Two concetric coplanar circular conducing loops have radii R and r `(R gt gt r)` . Their mutual inductance is proportional to

Options

A. `(r )/(R )`
B. `(R )/(r )`
C. `(r^(2))/(R )`
D. `(R^(2))/(r )`

Answer

C

Explanation

Let us consider a coll having `N_(1)` turns and radius r is surrounded by another coll having `N_(2)` turns and of radius R as shown in the figure below

magnetic field due to coll 2 at the centre
`B_(2) = (mu_(0)N _(2) I)/( 2R )`
Magnetic flux through the coil
`phi_(B_(1)) = N_(1) B_(2) A = (N_(1) mu_(0) N_(2) l . pi r^(2))/( 2R )`
But `phi_(B _(1)) = MI`
From Eqs . (i ) and (ii ) we get
`MI= (N_(1) mu_(0) N_(2) I. pi r^(2))/( 2R ) rArr M = (N_(1) mu_(0) N_(2). pi r^(2))/( 2R ) rArr M prop (r^(2))/( R )`