Moving Coil Galvanometer

Moving Coil Galvanometer

When a coil of N turns, area A carrying current i is suspended in a magnetic field of induction B, the torque acting on it is given by

$\tau$ = M x B = M B sin$\theta$ where M = magnetic moment of the coil = NiA

$\therefore$ $\tau$ = NiAB sin$\theta$ where $\theta$ is the angle between B and the normal to the coil.

= NiAB since the field is radial.

This torque is balanced by the restoring couple 'C$\theta$' of the suspending wire.

$\therefore$ NiAB = C$\theta$ $\Rightarrow$ i = $\left(\frac{C}{NAB}\right)\theta$ = k$\theta$

Where k is the galvanometer constant

i $\propto$ $\theta$ in a moving coil galvanometer and by measuring $\theta$, current i can be measured.

Current sensitivity, $\frac{\theta }{i}=\frac{NAB}{C}$

Illustration 1: A coil in the shape of an equilateral triangle of side 0.02 m is suspended from a vertex such that it is hanging in a vertical plane between the pole pieces of a permanent magnet producing a horizontal magnetic field of 5 × 10 ^{– 2} tesla. Find the couple acting on the coil when a current of 0.1 ampere is passed through it and the magnetic field is parallel to its plane.

Solution: The couple acting on a closed loop is given by $\tau =NiAB\sin \theta$

Where, N = number of loops, A = area of loops, B = magnetic field and

$\theta$ = angle between magnetic field and normal to surface of the loop.

Here N =1, i = 0 amp, B = 5 x 10^–2 tesla, $\theta$ = 90º and

A = $\frac{1}{2}\left(Base\times height\right)$

$=\frac{1}{2}\left(0.02\times 0.01732\times {10}^{-4}\right)m^2$

$\therefore \tau =1\times 0.1\times \left(1.732\times {10}^{-4}\right)\times \left(5\times {10}^{-2}\right)\times 1$

$=8.66\times {10}^{-7}newtons/meter.$

AMMETER

A galvanometer with a small parallel resistance S (called shunt) is an Ammeter and is used in series for measuring current in a circuit.

The shunt required to increase the range of the ammeter from i_g to i is given by

S = $\frac{G}{\left(\frac{i}{i_g}\right)-1}=\frac{G}{n-1}$.

Where G is the resistance of the galvanometer, i_g is the maximum permissible current through the galvanometer and i is the range of the ammeter.

The resistance of an ideal ammeter should be zero, otherwise it introduces error in the measurement of current.

If r is the resistance of the ammeter and R is the external resistance the fractional error is $\frac{-r}{R+r}$.

VOLTMETER

A galvanometer with a high series resistance is called a voltmeter. It is connected in parallel between two points to measure the potential difference between those points.

The resistance required for converting a galvanometer of resistance G and maximum permissible current I_g into a voltmeter of range V is given by

R = $\frac{V}{i_g}-G$

The resistance of an ideal voltmeter should be infinite, otherwise it introduces errors in the measurement of potential difference.

It R is the resistance of the voltmeter and r is the resistance across which potential difference is to be measured, the fractional error is given by -$\left(\frac{r}{R+r}\right)$.

TANGENT GALVANOMETER

A coil of n turns carrying a current i produces a magnetic field along the axis of the coil given by

B = $\frac{{\mu }_{o}ni}{2r}$ where r is the radius of the coil.

If this field is horizontal and perpendicular to B_H, a magnetic needle will align itself at an angle $\theta$ to B_H given by

$\frac{B}{B_H}$ = tan$\theta$ $\Rightarrow$ i = $\left(\frac{2r{B}_H}{{\mu }_{o}n}\right)\tan \theta$

$\Rightarrow$ i = k tan$\theta$

By measuring $\theta$, the current through the coil can be measured. k is called the reduction factor of the galvanometer.

Illustration 2: Two tangent galvanometers of the same radius of coils have their number of turns in the ratio 2:3. Which galvanometer has large value of reduction factor?

Solution: Let K₁ and K₂ be the reduction factors of the two tangent galvanometers. Then

$K_1=\frac{2a}{(2n){\mu }_0}{B}_{H}and{K}_2=\frac{2a}{(3n){\mu }_0}{B}_H$

So, the galvanometer having lesser number of turns has large reductio factor.

Comparison Table for Electrostatic quantity and Magneto static quantity