Question
Question: The sensitivity of tangent galvanometer is maximum when deflection is (A) \({45^o}\) (B) \({90^o...
The sensitivity of tangent galvanometer is maximum when deflection is
(A) 45o
(B) 90o
(C) 30o
(D) 60o
Solution
Hint The angle of deflection per unit current flowing through the galvanometer is known as the sensitivity of the galvanometer. Therefore consider an elementary current dI through a deflection of θ and apply the tangent law.
Compete Step by step solution
According to the tangent law,
B=BHtanθ
where, B is the magnetic field produced inside the coil due to the current I passing through it.
BH is the horizontal component of the magnetic field.
For a current- carrying conductor,
B=2anμoI
where, n is the number of loops.
Therefore equating the above two equations we get
2anμoI=BHtanθ
⇒I=nμo2aBHtanθ
Let us consider K=GBH where G=2anμo
∴I=Ktanθ
Let us differentiate both sides.
∴dI=Ksec2θdθ
Here dI stands for an elementary change in the current flowing through the galvanometer, and dθ stands for an elementary change in the angle of deflection, or the angle through which the needle will turn inside the galvanometer.
Therefore we can say that dIdθ is the sensitivity of the galvanometer.
Dividing both sides by Iwe get,
IdI=IKsec2θdθ
⇒IdI=KtanθKsec2θdθ
⇒IdI=sinθcosθdθ=2sinθcosθ2dθ
We know that sin2θ=2sinθcosθ
Substituting this value within the above equation we get,
IdI=sin2θ2dθ
⇒dθ=2sin2θIdI
The maximum sensitivity of the tangent galvanometer means the maximum possible value of dθ.
From the above equation, we can see that dθ is maximum when sin2θ is maximum.
∴sin2θ=1
⇒2θ=90o
⇒θ=45o
Therefore, option A. is the correct answer.
Additional Information
For the measurement of small electric currents, tangent galvanometers are used. Hence a tangent galvanometer is very sensitive and accurate. For a fractional change in the value of current, there is a large deflection in the galvanometer. It consists of a coil made from an insulated copper wire wound on a circular non-magnetic frame.
Note A tangent galvanometer is dependent on the law of tangent magnetism, hence it should not be confused with a moving coil galvanometer. A moving coil galvanometer works when it is acted upon by a torque.