Electric Potential: Electric charge surrounded by two spherical shells

[Bryon]

Homework Statement

A point charge q = -8 μC is surrounded by two thick, conducting spherical shells of inner and outer radii
a1 = 0.3 m, a2 = 0.4 m, a3 = 0.7 m, and a4 = 0.8 m respectively. The inner shell is uncharged; the outer
shell has a net charge Q = -10 μC. At this point in the problem, the potential at infinity is unspecified.

(c) V(a2) - V(a3) = ____________________ V

Homework Equations

E = kQ/r^2

Gauss’ Law: ϕ= ʃE·dA

V0 = - ʃE·dl

HELP: Identify the equivalent point-charge that gives the same electric fields everywhere in the region a2 < r < a3 as the specified charge distribution.
HELP: Use the electric potential function that corresponds to the equivalent point-charge problem.

The Attempt at a Solution

Electric field of the sphere at a2:
E = k(-8uC)/(.4^2) = -449500 N/C

Im stuck. I figured out the first two problems but I am not sure how to relate the electric field to the potential, and I am not sure how to get the equivalent point charge. I assume I need to use Gauss' law to find the equivalent point charge but my queastion is do I just use the difference of a2 and a3 for the radius?

 

[SammyS]

Assuming that the spherical shells are centered at the origin:

What is the electric field a distance r from the origin, where a₂ < r < a₃ ?

For a₂ < r < a₃:

$$V(r)-V(a_2)=-\int_{a_2}^{\,r}E(r)\,dr\ .$$

 

[Bryon]

Ah. So I just integrate from 0.4 to 0.7? I feel stupid sometimes lol.

Va-Vb = - ʃ(kQ/r^2)dr = kQ((1/a)-(1/b))

Thanks for the help by the way!