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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).
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Solution
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)]`
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