# Capacitance per unit length - two cylindrical conductors

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1. Aug 14, 2015

### gruba

1. The problem statement, all variables and given/known data
Two cylindrical conductors, of distance between them d and radius a (a<<d), have dielectric layer of relative permitivitty εr and thickness a. Calculate capacitance per unit length of this system.

2. Relevant equations
Capacitance per unit length, C'=Q'/U
Gauss law, cylindrical symmetry, E=Q'/(2πεr2)

3. The attempt at a solution
I have started with the equation C'=Q'/U.
Voltage integration limits: (a - 2a) + (2a - (d-2a)) + ((d-2a) - (d-a))
After calculating the partial integrals, voltage U is:
U=Q'(ln2/εr+ln((d-2a)/(2a))+ln((d-a)/(d-2a))/εr)

Applying voltage in the expression for capacitance per unit length gives
C'=2πε0/(ln2/εr+ln((d-2a)/(2a))+ln((d-a)/(d-2a))/εr)

In my books solution, in the numerator there is πε0 for C'.

Are my limits of integration for voltage correct? Maybe it is a mistake in books solution.
Thanks for replies.

2. Aug 14, 2015

### Staff: Mentor

I don't understand that part.

As far as I understand the problem statement, you have two separate "cables"? Then the problem does not have a cylinder symmetry. Otherwise: do you have a sketch of the layout?