Question

Energy stored in the form of electric field within spherical region of a uniformly charged sphere of charge $Q$ is _______. (Charge is distributed within volume of sphere)

• A. $\frac{Q^2}{32\pi {\epsilon }_{o}R}$
• B. $\frac{Q^2}{8\pi {\epsilon }_{o}R}$
• C. $\frac{Q^2}{20\pi {\epsilon }_{o}R}$
• D. $\frac{Q^2}{40\pi {\epsilon }_{o}R}$

Answer 1

The correct option is D $\frac{Q^2}{40\pi {\epsilon }_{o}R}$


We know that energy stored per unit volume in the form of electric field is given by ${\eta }_E=\frac{energy}{volume}=\frac{1}{2}{ϵ}_0{E}^2$

$U=\int dU=\int \frac{1}{2}{\epsilon }_o{E}^2\left(dV\right)U={\int }_0^R\frac{1}{2}{\epsilon }_o{E}^2.4\pi r^2.dr$

We know that,
Inside a sphere with uniform charge distribution, the electric field is given by:
$E=\frac{Qr}{4\pi {\epsilon }_o{R}^3}$
So,
$U=\frac{{\epsilon }_o}{2}4\pi \int (\frac{Qr}{4\pi {\epsilon }_o{R}^3}{)}^2.r^2.drU=\frac{Q^2}{40\pi {\epsilon }_{o}R}$