Capacitance per unit length - two cylindrical conductors

[gruba]

Homework Statement

Two cylindrical conductors, of distance between them d and radius a (a<<d), have dielectric layer of relative permitivitty ε_r and thickness a. Calculate capacitance per unit length of this system.

Homework Equations

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

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/ε_r+ln((d-2a)/(2a))+ln((d-a)/(d-2a))/ε_r)

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

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.

 

[mfb]

Voltage integration limits: (a - 2a) + (2a - (d-2a)) + ((d-2a) - (d-a))

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?