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A wire of length l, mass m and resistanc...

A wire of length `l`, mass `m` and resistance `R` slides without any friction down the parallel conducting rails of negligible resistance . The rails are connected to each other at the bottom by a resistanceless rail parallel to the wire so that the wire and the rails form a closed rectangualr conducting loop. The plane of the rails makes an angle `theta` with the horizontal and a uniform vertical magnetic field of the inducetion `B` exists throughout the rregion. Find the steady state velocity of the wire.

Text Solution

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Force down the plane = mg sintheta At any instant if the velocity is v the induced
Current in the loop -`(IB cos thetav)/(R)`
Force on the conductor in the horizontal direction  =`(nuBlcostheta)/(R)xxBxxl`
Component parallel to the incline =` (B^(2)l^(2)costheta)/(R)xxvxxcostheta`
If v is constant this force should be angle to mg sin thetatherefoer `v=(mRgsintheta)/B^(2)l^(2)cos^(2)theta)`
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