(a) Obtain an expression for the mutual inductance between a long straight wire and a square loop of side a as shown in Fig.
(b) Evaluate the induced emf in loop if the wire carries a current of 50 A and the loop has an instantaneous velocity `upsilon = 10 ms^(-1)` at the location x = 0.2 m, as shown. Take a = 0.1m and assume that the loop has a large resistance.

Explanation

(a) Consider a strip of width dx (of the square loop) at a distance x form the wire carrying current.
Magnetic field due to current carrying wire at a distance x from the wire is `B = (m_(0) I)/(2 pi x)`
Small amount of magnetic flux associated with the strip `d phi = B dA = (mu_(0) I)/(2 pi x) (a dx)`
Magnetic flux linked with the square loop
`phi = (mu_(0) Ia)/(2 pi) int_(x = b)^(x = a + b) (dx)/(x) = (mu_(0) Ia)/(2 pi) [log_(e) x]_(x = b)^(x = a + b) = (mu_(0) Ia)/(2 pi) log_(e) ((a + b )/(b)) = (mu_(0) Ia)/(2 pi) log_(e) ((a)/(b) + 1)`
As `phi = MI :. MI = (mu_(0) Ia)/(2 pi) log_(e) ((a)/(b) + 1) = (mu_(0) a)/(2 pi) log_(e) ((a)/(b) + 1)`
(b) Induced emf in the loop
`e = B l upsilon = ((mu_(0) I)/(2 pi x)) l upsilon = (4 pi xx 10^(-7) xx 50)/(2 pi xx 0.2) xx 0.1 xx 10 = 5 xx 10^(-5)` volt