Question

Derive an expression for Mutual Inductance between two long co-axial solenoids.

Answer 1

 Mutual inductance between two long co-axial solenoids:

Consider two long co-axial solenoids of same length 1. The length of these solenoids is large when compared to their radii so that the magnetic field produced inside the solenoids is uniform and the fringing effect at the ends may be ignored. Let A₁ and A₂ be the area of cross section of the solenoids with A₁ being greater than A₂ . The turn density of these solenoids are n₁ and n₂ respectively.

Mutual inductance of two long co-axial selonoids

Let i₁ be the current flowing through solenoid 1, then the magnetic field produced inside it is B₁ = μ₀n₁i_1.

As the field lines of \(\vec B\) are passing through the area bounded by solenoid 2, the magnetic flux is linked with each turn of solenoid 2 due to solenoid 1 and is given by

since θ = 0° 

The flux linkage of solenoid 2 with total turns N₂ is

N₂ Φ₂₁ = (n₂ l)(μ₀n₁i₁ )

since N₂ = n₂l 

N₂ Φ₂₁ = (μ₀n₁n₁ A₂l)i₁ ….. (1) 

From equation of mutual induction 

N₂Φ₂₁ = M₂₁i₁ …… (2) 

Comparing the equations (1) and (2), 

M₂₁ = μ₀n₁n₁ A₂l ….. (3)

This gives the expression for mutual inductance 

M₂₁ of the solenoid 2 with respect to solenoid

1. Similarly, we can find mutual inductance M₂₁ of solenoid 1 with respect to solenoid 2 as given below.

The magnetic field produced by the solenoid 2 when carrying a current i₂ is

B = μ₀n₂i₂ 

This magnetic field B₂ is uniform inside the solenoid 2 but outside the solenoid 2, it is almost zero. Therefore for solenoid 1, the area A₂ is the effective area over which the magnetic field B₂ is present; not area A₂ Then the magnetic flux Φ₁₂ linked with each turn of solenoid 1 due to solenoid 2 is

The flux linkage of solenoid 1 with total turns N₁ is 

[Since N₁ = n₁l]

Therefore, we get 

∴ M₁₂ = μ₀n₁n₂A₂l ……. (4) 

From equation (3) and (4), we can write 

M₁₂ =M₂₁=M ……. (5) 

In general, the mutual inductance between two long co-axial solenoids is given by

M= μ₀n₁n₂ A₂l ……. (6) 

If a dielectric medium of relative permeability’ pr is present inside the solenoids, then 

M = μn₁n₂A₂l 

or M = μ₀μ_r n₁n₂ A₂l