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A rectangular loop of wire ABCD is kept ...

A rectangular loop of wire ABCD is kept close to an infinitely long wire carrying a current I (t) `= I_(0) (1 - t//T)` for `0 le t le`T and I(0) = 0 for t>T, Fig. Find the total charge passing through a given point in the loop, in time T. resistance of the loop is R

Text Solution

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The emf induced can be obtained by differentiating the expression on magnetic flux linked wrt t and then applying Ohm's law
`I=(E)/(R)=(1)/(R)(dphi)/(dt)`
We know that electric current `I(t)=(d Q)/(dt)or (Dq)/(dt)=(1dphi)/(R dt)`
Integrating the variable separable form of differential equation for finding the charge Q that passed in time t, we have
`Q(t_1)-Q(t_2)=(1)/(R)[phi(t_1)-phi(t_2)]`
`phi(t_1)=L_1(mu_0)/(2pi)int_x^(L_2+x)(dx^1)/(x^1)I(t_1)` [Refer to the Eq. (i) of answer no. 25]
`=(mu_0L_1)/(2pi)I(t_1)In(L_2+x)/(x)`
The magnitude of charge is given by , `=(mu_0L_1)/(2pi)In (L_2+x)/(x)[I_0=0]`
`=(mu_0L_1)/(2pi)I_1In ((L_2+x)/(x))`
This is the required expression .
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