Charged Sphere and its own energy.

Consider a sphere of radius $x$ and uniform charge density $\rho$ . Now, we want to add a small charged shell around this sphere, and calculate the change in potential energy when that happens.

$dU = -dW_{c,i} = \dfrac{1}{4 \pi \epsilon_{0}} \dfrac{4 \pi x^2 dx \rho \cdot \frac{4}{3}\pi x^3 \rho}{x}$

$\int_0^U dU = \dfrac{1}{4 \pi \epsilon_{0}} \dfrac{16 \pi^2 \rho^2}{3}\int_0^R x^4 dx$

And we know:

$\rho = \dfrac{q_{0}}{\frac{4}{3} \pi R^3}$

$\Longrightarrow U =\dfrac{1}{4 \pi \epsilon_{0}} \dfrac{3 q_{0}^2}{5R}$

Substituting the required values, we obtain $\boxed{6}$ Joules as the final answer.