Field due to uniformly charged sphere

Application Of Gauss Law:-

(a)- Field outside the sphere:-

Suppose the charge q is uniformly distributed over a sphere of radius R as shown in figure.

Let us find the electric field E at the point P outside the sphere for which we draw a Gaussian surface shown by dotted curve at a concentric sphere of radius OP=r.

So by Gauss theorem

The flux through the surface

∮E.dS = q/ε_o

Or                E∮dS = q/ε_o

Or                E.4πr²= q/ε_o

E = 1/4πε_o q/r²

Which is same as electric field at distance r from the point P & charge q. Hence it is clear that the charge may be concentrated at the centre.

(b)- At a point on the sphere:-

We put r=R then,

Electric field on the surface of sphere

E=1/4πε_o q/R²     

(c)- Electric field inside the sphere:-

Let us find the electric field E at the point P inside the charged sphere. We draw a Gaussian surface of radius r₁.

The outward flux throw the surface of sphere.

∮E.dS = E.4πr₁²………………..(1)

Now the total charge inclosed by the Gaussian surface.

q’ = volume inclosed by the Gaussian surface X charge density

     = 4/3 πr₁² X ρ………………………….(2)

Since    ∮E.dS = q/ε_o

Hence from (1) & (2) we have

EX4πr₁² = 4/3 πr₁³ .ρ

E = ρr₁/3ε_o