Question

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

Answer 1

Hint: The mutual inductance between the solenoid is given by the relation \[M = \dfrac{{{\mu _0} \times {N_1} \times {N_2} \times A}}{l}\]. Use this relation to find the induced emf, \[M\dfrac{{di}}{{dt}}\].

Complete step by step answer:
Let \[{N_1}\] = number of turns in the primary coil, here \[{N_1} = 500\]
\[{N_2}\]= number of turns in the secondary coil, here \[{N_2} = 100\]
A = The area of the cross-section of the solenoid, here A = ${10^{ - 4}}{m^2}$
The mutual inductance between the solenoid is given by the relation
\[M = \dfrac{{{\mu _0} \times {N_1} \times {N_2} \times A}}{l}\], where I is the length of the solenoid and \[{\mu _0}\] 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 \[M\dfrac{{di}}{{dt}}\]
\[ \Rightarrow e = \dfrac{{4\pi \times 500 \times 100 \times {{10}^{ - 4}}}}{1} \times \dfrac{2}{{3.14 \times {{10}^{ - 3}}}} = 4\,mV\]

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.