An infinitely long coaxial cable consists of an inner conduc

An infinitely long coaxial cable consists of an inner conductor of radius a and outer conductor of inner radius b and thickness. The non-magnetic conductors arc having their axes along z-axis. The space between the conductors is filled with a non-magnetic dielectric with permeability epsilon = epsilon_0epsilon_r and conductivity sigma. Make a good sketch of the cross sectional diagram of the coaxial cable showing the dimensions and location of the dielectric space. A. The inner conductor maintains a potential V_o and the outer conductor is grounded. Using Laplace\'s equation, calculate the potential distribution V(rho) in the dielectric regiona

Solution

The coax has an outer diameter b, and an inner diameter a. The space between the conductors is filled with dielectric material of permittivity . Say a voltage V0 is placed across the conductors, such that the electric potential of the outer conductor is zero, and the electric potential of the inner conductor is V0.

The potential difference between the inner and outer conductor is therefore V0 – 0 = V0 volts.

Poisson’s equation reduces to Laplace’s equation: ( ) 2 V r 0 = This particular problem (i.e., coaxial line) is directly solvable because the structure is cylindrically symmetric. Rotating the coax around the z-axis (i.e., in the ˆa direction) does not change the geometry at all. As a result, we know that the electric potential field is a function of only ! I.E.,: V (r ) =V ( ) This make the problem much easier. Laplace’s equation becomes: