Question

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 λ 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 λ and −λ
parallel to and at distance d2−a2−−−−−−√ 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=λ2πϵ0ln(d+(√d+a)a) .
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?

Answer 1

To derive the potential distribution between the cylinder and the conducting plane, you can use Gauss' law and consider an infinitesimally small Gaussian cylinder surrounding a point on the surface of the conductor.

Step-by-step explanation:

To derive the potential distribution between the cylinder and the conducting plane, you can consider an infinitesimally small Gaussian cylinder surrounding a point on the surface of the conductor. The total charge inside the Gaussian cylinder is equal to the charge density multiplied by the cross-sectional area of the cylinder.

The electric flux crosses only the outer end face of the Gaussian surface and can be expressed as E multiplied by the cross-sectional area. Using Gauss' law, you can relate the electric field and the charge density to obtain the potential distribution.

Regarding the representation of the cylinder with a line of charge at its center, it is not valid for this particular scenario. The cylinder with charge density per unit length λ is a cylindrical shell, not a solid cylinder.

By considering infinitely long line charges arranged in a circle and reflecting each one in the plane, you can create an image of a corresponding cylinder. This is a valid method to analyze the potential of the cylinder.