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If you have a charged point ($q_1=+1$) inside an initially neutral spherical shell ($q_2=0$), how would you find the following:

(A) Potential at the outer surface of the spherical shell? Would it be just $k(q_1+q_2)/r$?

(B) Potential inside the shell itself? Would this potential be equal to the potential of the outside surface since $E=0$ inside the shell or would it equal the potential due to the inner point charge inside the spherical shell?

(C) Potential between the sphere and the shell?How would you integrate to find this using $\Delta V = -\int_a^b \vec{E}\cdot d\vec{r}$?

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Generally, you'll want to calculate the electric field in each region of space (both inside and outside the shell) using Gauss' Law. When point charges are involved, since the potential and electric field tend to diverge at the location of the point charge, you'll want set the potential equal to zero at infinity, and integrate the electric field as you've suggested in part (C) from infinity to the point $r$ that you want to know the potential at. If you know the electric field in each region of space, then you can integrate all the way in from infinity.

As for part B, think about Gauss' Law. You should not expect the electric field to be zero inside the shell if there is a point charge in there.

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