A circular wire loop of radius R is placed in the x-y plane centered at the origin O. A square loop of side a (`altltR`) having two turns is placed with its centre at `=sqrt(3)R` along the axis of hte circular wire loop, as shown in figure. The plane of the square loop makes an angle of `45^(@)` with respect to the z-axis. If the mutual inductance between the loops is given bu `(mu_(0)a^(2))/(2^(p//2)R)`, then the value of p is

Options

A. 2
B. 4
C. 7
D. 9

Answer

C

Explanation

In fig. we have shown a circular wire loop of radiux R centred at the origing O. A square loop of side a lt lt R having two turns is held at
`z = 3 sqrt3 R` along the axis of circular loop. Plane of square loop makes an angle of `45^(@)` w.r.t. z-axis.
Due to current I in the circular loop, magnetic field at the centre of square loop is
`B = (mu_(0) IR^(2))/(2[R^(2) + (sqrt3 R)^(2) ]^(3//2)) = (mu_(0) IR^(2))/(2 xx 4 R^(2) xx 2 R)`
`(mu_(0) I)/(16 R)`
Magnetic flux linked with the square loop
`phi = 2 BA cos 45^(@) = 2 ((mu_(0) I)/(16 R)) (a^(2)) (1)/(sqrt2)`
`phi = (mu_(0) I a^(2))/((2)^(7//2) R)`
As `M = (phi)/(I) = (mu_(0) a^(2))/((2)^(7//2) R) = (mu_(0) a^(2))/(2^(p//2) R) :. P = 7`