Question
(a) Explain the meaning of the term mutual inductance. Consider two concentric circular coils, one of radius r1 and the other of radius r2 (r1 < r2) placed coaxially with centres coinciding with each other. Obtain the expression for the mutual inductance of the arrangement.
(b) A rectangular coil of area A, having number of turns N is rotated at ‘ f ’ revolutions per second in a uniform magnetic field B, the field being perpendicular to the coil. Prove that the maximum emf induced in the coil is 2
f NBA.
Solution
a) Mutual induction is the property of a pair of coils due to which an emf induced in one of the coils is due to change in the current in the other coil.
Mathematically,
Consider two circular coils S1 and S2 of the same length l, such that both their centres coincide with each other.
Let,
n1 is the number of turns per unit length of S1
n2 is the number of tuns per unit length of S2
I1 is the current passed through the solenoid S1
is the flux linked with S2 due to current flowing through S1
We have,
where,
M21 is the coefficient of mutual induction of the two coils.
When current is passed through S1 an emf is induced in S2.
Magnetic field produced inside S1 on passing current through it is given by,
Magnetic flux linked with each turn of S2 will be equal to B1 times the area of cross section of S1.
Magnetic flux linked with each turn of S2 = B1A
Therefore,
Total magnetic flux linked with each turn of the S2 is,
Similarly, mutual induction between the two coils, when current is passed through coil S2 and induced emf is produced in coil S1 is given by,
Hence, coefficient of mutual induction between the two coil will be,
b) Flux is given by,
We know that,
Hence proved.
Mathematically,
Consider two circular coils S1 and S2 of the same length l, such that both their centres coincide with each other.
Let,
n1 is the number of turns per unit length of S1
n2 is the number of tuns per unit length of S2
I1 is the current passed through the solenoid S1
is the flux linked with S2 due to current flowing through S1We have,
where,
M21 is the coefficient of mutual induction of the two coils.
When current is passed through S1 an emf is induced in S2.
Magnetic field produced inside S1 on passing current through it is given by,
Magnetic flux linked with each turn of S2 will be equal to B1 times the area of cross section of S1.
Magnetic flux linked with each turn of S2 = B1A
Therefore,
Total magnetic flux linked with each turn of the S2 is,
Similarly, mutual induction between the two coils, when current is passed through coil S2 and induced emf is produced in coil S1 is given by,
Hence, coefficient of mutual induction between the two coil will be,
b) Flux is given by,
We know that,
Hence proved.
