Question

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

Answer 1

To approach these types of problems integration is very useful to find the total magnetic flux linked with the loop.

Let us first consider an elementary strip of length l and width dr at a distance r from an infinite long current carrying wire. The magnetic field at strip due to current carrying wire is given by

`l = B(r) = (mu_0I)/(2pir) l.B(r)`

∴ Flux in strip `phi = (mu_0I)/(2pi) l int_(x_0)^x (dr)/r`

`phi = (mu_0Il)/(2pi) [log_e r]_(x_0)^x = (mu_0Il)/(2pi) log_e  x/x_0`

`ε = (-dphi)/(dt)`

So `IR = (dphi)/(dt)`

`I = 1/R d/(dt) [(mu_0Il)/(2pi) log_e  x/x_0] = (mu_0l)/(2piR) . log_e  x/x_0 (dl)/(dt)`

`I = (mu_0λl)/(2piR) log_e  x/x_0`  .....`[because (dl)/(dt) = λ (given)]`