# Electrostatics: Cylinder and conducting plane question (method of images)

I was looking at some problems/questions about electrostatics and came across this question:

An infinitely long cylinder of radius $a$ and with charge density per unit length $\lambda$ is placed with its axis a distance $d$ from an infinite conducting plane which is at zero potential. Show that two line charges with charges $\lambda$ and $-\lambda$ parallel to and at distance $\sqrt{{d^2 - a^2}}$ either side of the plane, give rise to the potential distribution between the cylinder and plane.

Hence show that the potential of the cylinder is given by $$V = \frac{\lambda}{2\pi\epsilon_0}ln\left(\frac{d + \sqrt(d+a)}{a}\right) \ .$$

The main issues I am having are to do with the first part and showing rather than verifying the configuration. Is there a way of deriving this and how would you go about deriving it (without simply 'guessing')?
I have also tried to think of the cylinder as many, infinitely-long line charges arranged in a circle, and then reflected each one in the plane to produce an image of a corresponding cylinder. Is it valid to represent the cylinder with a line of charge at its centre?

• Is $a$ the radius of the cylinder? – Floris Jan 30 '16 at 19:41
• yes sorry I forgot to mention that, I'll edit it now – Hakkihan Tunbak Jan 30 '16 at 22:52