Question

Three infinitely long thin wires, each carrying current i, in the same direction, are in the x– y plane of gravity free space. The central wire is along the y–axis while other two are along x = ±d . If the central wire is displaced, along the z–direction, by a small amount, and released, it starts executing simple harmonic motion. If the linear density of wire is λ. The frequency of oscillations will be

(1) \(\frac{i}{2\pi d}\sqrt{\frac{μ_0}{\pi λ}}\)

(2) \(\frac{i}{d}\sqrt{\frac{μ_0}{\pi λ}}\)

(3) \(\frac{i}{\pi d}\sqrt{\frac{μ_0}{\pi λ}}\)

(4) \(\frac{2i}{\pi d}\sqrt{\frac{μ_0}{\piλ}}\)

Answer 1

 (1) \(\frac{i}{2\pi d}\sqrt{\frac{μ_0}{\pi λ}}\)

Let the +y–axis be perpendicular (into) the plane of paper x and z axis are in the plane of the paper. After displacing the central wire, along the z axis a little, through a distance, ‘z’, say

Force/length, acting on the central wire, is

Let ‘a’ be the acceleration in the wire in this position then