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## Homework Statement

An infinite solid cylinder of radius A and uniform charge distribution ρ is surrounded by a thin cylindrical envelope of radius B and linear charge distribution λ. The two cylinders are co-axial.

Find the potential V(r) as a function of r from r=0 to r=∞.

## Homework Equations

V(r)-V(r

_{0})=-∫

_{r0}

^{r}E⋅dl

## The Attempt at a Solution

Quick notes:

- Do not worry about the constants in front of the integrals; my questions do not concern them, so I will omit them.

- Ignore the V(A) I have in my solutions, it's another unimportant constant.
- I am trying to find the potential
__function__, not the potential difference between specific points.

^{-1}. I also have no problem finding V(0<r<A), it's just the two other domains that give me trouble.

What I am having trouble with is that I don't know how to structure my integral in the regions A<r<B and r>B such that V goes to zero as r increases. Currently, my solutions increase without bound. I do not know how to mathematically "choose" V(∞)=0 in this situation. My first thought was to integrate from r=A to r=r in the first region and r=B to r=r in the second, but this is what I got:

ΔV(A<r<B)=-∫

_{A}

^{r}E(A<r<B)⋅dl

V(A<r<B)-V(A)=-∫

_{A}

^{r}(1/r)dr

V(A<r<B)=V(A)+ln(A/r)

==> V(r>B)=V(B)+ln(B/r)=V(A)+ln(A/B)+ln(B/r)

Which doesn't work because the functions in both regions increase without bound (or at least the function in the first region would increase without bound had its domain not been restricted). But if I try to choose r=∞ as my reference point, I get

V(A<r<B)=V(A)+ln(∞/r)

==> V(r>B)=V(B)+ln(∞/r)=V(A)+ln(∞/B)+ln(∞/r)

Which also doesn't work because ∞ is in the actual expression and the constant ln(∞/b) makes no logical sense.

What am I doing wrong?

Again, I am trying to find the potential function for these regions, so structuring an integral from r=A to r=B does not help me because that gives the potential difference.