Question

Problem 2.12 Use Gauss's law find the electric field inside a uniformly charged sphere (charge density p). Compare your answer to Prob. 2.8.

Answer 1

To find electric field inside a charged solid sphere using Gauss's law.

Consider a charged sphere of radius R and uniform charge density ρ, q is the charge on sphere where $ρ=\frac{q}{V}=\frac{q}{\frac{4}{3} \pi R^3}$

Consider a point P at a distance r inside the sphere.

Now draw a Gaussian surface with radius r<R

Using Gauss’s theorem

$Φ = \oint \vec{E}\cdot \vec{ds} = \frac{q_{enclosed}}{ε_0} \\ q_{enclosed} = ρV_{enclosed} = (\frac{4}{3} \pi R^3)(\frac{4}{3} \pi r^3) \\ q_{enclosed} = q(\frac{r^3}{R^3}) \\ \oint \vec{E}\cdot \vec{ds} = \frac{qr^3}{ε_0R^3} \\ E4 \pi r^2 = \frac{qr^3}{ε_0R^3} \\ E= \frac{1}{4 \pi ε_0} \frac{qr}{R^3}$

In terms of charge density

$E= \frac{1}{4 \pi ε_0} \frac{(ρ \frac{4}{3} \pi R^3)r}{R^3} \\ E = \frac{ρr}{3ε_0 }$