Question

A moving coil galvanometer has a coil of area A and number of turns N. A magnetic field B is applied on it. The torque acting on it is given by $\tau =ki$ where i is current through the coil. If moment of inertia of the coil is I about the axis of rotation.
A charge Q is passed almost instantaneously through coil.

If the maximum angular deflection in it is $\theta =Q\sqrt{\frac{\pi NAB}{xI{i}_0}}$. Find x.

Answer 1

If $q_m$ is the maximum deflection, then from conservation of energy:
$\frac{1}{2}C{\theta }_m^2=\frac{1}{2}I{\omega }^2\Rightarrow {\theta }_m=\sqrt{\frac{I}{C}}\omega$ (i)
We have, $\tau =NiAB$
Put, $\tau =\frac{dL}{dt}$ where L is angular momentum.
$\frac{dL}{dt}=N\left(\frac{dQ}{dt}\right)AB$ or $dL=NABdQ$

Integrating ${\int }_0^{L}dL=NAB{\int }_0^{Q}dQ\Rightarrow L=NABQ$
If $\omega$ is angular velocity,
put $L=I\omega$
$I\omega =NABQ$
$\omega =\frac{NABQ}{I}$ (ii)
Substituting this value in Eq. (i), we get
$\theta =\sqrt{\frac{I}{C}}\cdot \frac{NABQ}{I}=\frac{NABQ}{\sqrt{\left\{\frac{2NAB{i}_{0}I}{\pi }\right\}}}=Q\sqrt{\frac{\pi NAB}{2I{i}_0}}$