Question

Two large metallic plates having area $A$ and charge $Q$ and $-Q$ each are separated by a small distance ${}^{\prime }{d}^{\prime }$ and are connected by two springs as show in the figure. The energy stored in any one of the springs in equilibrium condition is

• A. $\frac{Q^4}{4A^2{\epsilon }_0^{2}k}d$
• B. $\frac{Q^4}{2A{\epsilon }_0^2}d$
• C. $\frac{Q^4}{8A^2{\epsilon }_0^{2}k}$
• D. $\frac{Q^2}{A{\epsilon }_0}$

Answer 1

Answer: (A), (C)

The correct options are
A $\frac{Q^4}{4A^2{\epsilon }_0^{2}k}d$
C $\frac{Q^4}{8A^2{\epsilon }_0^{2}k}$

Force on left plate per unit area

$\left(\frac{\sigma }{2{\epsilon }_0}\right)\sigma =\frac{{\sigma }^2}{2{\epsilon }_0}$

$=\frac{Q^2}{2{\epsilon }_0{A}^2}$

$\frac{Q^2}{2\epsilon A^2}=kx$

$x=\frac{Q^2}{2k{\epsilon }_{0}A}$

spring energy $=\frac{1}{2}k{x}^2$

$=\frac{1}{2}k{\left(\frac{Q^2}{2k{\epsilon }_{0}A}\right)}^2$

$=\frac{Q^4}{8k{\epsilon }_0^2{A}^2}$