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In the figure the wires AB and PQ carry ...

In the figure the wires `AB` and `PQ` carry constant current `l_(1)` and `l_(2)` respectively.`PQ` is of uniformly distributed mass `m` and length `l`.`AB` and `PQ` are both horizontal and kept in the same vertical plane.The `PQ` is in equilibrium at height `h`.Find
(i) `h` in terms of `I_(1),I_(2),l,m,g` and other standard constants.
(ii) If the wire `PQ` is displaced vertically by small distance prove that it performs `SHM`.Find its time period in terms of `h` and `g`.

Text Solution

Verified by Experts

(i) Magnetic repulsive force balances the weight .
`(mu_(0)I_(1)I_(2))/(2pih)l mg rArr h=(mu_(0)I_(1)I_(2)l)/(2pimg)`
(ii) Let the wire be displaced downward by distance `x(lt lt h)`.
Magnetic force on it will increases, so it goes back towards its equilibrium position.Hence it performs oscillations.
`F_(res)=(mu_(0)I_(1)I_(2))/(2x(h-x))l-mg`
`=(mgh)/(h-x)-mg=(mg(h-h+x))/(h-x)`
`=(mg)/(h-x)~=(mg)/hx "for"xltlt h`
`:.T=2pisqrt(m/(mg//h))=2pisqrt(h/g)`
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