Question

Define the term mutual inductance. Deduce the expression for the mutual inductance of two long coaxial solenoids of the same length having different radii and different number of turns.

Hint:

Mutual inductance quantifies the relationship between two coils and is directly proportional to the number of turns in each coil and the cross-sectional area of the second coil.

Answer 1

Mutual inductance \( M \) between two coils is the property of the system that quantifies how much the magnetic flux produced by one coil links with the second coil. It is defined as the ratio of the induced emf in coil 2 to the rate of change of current in coil 1. The mutual inductance between two coaxial solenoids can be derived as follows: Let: 
- \( N_1 \) be the number of turns of the first solenoid, 
- \( N_2 \) be the number of turns of the second solenoid, 
- \( L \) be the length of each solenoid, 
- \( R_1 \) be the radius of the first solenoid, 
- \( R_2 \) be the radius of the second solenoid, 
- \( \mu \) be the permeability of the material inside the solenoids. The magnetic field produced by the first solenoid inside it is given by Ampere’s Law: \[ B_1 = \frac{\mu N_1 I_1}{L} \] The flux linkage of the second solenoid is: \[ \Phi_2 = B_1 \cdot A_2 \] where \( A_2 \) is the cross-sectional area of the second solenoid, given by \( A_2 = \pi R_2^2 \). Thus, the flux linkage becomes: \[ \Phi_2 = \frac{\mu N_1 I_1 \pi R_2^2}{L} \] The induced emf in the second solenoid is: \[ \mathcal{E}_2 = -N_2 \frac{d\Phi_2}{dt} \] The mutual inductance \( M \) is then defined as: \[ M = \frac{\mathcal{E}_2}{dI_1/dt} \] Substituting for \( \mathcal{E}_2 \): \[ M = \frac{\mu N_1 N_2 \pi R_2^2}{L} \] This is the expression for the mutual inductance between two coaxial solenoids.