3 infinitely long thin wires each carrying current i in the same direction, are in the x - y plane in a gravity-free space. The central wire is along the y -axis while the other two are along with x = ∓ d. If the central wire is displaced along the z -direction by a small amount & released, the wire executes the simple harmonic motion. If the linear density of the wire is λ , find the frequency of oscillation.

Step-by-step solution

When wire (2) is displaced by x along y-axis force is taken per unit length F → 2 = F → 1 2 + F → 2 3 since F 1 2 = F 2 3 = μ 0 4 π 2 i 2 d 2 + x 2 F 2 = 2 F 1 2 sin θ F = 2 μ 0 4 π 2 i 2 d 2 + x 2 × x d 2 + x 2 = μ 0 π i 2 x d 2 + x 2 x < < d F R = - μ 0 π i 2 x d 2     comparing with    F R = - Kx ⇒ K = μ 0 π i 2 x 2 d 2 n = 1 2 π K m = 1 2 π μ 0 i 2 π d 2 × λ = i 2 &#