JEE Advanced · 2015 · 24 May 2015 · Paper 2 · Units & Measurement

(A) Energy stored in an inductor: $U_L=\frac{1}{2}L{I}^2=\frac{1}{2}{\mu }_0\frac{N^{2}A}{\ell }{I}^2\propto {\mu }_0\frac{A}{\ell }{I}^2$

Energy stored in a capacitor: $U_C=\frac{1}{2}C{V}^2=\frac{1}{2}{\epsilon }_0\frac{A}{d}{V}^2\propto {\epsilon }_0\frac{A}{d}{V}^2$

Both have dimensions of energy, so ${\mu }_0{I}^2$ and ${\epsilon }_0{V}^2$ are dimensionally equivalent. ✓

(C) From $Q=CV$: $I=\frac{dQ}{dt}=C\frac{dV}{dt}$

For a parallel plate capacitor: $C={\epsilon }_0\frac{A}{d}$

If we consider $\frac{dV}{dt}\sim \frac{V}{t}$ and $\frac{d}{t}\sim c$ (speed of signal propagation):
$I\sim {\epsilon }_0\frac{A}{d}\cdot \frac{V\cdot d}{A}\cdot c={\epsilon }_{0}cV$ ✓

Alternatively, displacement current density: $J_d={\epsilon }_0\frac{\partial E}{\partial t}$, and with $E\sim V/d$ and characteristic time $\sim d/c$, we get $I\sim {\epsilon }_{0}cV$.