Consider an infinitely long wire carrying a current I(t), with `(dI)/(dt)=lambda` = constant. Find the dt current produced in the rectangular loop of wire ABCD if its resistance is R as in figure.

Explanation

Consider small area element having thickness dr and length l on closed rectangular loop ABCD at distance l from very long current carrying wire as shown in figure.
Magnetic field at this strip, due to very long current carrying wire,
`B=(mu_0I)/(2pir)` (Outside the plane)
Magnetic flux linked with this strip,
`d phi=Bda=Bldr`
`d phi=(mu_0 I ldr)/(2pir)`
Net flux linked with ABCD loop,
`phi=int_(x_0)^x (mu_0Il)/(2pir)dr`
`=(mu_0Il)/(2pi)[ln r]_(x_0)^x`
`phi=(mu_0Il)/(2pi)ln (x/x_0)`
Induced emf,
`epsilon=(dphi)/(dt)=d/(dt)[(mu_0Il)/(2pi) ln (x/x_0)]`
`=(mu_0l)/(2pi) ln (x/x_0)(dI)/(dt)`
`epsilon=(mu_0l lambda)/(2pi) ln (x/x_0) (because (dI)/(dt)=lambda)`
Induced current,
`I=epsilon/R=(mu_0llambda)/(2piR)ln (x/x_0)`