The number of turns in the two coupled coils are 500 and 1500 res

SOLUTION

Concept: 

The self-inductance is a magnetic flux through the coil due to the current in the coil itself. It is given by the expression

\(L = \frac{{N\phi }}{I}\)

Consider two coils having self-inductance L1 and L2 placed very close to each other. Let the number of turns of the two coils be N1 and N2 respectively. Let coil A carries current I1 and coil B carries current I2.

Due to current I1, the flux produced is ϕ1 which links with both the coils. Then the mutual inductance between two coils can be written as

\(M = \frac{{{N_1}{\phi _{12}}}}{{{I_1}}}\)

Here, ϕ12 is the part of the flux ϕ1 linking with the coil 2

\(M = K\;\sqrt {{L_1}{L_2}} \)

Calculation:

Coil A:

N1 = 500

ϕ1 = 0.6 mWb

I1 = 5 A

\({L_A} = \frac{{{N_1}\;{\phi _1}}}{{{I_1}}}\)

LA = 0.06 H

\(\frac{{{L_A}}}{{{L_B}}} = {\left( {\frac{{{N_A}}}{{{N_B}}}} \right)^2}\)

LB = 0.54 H

Coil B:

N2 = 1500

ϕ12 = 0.3 mWb

\(M = \frac{{{1500}{\times 0.3\times10^{-3}}}}{{{5}}} = 90 mH\)

Coupling coefficient = K

We know that, \(M = K\sqrt {{L_A}{L_B}}\)

∴\(K = {{90\times10^{-3}} \over {(0.06\times 0.54)}^{0.5}}\)

K = 0.5

Alternate Solution:.

The coupling coefficient K is a measure of the magnetic coupling between two coils.

The orientation of two coils relative to each other affects the mutual induction between the two coils.

This orientation determines the coefficient of coupling (K) which is the fraction of magnetic flux induced in a coil due to the current in the neighboring coil and is given by

          \(K=\sqrt{\frac{M}{L_1L_2}} \)

Also, \(K = {{\phi_{12} } \over \phi_1} = {{0.3\times10^{-3} } \over 0.6\times10^{-3}}\)

∴ K = 0.5

 Where, M is the mutual inductance between two coils and L1 and L2 are the respective self-inductances of the two coils.

0 ≤ K ≤ 1

K < 0.5 loosely coupled

K > 0.5 tightly coupled

K = 1 magnetically tightly coupled or ideal coupled