- #1

- 135

- 1

## Homework Statement

To find the electric field of a uniformly charged sphere at some distance using the result of the field at an axial of a charged disc.

## Homework Equations

## The Attempt at a Solution

Well,I approached this proble m in the following way:

You may verify that

E’=(σz’/2ε)[(1/z’)-{1/√(z’^2+r’^2)}is the expression for electric field at an axial point due to the charged disc. We use primed co-ordinates to any such disc located at an arbitrary place in the sphere. z axis is taken vertical, r’ is the radius of a disc.

Now consider the sphere to be made up of such charged discs, in succession; i.e. one placed on another and so on. For a very large number of discs, we may assume contribution from each disc to be dq and in that case, it is easier to think of a volume charge density ρ distributed all over the volume.

Then, dE={(dq/πr’^2)z’/2ε}[1/√(z’^2+r’^2)]

Putting z’=(z-Rcosθ) and r’=Rsinθ in the expression we get a formidable expression. However, dq has not yet been replaced. We Should replace it with ρ dV where dV is the standard volume element in spherical polar co-ordinates.

Now, the problem is how to evaluate the integration. Ф integration is not a problem. Θ integration is bizarre. What about radial co-ordinate integration? I thought only ∫dV will suffice. My friend says that each and every R over here: dq/πr’^2 and over here: [1/√(z’^2+r’^2)] are to be integrated.

I will simply die then.

However, it’s not any homework problem as you could easily see from its level of difficulty. I thought there should exist a way to approach in this path.