Obtain an expression for the torque on a loop placed in a uniform magnetic field. Describe the construction and working of moving coil galvanometer.

a) Torque on a current loop or coil in a uniform magnetic field :
1) Let a rectangular current loop ABCD of length AB = CD = $l$ and breadth AD = BC= b and carrying current 'i' be suspended in a uniform magnetic field of strength 'B'.
2) Let the normal to the plane of the coil making an angle ' $\mathrm{\theta }$ ' with the field direction.
3) Forces $\mathrm{i}(\overline{\mathrm{b}}\times \overline{\mathrm{B}})$ on arms AD and BC act in opposite direction along vertical axis of suspension xy and hence cancel.
4) Force on arms AB and DC are ilB act at the middle points 'P' and 'Q' as shown in fig form a couple of arm $\mathrm{PR}=\mathrm{PQsin}\mathrm{\theta }=\mathrm{bsin}\mathrm{\theta }$

The moving coil galvanometer is made up of a rectangular coil that has many turns and it is usually made of thinly insulated or fine copper wire that is wounded on a metallic frame. The coil is free to rotate about a fixed axis. A phosphor-bronze strip that is connected to a movable torsion head is used to suspend the coil in a uniform radial magnetic field.

Working of Moving Coil Galvanometer
Let a current I flow through the rectangular coil of n number of turns and a cross-sectional area A. When this coil is placed in a uniform radial magnetic field B, the coil experiences a torque τ.

Let us first consider a single turn ABCD of the rectangular coil having a length l and breadth b. This is suspended in a magnetic field of strength B such that the plane of the coil is parallel to the magnetic field. Since the sides AB and DC are parallel to the direction of the magnetic field, they do not experience any effective force due to the magnetic field. The sides AD and BC being perpendicular to the direction of field experience an effective force F given by F = BIl

We know that torque τ = force x perpendicular distance between the forces

τ = F × b

Substituting the value of F we already know,

Torque τ acting on single-loop ABCD of the coil = BIl × b

Where lx b is the area A of the coil,

Hence the torque acting on n turns of the coil is given by

τ = nIAB

The magnetic torque thus produced causes the coil to rotate, and the phosphor bronze strip twists. In turn, the spring S attached to the coil produces a counter torque or restoring torque kθ which results in a steady angular deflection.

Under equilibrium condition:

kθ = nIAB

Here k is called the torsional constant of the spring (restoring couple per unit twist). The deflection or twist θ is measured as the value indicated on a scale by a pointer which is connected to the suspension wire.

θ= ( nAB / k)I

Therefore θ ∝ I

The quantity nAB / k is a constant for a given galvanometer. Hence it is understood that the deflection that occurs the galvanometer is directly proportional to the current that flows through it.