In a moving coil galvanometer, torque on the coil can be experessed as `tau = ki`, where `i` is current through the wire and `k` is constant . The rectangular coil of the galvanometer having number of turns `N` , area `A` and moment of interia `I` is placed in magnetic field `B`. Find
(a) `k` in terms of given parameters `N,I,A andB`
(b) the torsion constant of the spring , if a current `i_(0)` produces a deflection of `(pi)//(2)` in the coil .
(c) the maximum angle through which the coil is deflected, if charge `Q` is passed through the coil almost instaneously. ( ignore the daming in mechinal oscillations).

Answer

A, B, C

Explanation

The torque acting on a rectangular coil placed in a uniform magnetic field is given by,
` vec(tau) = vec(M)xxvec(B) rArr tau = MB sin theta`
But ` M = N I A and theta = 90^(@)` (for moving coil galvanometer)
:. `tau = N I A B sin 90^(@)`
rArr ` tau = N i A B`
But ` tau = k i` (given)
rArr ` k = NAB`
(b) The torsion constant is given by
:. ` C = ( 2 N i_(0) A B)/( pi)` .....(i)
(c) We know that angular Impulse
` = int tau dt = int NiAB dt = NA B int i dt`
` = NABQ` .......(ii)
This angular impulse creates an angular momentum
` int i dt = I omega` ....(iii)
From (ii) and (iii)
` I omega = (NABQ)/(I)`
This is the instantaneous angular momentum due to which the coil starts rotating . Let us apply thwe law of energy conservation to find the angle of rotation .
Rotational kinetic energy of coil
` = (1)/(2) I omega^(2) = (1)/(2) ( IN^(2)A^(2) B^(2) Q^(2))/( I^(2)) = ( N^(2)A^(2)B^(2)Q^(2))/( 2I)`
`(1)/(2) C theta_(max)^(2) = ( N^(2)A^(2)B^(2)Q^(2))/( 2I)`
rArr `theta_(max)^(2) = ( N^(2)A^(2)B^(2)Q^(2))/( CI) = ( N^(2)A^(2)B^(2)Q^(2))/( 2 Ni_(0) ABI)xx pi`
rArr ` theta_(max)^(2) = (pi NABQ^(2))/( 2 i_(0)I) rArr theta_(max) = Q sqrt((NABpi)/( 2 Ii_(0)))`