Question

3. Inside an isolated sphere of radius R the electric charge Q is evenly distributed. After a

redistribution, the density of the load becomes:

ρ (r) = k (2 * r / R-r ^ 3 / R ^ 3)

Determine the value of the constant k.

Answer 1

$\rho =k(\frac{2r}{R}-\frac{r^3}{R^3})=\frac{kr(2R^2-r^2)}{R^3},$

$\text{if}~r=0~\text{then}~ \rho=0,$

$\text{if}~r=R~\text{then}~ \rho=k,$

$\rho=\frac QV,$

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

$\rho(r)=\frac{3Qr(2R^2-r^2)}{4\pi R^6}.$