A transmission line consists of a cylindrical conductor of radius r at a distance d in air from a conducting plane (r >>d).
Derive the capacitance per unit length C and the inductance per unit length L and
check that 1/sqrt(LC) = c.
The Attempt at a Solution
I thought I could just give the cyclinder a charge -Q and the plane a charge +Q and superimpose the fields. So that along the line through the centre of the cylinder and normal to the plane, the field is:
E = -Q/(2.pi.r.l.e0) -Q/(2.l.m.e0)
Where l is the length of the cylinder and l, m are the dimensions of the plane.
Then integrating to find the potential difference and dividing by Q and multiplying by l gives:
1/C = (1/2.pi.r.e0)*ln[(d-r)/r] + (1/2e0)*(d-r)/m
Then the B due to the cylinder would be vI/2.pi.r where I am using v as permeability of free space. And on the normal through the centre line, it would be perpendicular to the line.
I think that on the line, the field from all the elements on the plane would superimpose to produce a field perpendicular to the line, which Ampere would then give as vI/m
B = vI/2.pi.r + vI/m
Then flux is the integral of that, and dividing by I and l gives L:
L = (v/2.pi)*ln[(d-r)/r] + v(d-r)/m
But this doesn't seem right.
Any help? Thanks.