The magnetic moment of a coil having n turns, area A and current i is given by $$\overrightarrow \mu = ni\overrightarrow A $$. The torque on this coil when placed in a magnetic field $$\overrightarrow B $$ is given by $$\overrightarrow \tau = \overrightarrow \mu \times \overrightarrow B $$. In a moving coil galvanometer, cylindrical magnets ensure that $$\overrightarrow A $$ and $$\overrightarrow B $$ are perpendicular to each other. Thus, deflection torque on the coil is $$\tau$$_d = niAB. The restoring torque on a coil (at deflection angle $$\theta$$) due to the wire of torsional constant k is $$\tau$$_r = k$$\theta$$. In equilibrium, deflection torque is balanced by the restoring torque i.e., $$\tau$$_d = $$\tau$$_r, which gives

$$i = {k \over {nAB}}\theta $$.

The maximum deflection current of the galvanometer (i_g) occurs at full scale deflection $$\theta$$_max i.e.,

$${i_g} = {{k{\theta _{\max }}} \over {nAB}} = {{({{10}^{ - 4}})(0.2)} \over {(50)(2 \times {{10}^{ - 4}})(0.02)}} = 0.01$$ A.

A galvanometer of resistance G is converted to an ammeter by connecting a small shunt resistance S in parallel.

Kirchhoff's loop law gives

$${i_g}G - (i - {i_g})S = 0$$, $$ \Rightarrow S = {i_g}G/(i - {i_g})$$.

The maximum deflection current of galvanometer sets upper limit on the current measured by this ammeter (i_max). Substitute the values to get

$$S = {{{i_g}G} \over {{i_{\max }} - {i_g}}} = {{(0.1)(50)} \over {1 - 0.1}} = {5 \over {0.9}} = 5.55\Omega $$.