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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 isIϕ1= 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, thenSelf 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 isM=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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