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Consider an infinitely long wire carrying a current I(t), with \frac{dl}{dt}=\lambda =constant.. Find the current produced in the rectangular loop of wire ABCD if its resistance is R (figure).

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

\vec{B}(r)=\frac{\mu _{0}I}{2\pi r} (Out of paper)

Area of the elementary strip is, dA=l.dr

So, total flux through the loop is

\phi _{m}=\vec{B}.\vec{A}=\frac{\mu _{0}I}{2\pi }l\int_{x_{0}}^{x}\frac{dr}{r}=\frac{\mu _{0}Il}{2\pi}ln\frac{x}{x_{0}}\; \; \; \; \; \; \; \; \; .....(i)

The emf induced can be obtained by differentiating the eq. (i) w.r.t. t and then applying Ohm's law

I=\frac{\varepsilon }{R}      and \left | \varepsilon \right |=\frac{d\phi }{dt}

We have, induced current

I=\frac{1}{R}\frac{d\phi }{dt}=\frac{\mu _{0}l}{2\pi}\frac{\lambda }{R}ln\frac{x}{x_{0}}\left ( \because \frac{dI}{dt} =\lambda \right )


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