- Consider a long solenoid of length l and area of crossection A containing N
_{p}turns in its primary coil

- Let a shorter secondary coil having N
_{2}number of turns be wounded closely over the central portion of primary coil as shown below in the figure.

- If I
_{p}is the current in the primary coil then magnetic field due to primary coil would be

- So flux through each turn of secondary coil would be

where A is the area of crossection of primary coil

- Total magnetic flux through secondary coil is

- Emf induced in secondary coil is

Thus from equation 24

So

- Consider two coils of same length l and same area of cross-section placed near each other as shown below in the figure

- Let there are N
_{1}number of turns in primary coil and N_{2}number of turns in secondary coil

- A current I
_{1}in the primary coil produces a magnetic field

which in turns gives rise to flux?

in primary coil and

in the secondary coil due to current in primary coil.

- By the definition of self induction

and by definition of mutual induction

- Reversing the procedure if we first introduce the current I
_{2}in secondary coil then we get

- So L
_{1}is the self inductance of primary coil,L_{2}is the self induction of secondary coil and M_{21}=M_{12}=M is the mutual inductance between two coils

- Product of L
_{1}and L_{2}is

- In practice M is always less than eq due to leakage which gives

Where K is called coefficient of coupling and K is always less then 1.

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