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Question

The self inductance of two coils P and S are L1 and L2. The coupling between them is ideal. Show that the mutual inductance between these coils is M=L1L2.

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Solution

Relationship between mutual inductance between two coils and their self inductance:
Let N1 be the number of turns in one solenoid P. Its length is l and its area of cross section is A. On passing current I in it, the magnetic flux linked with it is
Iϕ1= magnetic field × effective area
=μ0N1Il×(N1A)
Self inductance of solenoid P is L1=ϕ1I=μ0N21Al .......(1)
Similarly, if N2 is the number of turns of the other solenoid S and its length is l, its area of cross section is A, then
Self inductance of the solenoid S is L2=μ0N22Al .....(2)
Now on passing current I in the primary solenoid P, the magnetic flux linked with the secondary solenoid S is ϕs=μ0N1Il×(N2A)
Hence mutual inductance between the two solenoids is
M=ϕsI=μ0N1N2Al .....(3)
From equations (1) and (2)
L1L2=μ0N21Al×μ0N22Al=μ0N1N2Al .......(4)
Hence from equations (3) and (4)
M=L1L2

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