# Transmission line: leakage current differential equation

• Granger
In summary, the conversation involves a coaxial cable with internal and external conductors of different radii and a material conductivity of ##\sigma_1##. There is an imperfect dielectric with a conductivity of ##\sigma_2## between the conductors. The goal is to determine the evolution of the cable current intensity along the longitudinal coordinate z, caused by leakage currents crossing the dielectric. The approach involves using the fundamental equation div J = 0 and integrating over a section of the cable to obtain the constant ##K##, which must be proven to be equal to the per-unit length transverse conductance of the dielectric medium, ##GU##. The individual seeking help is struggling with this last step of the derivation and is

## Homework Statement

I have a coaxial cable with internal conductor of radius r1 and external conductor of radii r2 and r3. The material of the conductors has a conductivity ##\sigma_1##. Between the conductors there is a imperfect dielectric of conductivity ##\sigma_2##.

Consider the approximation that cable conductors are perfect (that is, cable voltage U is constant along the longitudinal coordinate z). Determine the evolution of the cable current intensity along z, a consequence of the leakage currents crossing the imperfect dielectric.

## Homework Equations

3. The Attempt at a Solution [/B]
So, my attempt was to use the fundamental equation div J = 0. By applying the divergence in cylindrical coordinates, you obtain ##\frac{dJ_y}{dy}=0##. I thought about now integrating over a section of the cable I obtain ##\frac{dI}{dy}=K##. My question now is how do I prove that this constant should be equal to GU, where G is the per-unit length transverse conductance of the dielectric medium.

I'm only having trouble in that last step of the derivation. Can someone help me?
<mentor edit: fix latex, add ##>

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Really can't anyone help me? I've been around this problem for an entire day I'm losing my mind with this, it must be a simple math trick I have no freaking idea about...

Granger said:
, you obtain ##\frac{dJ_y}{dy}=0##. I thought about now integrating over a section of the cable I obtain ##\frac{dI}{dy}=K##. My question now is how do I prove that this constant should be equal to GU, where G is the per-unit length transverse conductance of the dielectric medium.

I'm only having trouble in that last step of the derivation. Can someone help me?

I can't except except to recommend that you make imore readable by using ## (or maybe ) in place of \$ everywhere so your text will look like above