Spherical capacitor-RC system- determine steady state charges

[palaphys]

Homework Statement

Three uncharged metallic balls of radii a, b and a respectively are connected to terminals A, B and C with the help of long thin conductors as shown in the circuit. Find charges established on each of the balls, when a steady state is reached after the switch is closed. Consider the balls to be at great distances from each other as well from the circuit and neglect internal resistance of the battery.

Relevant Equations

Q=CV ,V=iR

This is the diagram given for the problem.
Now I was able to identify, that the fact that the capacitance of a spherical capacitor, with one plate and the other at an infinite distance, is somehow to be used in this problem, i.e $C= 4\pi\epsilon_0R$

IF I can replace all the spherical capacitors with parallel plate capacitors with the same capacitance, I can solve the problem very easily. Though this seems intuitive, it yields the incorrect result.
Looking forward on how to approach this problem.

 

[haruspex]

Though this seems intuitive, it yields the incorrect result.

Please post your result and the official result if known.

 

[kuruman]

Here is your approach. Look at the figure on the right. Each colored region of space encloses equipotential conductors in the steady state.

Start by finding expressions for the potential difference between each sphere and infinity. Then find expressions for the potential differences $V_{AB}$ and $V_{BC}.$