A long solenoid (black coil) with cross - sectional area A and length L is wound

Question

A long solenoid (black coil) with cross - sectional area A and length L is wound with n_1 turns of wire. A time - varying current i_1 flows through this wire. A shorter coil (blue coil) with n_2 turns of wire surrounds it. Use mu_0. Find the value M of the mutual inductance. Note: The current in black coil is constantly changing. However, when using the hints it may help you to consider the instant at which the current in the black coil is i_1. Express your answer symbolically using the terms given in the question or numerically (in henry) if given numerical values. M =

Explanation / Answer

Here

For a solenoid:
B=0n1i/L

B:Magnetic Induction
0:Magnetic Permeability

i:Currnet through Solenoid
L:Lenght of solenoid

Fulx=B*A
Flux=0n1iA/L--------------------(1)
Emf induce in solenoid of truns n2 due to change in current inn1:
e=Mdi/dt--------------------(2) (M:Multual Inductance)
But by Faraday's Law:
EMF=Number of turns*Rate of Change of flux with time
e=n2d(flux)/dt
From (1)
e=n2d/dt(0n1iA/L)
e=(0n1n2A/L)*di/dt--------------------…
Equating (2) and (3)
Mdi/dt=0n1n2A/Ldi/dt
M=0n1n2A/L
M=0*n1*n2*A/n1iA/L--------------------(1)
Emf induce in solenoid of truns n2 due to change in current inn1:
e=Mdi/dt--------------------(2) (M:Multual Inductance)
But by Faraday's Law:
EMF=Number of turns*Rate of Change of flux with time
e=n2d(flux)/dt
From (1)
e=n2d/dt(0n1iA/L)
e=(0n1n2A/L)*di/dt--------------------…
Equating (2) and (3)
Mdi/dt=Pn1n2A/Ldi/dt
M=0n1n2A/L

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