A transmission line consisting of two concentric circular cylinders of metal, with radii a, and b, with b > a, is filled with a uniform dielectric material with permittivity ε and permeability μ0. A TEM mode is propagated along the line and the peak value of magnetic field when rho = a is B0. The cylinders are symmetric about the z-axis. Calculate the time-averaged power flow, P, along the line.

P = √( μ / ε ) × πa^2 |Jo|^2 ln {b/a}.

Explanation:

So, we will be making use of the data or parameters given in the question above;

=>" radii a, and b, with b > a, is filled with a uniform dielectric material with permittivity ε and permeability μ0."

=> " A TEM mode is propagated along the line and the peak value of magnetic field when rho = a is B0."

So, we will be making use of the two equations below;

Ë = ( λ/ 2πEP) × P'. (1)..

B' = √ μE × ( λ/ 2πEP) × P' (2).

Where equation (1) and (2) represent Gauss' law and magnetic field equation respectively.

Jo = 1/√ μE × λ/2 × π × a.

When we solve for charge per unit length, we have;

λ = 2 × π × Jo × a × √ μE.

The energy flux,s = E' × J'= √μE× |Jo(Z) |^2 × a^2/b^2 { cos^2 kz - wt + Avg Jo} Z'.

Hence, the time. Average power flux = 1/2× √ μE× |Jo(Z) |^2 × a^2/b^2 × Z'.

Therefore, P = ∫z' . <s> da

P = ∫ ∫ 1/2× √ μE× |Jo(Z) |^2 × a^2/b^2 × Z' pd pd p Θ.

(Take limit on the first at second integration as : 2π,0 and b,a).

P = √( μ / ε ) × πa^2 |Jo|^2 ln {b/a}.