Question

Consider two concentric circular coils, one of radius r₁ and the other of radius r₂ (r₁ < r₂) placed coaxially with centers coinciding with each other. Obtain the expression for the mutual inductance of the arrangement.

Answer 1

Coefficient of mutual induction − consider two coils P and S. Suppose that a current I is flowing through the coil P at any instant i.e.,

Φ ∝ I

Φ = MI… (i)

If ‘e’ is the induced emf produced in the S-coil, then

`e=(dphi)/dt=-d/dt(MI)=-M(dl)/dt`

Mutual Inductance of two concentric coils, one of radius r₁ and the other of radius r₂ (r₁ < r₂) placed coaxially with centers coinciding with each other:

Consider two circular coil S₁ and S₂ of same length l, such that coil S₂ surrounds coil S₁ completely

Let

n₁ − Number of turns per unit length of S₁

n2 − Number of turns per unit length of S2

I₁ − Current passed through solenoid S₁

Φ21 − Flux linked with S2 due to current flowing through S1

Φ₂₁ ∝ I₁

Φ₂₁ = M₂₁I₁

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,

B₁ = μ₀n₁I₁

Magnetic flux linked with each turn of S₂ will be equal to B₁ times the area of the cross-section of S₁.

Magnetic flux linked with each turn of the  S₂ = B₁A

Therefore, total magnetic flux linked with the S₂,

Φ₂₁ = B₁A × n₂l = μ₀n₁I₁ × A× n₂l

Φ₂₁ = μ₀n₁n₂AlI₁

∴ M₂₁ = μ₀n₁n₂Al

Similarly, the mutual inductance between the two coils, when current is passed through coil S₂ and induced emf is produced in coil S₁, is given by

M₁₂ = μ₀n₁n₂Al

∴M₁₂ = M₂₁ = M (say)

Hence, coefficient of mutual induction between the two coil will be

`M=mu_0n_1n_2Al`