Three very long straight current carrying conductors are placed parallel to each other as shown in. The conductors A and C are fixed whereas conductors B is free to move. If the conductor B is pulled towards right through a very small distance r, then what is the net force acting on it and the angular frequency of the resulting oscillation.

Explanation

The net (restoring) force acting on the conductor after displacing it through a very small distance x is given as

`F = |F_(1) - F_(2)|`
`= (mu_(0) Ii_(0))/(2 pi (d - x)) - (mu_(0) Li_(0))/(2 pi (d + x)) = (mu_(0) Ii_(0))/(2 pi) [((d + x) - (d - x))/(d^(2) - x^(2))]`
For x < < d, `F = (mu_(0) Ii_(0) x)/(pi d^(2))`
`implies M omega_(osc)^(2) x = (mu_(0) Ii_(0) x)/(pi d^(2)) x`
`implies omega_(osc) = sqrt((mu_(0) Ii_(0))/(pi md^(2)))`