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Question: An ideal solenoid of cross- sectional area \({10^{ - 4}}{m^2}\) has \[500{\rm{ }}turns\] per metre. ...

An ideal solenoid of cross- sectional area 10−4m2{10^{ - 4}}{m^2} has 500turns500{\rm{ }}turns per metre. At the centre of this solenoid, another coil of 100turns100{\rm{ }}turns is wrapped closely around it, if the current in the coil changes from 0 to 2 A in 3.14 ms3.14\,ms, the emf developed in the second coil is:
A. 1 mV1\,mV
B. 2 mV2\,mV
C. 3 mV3\,mV
D. 4 mV4\,mV

Explanation

Solution

The mutual inductance between the solenoid is given by the relation M=μ0×N1×N2×AlM = \dfrac{{{\mu _0} \times {N_1} \times {N_2} \times A}}{l}. Use this relation to find the induced emf, MdidtM\dfrac{{di}}{{dt}}.

Complete step by step answer:
Let N1{N_1} = number of turns in the primary coil, here N1=500{N_1} = 500
N2{N_2}= number of turns in the secondary coil, here N2=100{N_2} = 100
A = The area of the cross-section of the solenoid, here A = 10−4m2{10^{ - 4}}{m^2}
The mutual inductance between the solenoid is given by the relation
M=μ0×N1×N2×AlM = \dfrac{{{\mu _0} \times {N_1} \times {N_2} \times A}}{l}, where I is the length of the solenoid and μ0{\mu _0} is the absolute permeability of free space. Its value depends on the system of units chosen for the measurement of various quantities and also on the medium between a point and the electric current.
Emf developed in the secondary coil is given by the relation MdidtM\dfrac{{di}}{{dt}}
⇒e=4π×500×100×10−41×23.14×10−3=4 mV\Rightarrow e = \dfrac{{4\pi \times 500 \times 100 \times {{10}^{ - 4}}}}{1} \times \dfrac{2}{{3.14 \times {{10}^{ - 3}}}} = 4\,mV

Hence the correct option is (D).

Note: Mutual inductance is the property of two coils by virtue of which each opposes any change in the strength of the current flowing through the other by developing an induced emf. When a time varying current flows through the primary coil of a solenoid, it sets up a time varying magnetic flux through the primary coil. When magnetic flux linked with the primary coil increases, an emf is induced in the secondary coil as the secondary coil is placed nearby to the primary coil. According to Lenz’s law the induced current in the secondary coil would opposes any increase in the current flowing through the primary coil.