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Obtain the expression for mutual inductance of a pair of long coaxial solenoids each of length *l* and radii *r*_{1} and *r*_{2} (*r*_{2} >> *r*_{1}). Total number of turns in the two solenoids are N_{1} and N_{2}_{,} respectively.

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#### Solution

Suppose a current i is passed through the inner solenoid S_{1}.

A magnetic field B=μ_{0}n_{1}i is produced inside S_{1} , whereas the field outside it is zero.

The flux through each turn S_{2} is Bπr_{1}^{2}=μ_{0}n_{1}iπr_{1}^{2}

The total flux through all the turns in a length *l* of S_{2} is

ϕ=(μ_{0}n_{1}iπr_{1}^{2})n_{2}l=(μ_{0}n_{1}n_{2}πr_{1}^{2}l)i

⇒M=μ_{0}n_{1}n_{2}πr_{1}^{2}l

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