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Two concentric circular coils, one of sm...

Two concentric circular coils, one of small radius `r_(1)` and the other of large radius `r_(2)`, such that `r_(1)ltltr_(2)`, are placed co-axially with centres coinciding. Obtain the mutual inductance of the arrangement.

Text Solution

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Let a current `I_(2)` flow through the outer circular coil. The field at the centre of the coil is `B_(2)=(mu_(0)I_(2))/(2r_(2))`. Since `r_(1)ltltr_(2),B_(2)` may be considered constant over cross-sectional area of the other coil. Hence, flux linking with it is
`phi_(1)=pir_(1)^(2)B_(2)=(mu_(0)pir_(1)^(2))/(2r_(2))I_(2)=M_(12)I_(2)`
Thus, `M_(12)=(mu_(0)pir_(1)^(2))/(2r_(2))` from equation (3) and (6)
`M_(12)=M_(21)=(mu_(0)pir_(1)^(2))/(2r_(2))`
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