# Charge distribution on an irregular conducting surface

1. May 11, 2015

### Aman Chauhan

I recently read that the charge density is less on surfaces with greater radius of curvature on the surface of a charged irregular conducting body . if any one can provide a proof or explanation , please help!

2. May 11, 2015

### Staff: Mentor

That is not always true, but often it is a good approximation.
Consider two spheres of different size far away from each other at the same potential. If you calculate their electric field strengths at the surface, you will see that the smaller sphere has the stronger field, which corresponds to a larger charge density.

3. May 12, 2015

### officialmanojsh

Agreed! Because when there is same potential on both smaller and larger spheres, the charged atoms on smaller sphere will be very near where as due to more area of larger sphere, there is not space between charges atoms. Here both the potential difference and radius of curvature in case of spheres matters. :)

4. May 13, 2015

### Aman Chauhan

I am able to understand your explanation . But how can you say that this situation of 2 different sized spheres is equivalent to an irregular body which is just more curved at some points.

5. May 13, 2015

### Staff: Mentor

Every part of the surface is at the same potential. If the more curved parts are exposed, the situation is similar to the small sphere. If they are hidden somewhere in a notch, their charge density can be small.

6. May 14, 2015

### Aman Chauhan

Sorry I could not understand what you meant by "hidden somewhere in a notch". please elaborate a bit.

7. May 14, 2015

### Staff: Mentor

Like this:

8. May 14, 2015

### Aman Chauhan

Is this low charge density at the notch due to great repulsion from nearby charges (i suppose) Or is there some other reason ?

9. May 14, 2015

### Staff: Mentor

It is low due to the specific geometry here.

The electric potential is roughly the same everywhere in the notch (it is exactly the same everywhere at the surface!), which leads to low charge concentrations at the surface.

10. May 15, 2015

### Aman Chauhan

OK, So since the potential at every surface has to be the same and there is more surface nearer to a point in the notch so the charge density has to be less there in order to make the potential same as that of every point. Is that what you meant? (a rough approx. I mean)

11. May 15, 2015

### Staff: Mentor

The surface area has nothing to do with that, and there is no meaningful relation between potential (as absolute value) and charge density here.
The electric field is weaker, and surface charge density is proportional to the field at the surface.