Graphical variation of electric field due to a uniformly charged insulating solid sphere of radius R, with distance r from the centre O is represented by:

For a uniformly charged insulating solid sphere of radius RRR, the electric field varies with distance rrr from the center as follows

$$Inside\ the\ sphere\ (r\le R)$$

Using Gauss law:

Charge enclosed within radius r,

$$q_{\text{enc}}=\rho\frac{4}{3}\pi r^3$$

where

$$\rho=\frac{Q}{\frac{4}{3}\pi R^3}$$

Using Gauss law,

$$E(4\pi r^2)=\frac{q_{\text{enc}}}{\varepsilon_0}$$

Substituting,

$$E=\frac{Qr}{4\pi\varepsilon_0R^3}$$

Thus,

$$E\propto r$$

So electric field increases linearly from zero at the center to a maximum at the surface.

At

$$r=R$$

$$E_{\max}=\frac{Q}{4\pi\varepsilon_0R^2}$$
$$Outside\ the\ sphere(r>R)$$

Entire charge acts as if concentrated at the center:

$$E=\frac{Q}{4\pi\varepsilon_0r^2}$$

Thus,

$$E\propto\frac{1}{r^2}$$

So electric field decreases inversely as square of distance.

Graph

• From $$r=0\ to\ r=R$$: straight line increasing linearly
• At r=R : maximum field
• For r>R : decreasing inverse-square curve

So the graph starts from origin, rises linearly, then falls as a hyperbola-like curve.