In the figure shown the wires AB and PQ carry constant currents `I_(1)andI_(2)` respectively. PQ is of uniformly distributed mass ‘m’ and length 'I'. 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.

Explanation

(i) Magnetic repulsive force balances the weight.
`(mu_(0)I_(1)I_(2))/(2pih)l "mg"rArrh=(mu_(0)I_(1)I_(2)l)/(2pi"mg")`
(ii) Let the wire be displaced downward by distance `x(ltlth)`. Magnetic force on it will increase , so it goes back towards its equilibrium position . Hence it performs oscillations.
`F_("res")=(mu_(0)I_(1)I_(2))/(2pi(h-x))l-"mg"`
`=("mgh")/(h-x)-mg=(mg(h-h+x))/(h-x)`
`=(mg)/(h-x)x=(mg)/(h)x" for "xltlth`
`thereforeT=2pisqrt((m)/(mg//h))=2pisqrt((h)/(g))`