Question

Three very long straight current carrying conductors are placed parallel to each other as shown in the figure. The conductors 1 & 3 are fixed where as conductor 2 is free to move. If the conductor 2 is pulled towards right through a very small distance x, find the net force acting on it and angular frequency of the resulting oscillation.

Answer 1

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

$F=|F_1-F_2|$

$\frac{{\mu }_{0}Ii}{2\pi (d-x)}-\frac{{\mu }_{0}I{i}_0}{2\pi (d+x)}=\frac{{\mu }_{0}I{i}_0}{2\pi }\left[\frac{(d+x)-(d-x)}{d^2-x^2}\right]$

For x << d, $F=\frac{{\mu }_{0}I{i}_0}{\pi d^2}$

$\Rightarrow m{\omega }_{osc}^{2}x=\frac{{\mu }_{0}I{i}_0}{\pi d^2}x$

$\Rightarrow {\omega }_{osc}=\sqrt{\frac{{\mu }_{0}I{i}_0}{\pi m{d}^2}}$