Capacitance of Coaxial Cable Configuration with Multiple Conductors

[DRussian]

Homework Statement

I can't seem to understand this problem (I do see that the inner cylinder and outer shell will have the same charge, but I can't see what I'm supposed to assume about the middle shell):

A cable consists of three conductors, a solid inner cylinder and two thin cylindrical shields. All three conductors are coaxial, with the solid inner conductor, central shield, and outer shield having diameters 0.55 mm, 3.43 mm, and 5.88 mm, respectively. Assume the space between the conductors is filled with air. The inner cylinder and outer shield are connected at one end of the cable.

What is the capacitance per unit length of this configuration?

Homework Equations

ΔV = ∫E ds
C = Q/(ΔV)

The Attempt at a Solution

I tried finding the resulting change in voltage from the inner cylinder to the first inner shell.

Solving from Gauss's Law, I first got that the magnitude of the electric field of a cylindrical charge distribution having linear charge density λ is (k is Boltzman's constant) E = 2kλ/r.
Integrating from the radius of the center to the first inner shell, I get:

ΔV = ∫E ds = -2kλ*ln(b/a) (where b = radius from center of capacitor to first inner shell and a = radius of inner solid conductor).

This is where I got confused; how would I deal with the second, outer shell into here? How would I account that the E-field is in the opposite direction in the second space (between the two hollow shells) as in the first space (between the solid conductor and first shell)?

 

[haruspex]

I do see that the inner cylinder and outer shell will have the same charge,

No, the cylindrical core and the outermost shield will not have the same charge. What is it that will be the same for them?