the cylindrical distance between the wires.

Just use Gauss' law for a Gaussian cylinder of radius p:
\int E\cdot dA = \frac{q}{\epsilon_0}

Suppose the ~E field cannot be allowed to reach values above the value it reaches in the middle of the arrangement,
where p = (a+b)/2
2 . This can be achieved by shielding the inner cable with a dielectric of permittivity e.
a) What is the minimum value of e and the minimum thickness of dielectric coating that will ensure that the field
is everywhere below its maximal value?

The maximum field is the E determined in 1 where p = (a+b)/2. A dielectric of thickness t will reduce this by a factor \frac{1}{\epsilon}

from p=a to p = a+t. So work out the expression for E at p=a+t. The value of E from p=a+t decreases as 1/p.

AM