Question

Obtain the expressions for sensitivity and figure of merit of galvanometer giving its principle. On what factors does it depend?

Answer 1

Sensitivity of Galvanometer
That galvanometer is more sensitve; in which a small current when flowing in the coil or small voltage when applied at its ends the necessary deflection is noted.
Current Sensitivity
The current sensitivity of the galvanometer is the deflection per unit current. It is measured in div/amp From equation (2)
$I=\left[\frac{C}{NAB}\right]\varphi =k\varphi$
Hence, current sensitivity
$S_i=\frac{\varphi }{I}=\frac{NAB}{C}=\frac{1}{k}....\left(5\right)$
It is clear from the equation (5) that to increase the current sensitivity of the galvanometer the cross sectional area $A$ of the coil, number of turns $B$ of the coil and the magnetic field $B$ has to be more and the value of $C$ has to be less. The value of $N,A$ and $B$ Cannot be increased more because due to this the weight of the coil will increase.
Hence, to increase the sensitivity it is more relevant to decrease the value of $C$
Since, $C=\frac{\pi \eta r^4}{2l}$
where $\eta$ is the elasticity constant of the substance f the wire, $r$ is the radius of the wire and $l$ is the length of the wire.
For Phosphor-Bronze the value of $r$ is less. Hence, to increase the current sensitivity of the Galvanometer a thin long wire of Phosphor-Bronze is used. But due to the thin suspended wire there is more risk of breaking, hence this galvanometer is not portable.
Voltage Sensitivity
Voltage sensitivity is defined as the deflection per unit voltage Hence, voltage sensitivity;
$V_S=\frac{\varphi }{V}.....\left(6\right)$
If $G$ is the resistance in the coil of the galvanometer and $I$ is the current flowing then
$V=IG$
$\therefore$ $V_S=\frac{\varphi }{IG}$
$V_S=\frac{NAB}{CG}....\left(7\right)$
From equation (5) and (7)
$V_S=\frac{S_i}{G}....\left(8\right)$
Figure of Merit of a Galvanometer
The figure of merit of a galvanometer is the current required to produce a deflection of one division in the galvanometer scale. This is inversely proportional to the sensitivity of the galvanometer
$\therefore$ $X=\frac{1}{S_i}=\frac{I}{\varphi }$
or $X=\frac{C}{NAB}=k....\left(9\right)$