Figure shows a rectangular coil near a long wire. The mutual inductance of the combination is

Options

A. `(mu_(0)a)/(2pi) In(1-(b)/(c))`
B. `(mu_(0)a)/(2pi) In(1+ (b)/(c))`
C. `(mu_(0)a)/(pi) In(1+ (b)/(c))`
D. `(mu_(0)a)/(sqrt(2)pi) In(1+ (b)/(c))`

Answer

B

Explanation

Let current `i_(1)` in the straight wire be upward. Then the magnetic field due to the straight wire has magnitude `B_(1) = mu_(0)i_(1)//2 pir` at a distance `r`. In according with right hand rule, `B_(1)` points inward to the plane of page. We consider a differential strip of thickness `dr`, area `dA_(2) = a dr`. Magnetic flux through area `dA, dphi_(B)(a dr)`.
Total flux through the loop,
`phi_(B) = int B_(1) adr = int_(c)^(c + b) (mu_(0)i_(1))/(2pir) a dr`
`= (mu_(0)i_(1)a)/(2pi) int_(c)^(c + b) (dr)/(r ) = (mu_(0)i_(1)a)/(2pi) In((c + b)/(c))`
therefore mutual inductance,
`M = (phi)/(i_(1)) = (mu_(0)a)/(2pi) In (1 + (b)/(c))`