Little question on the potential in a conducting spherical shell

[Lisa...]

The question is:

A conducting spherical shell of inner radius b and outer radius c is concentric with a small metal sphere of radius
a < b. The metal sphere has a positive charge Q. The total charge on the conducting spherical shell is –Q. What is the potential of the metal sphere?

I thought: The potential of the metal sphere is measured when a < r < b and V= V_{metal sphere}+ V_{inner radius of shell}+V_{outer radius of shell}

1) The metal sphere is to be considered as a point charge, therefore $$V_s_p_h_e_r_e= \frac{kQ}{r}$$

2) The inner radius has charge -Q in order to compensate the charge of the metal sphere and if a< r < b then r < the inner radius of the metal sphere (b). Because the rule for a spherical shell is:

$$V= \frac{kQ}{r}$$ if r > R
$$V=\frac{kQ}{R}$$ if r < R

with R the radius of the spherical shell and r the distance from its center

V _{inner radius of shell}= $$\frac{kQ}{b}$$

3) The outer radius has charge 0 because a charge of -Q is induced on the inner sphere, a charge of +Q will be found on the outer spherical shell which already was -Q before the electrostatic equilibrium was established. Therefore Q _{outer radius}= 0 and V _{outer radius}=0

Summerizing:

V_total= $$\frac{kQ}{r} + \frac{kQ}{b} = kQ{\frac{1}{r} - \frac{1}{b}$$

though my textbook tells me V_total= $$kQ{\frac{1}{a} - \frac{1}{b}$$

So who is right and who is wrong?

 

[Doc Al]

How can you have an r in your answer? You are asked for the potential at a particular point, r = a.

(1) What's the potential difference between points r = b & r = a?

(2) What's the potential difference between r = b & r = c? (Don't forget it's a conducting shell.)

(3) What's the potential difference between r = c & r = infinity? (The outer surface has zero charge.)

 

[Lisa...]

Thank you I've figured it out now :)