Electric Field and Potential at a Point Near a Regular Charged Body

Question

While calculating the electric potential at a point near charged bodies such as a uniform ring, hollow shell and solid sphere, iv'e seen that the potential at a point is equal to:

V = KQ/D(avg)

Here D average denotes the average distance between point of potential measurement and the elements of the object. It is somewhat like the distance between the "Centre of charge of the body" and the point of measurement.

This seems to work out for my few cases which ive tried. It also sometimes works for electric field at a point by replacing D with D squared, but fails in some cases such as with a uniform ring.

My doubt: Is this approach of calculating "Centre of charge" always applicable for potential at a point? Why so? And if yes, why does it fail for electric field strength at a point?

Answer 1

$$\left(\frac{x+y}2\right)^{-1}\neq\frac{x^{-1}+y^{-1}}2$$ and thus your averaging cannot possibly work for most cases. Instead, it is proved, only for spherically symmetric cases, that this averaging yields the correct answer. But it explains why the sphere and the ball cases worked. Noumenclature standardised for both maths and physics: ball has insides, and sphere is just the skin of ball.

This averaging over the uniform ring will fail for both the potential and the electric field cases.