**Coefficient of Coupling:**

While two coils are coupled magnetically, it is significant to know how much of the flux generated by one coil is linking with the other. Coefficient of coupling gives an idea about that.

Let us consider again the magnetically coupled coils of Figure Following Eq. (9), inductance of coil1 and coil 2 may be written as :

L_{1} = N_{1} (φ_{1} /i_{1} )

L _{2}= N_{2 } (φ_{2} /i_{2} )

Let Φ_{1} be the total flux produced by current i_{1} in coil 1 and Φ_{12} be the part of Φ_{1} that is linked with coil 2.

Let φ_{12 }= k_{1} φ_{1}

From Eq. (20),

M = N (φ_{12} /i_{12} )= N k_{1} φ_{1 } /i_{1 }

Likewise, let Φ_{2} be the total flux produced by current i_{2} in coil 2 and Φ_{21} be the part of Φ_{2} that is linked with coil1.

Let φ_{21} = k_{2} φ_{2}

From Eq. (2.21),

M = N (φ_{21} /i_{12} ) = N_{1}( k_{2 }φ_{2}/ i_{12})

Multiplying Eqs. (24) and (25), we get,

M ^{2} = k_{1} k_{2 }N_{1} N_{2} φ_{1} φ_{2} / i_{1}i_{2}= k_{1} k_{2} =( N1 φ1 / i_{12})( N2 φ2 / i_{12} )= k_{1} k_{2} L_{1 }L_{2}

where ∴

So, From Eq. (26), we obtain,

Constant k is called as coefficient of coupling and this is defined as the ratio of mutual inductance M to the square root of the product of inductances of coil 1 and coil 2.

If k =1, you know that the flux because of one coil is fully connected with the other. If k = 0, the flux in one coil does not link with the other coil at all.