# Infinitely Long Wire Potential

1. Sep 20, 2015

### stefan10

I am doing homework for my Intermediate Electricity & Magnetism class. We are currently working on Electric Potential. There are a few problems I have attempted but I have gotten stuck on. I think I keep getting stuck when I have to find what |r-r'| is for uniformly charged objects to some point at which I am finding the potential. Often the geometry gets very much involved and the integrals ugly, but I will post a more simple one (at least I think.) Thank you.

1. The problem statement, all variables and given/known data

Find the potential (using the general expression for V obtained from Coulomb’s law) a distance s from an infinitely long straight wire that carries a uniform line charge density λ. (Hint: assume wire is length L, integrate, take the limit L → ∞, express in terms of a reference length s0, and discard a large constant.) Compute the gradient of your potential to check that it yields the correct field.

2. Relevant equations
$$v = \frac{1}{4\pi\epsilon_{0}} \int \frac{\lambda}{|r-r'|} dl$$

3. The attempt at a solution
I would've used Gauss Law to do this, but the prompt says to use Coulomb's.

Following the instructions stated in the prompt, I first try to draw a diagram with a wire with a point s away from the wire in the perpendicular direction. I assume the wire has length L. I then set λ = q/L, and dl = dx.

What I seem to be stuck on with a lot of these problems is finding |r-r'| though. For this one I thought it would be, $$\sqrt{s^2 + x^2}$$ because that is the magnitude of a hypotenuse of a triangle with one leg going to s, and one leg going to some distance x on the wire.

Since I was told to assume the wire is of length L, I made my limits of integration - L/2 and L/2.

This all gives me the integral

$$v = \frac{q/L}{4\pi\epsilon_{0}} \int_{-L/2}^{L/2} \frac{\lambda}{\sqrt{s^2 + x^2}} dx$$

When I solve this integral I get

$$= \frac{q/L}{4\pi\epsilon_{0}} (\ln({\sqrt{s^2 + \frac{L^2}{4}} + \frac{L}{2}}) - \ln({\sqrt{s^2 + \frac{L^2}{4}} + \frac{-L}{2}}))$$

When I take the limit at L -> infinity of this, I get 0. What did I do wrong/not consider?

2. Sep 25, 2015