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Consider an infinitely long wire carryin...

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

Text Solution

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In Fig. XY is infinitley long wise carrying current
I (t), such that `(dI)/(dt) = lambda =` constant.
At any distance r from the wire, strength of manetic field, `B (r ) = (mu_(0) I)/(2 pu r)`. It is directed out of the pape. Total magnetic flux linked with the rectangular loop
`phi = int_(x_(0))^(x) B (r ). dA = int_(x_(0))^(x) (mu_(0) I)/(2 pi r) ldr`
`phi = (mu_(0) Il)/(2 pi) (log_(e)) (x)/(x_(0)) = LI`, where L is self-inductance of the loop
`L = (mu_(0) l)/(2 pi) (log_(e)) (x)/(x_(0))`
Induced current, `I = (e)/(R ) = (L)/(R ) (dI)/(dt) = (L)/(R ) lambda = (mu_(0) l lambda)/(2 pi R) (log_(e)) (x)/(x_(0))`
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