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## Homework Statement

Two circular lines of charge are centred at the origin and lie on the xy plane. The inner loop has a radius of a and a total positive charge q. The outer loop has a radius of b and total negative charge -q.

(a) Use Coulomb's Law to calculate the electric field on the z-axis.

(b) Calculate the electric potential of the loops from first principles along the z-axis.

(c) Use the gradient of the electric potential in (b) to calculate the electric field.

(d) Consider a single circular line of charge of radius a, that is split to have a total charge q in the positive y direction, and -q in the negative y direction. Calculate the electric field on the z-axis.

## Homework Equations

[itex]E = \frac{1}{4\pi\epsilon}\ \int_{-q}^q \frac{\sigma\,dr(2{\pi}rz)}{(z^2 + r^2)^\frac{3}{2}}[/itex]

[itex]V = \frac{1}{4\pi\epsilon}\ \int_{-q}^q \frac{\sigma\,dr(2{\pi}r)}{\sqrt{z^2 + r^2}}[/itex]

## The Attempt at a Solution

(a) I just used the first formula for E

(b) I was confused about the first principles bit so I just used the formulae for V

(c) I differentiated my (b) answer to work back to a similar answer as my (a) answer.

(d) This is where I got stumped a bit. It seems as if it should be straight forward compared to the rest but I can't grasp it.

I would appreciate it if anybody could tell me if my method for all the parts is correct and if they are, if you could point me in the right direction for (d) that would be great. I'm new to Latex so that's why I didn't type up all my answers again. It took me a while to get the first two formulae in. Thanks in advance for any help.