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1. The problem statement, all variables and given/known data

Consider the cylindrical in free space as shown below. The resistor has diameter 2a, length l, is filled with a inhomogeneous material with a z dependent electrical conductivity σ(z) and is capped by 2 thin disks of materials with infinite conductivity. A static voltage φ_{0}is applied to the 2 disks with a battery.

Show that the potential inside the battery is given by

φ (z)= ∫ (C1/σ(z)) dz+ C2

C1,C2 are constants

2. Relevant equations

3. The attempt at a solution

I am trying to use poisson's equation in cylindrical coordinates.

But I am not sure, how to get the right hand side

The potential varies only along Z and is constant along r and φ

Therefore, poissons eqn will reduce to

d^{2}(φ)/dz^{2}= -ρ/ε

I need σ(z) on the right hand side, but I have ρ/ε. How do I arrive at σ(z)?

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# Homework Help: Scalar potential

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