Question

Consider two concentric circular coils,one of radius r1 and the other of larger radiusr2(r1＜＜r2),placed coaxially with centres coinciding with each other.obtain the expression for the mutual inductance of the arrangement.

Answer 1

n₁ = number of turns per unit length of inner coil
n₂ = number of turns per unit length of outer coil
r₁ = inner coil radius
r₂ = outer coil radius
μ₀ = permeability of free space
L = length of the coils (both)

Let i₁ be the current flowing in the inner coil.
   Magnetic field B due to this current = μ₀ n₁ i 
This field B is only inside the coil 1. Outside the coil 1, the field is 0.

Magnetic flux through 1 turn of coil 2 due to B is :
   Φ = B * Area = μ₀ n₁ i * π r² 

Total Number of turns in coil2 = n₂ * L
Total Magnetic flux through  all the turns of  coil 2
         = Φ = μ₀ n₁ i * π r² * n₂ L

Mutual inductance M is defined as :   Φ = M * i
     So M = μ₀ n₁ n₂ π r² L

Mutual inductance per unit volume of the coils:  μ₀ n₁ n₂