Question

A long horizontal wire AB, which is free to move in a vertical plane and carries a steady current of 20 A, is in equilibrium at a height of 0.01 m over another parallel long wire CD which is fixed in a horizontal plane and carries a steady current of 30 A, as shown in figure. Show that when AB is slightly depressed, it executes simple harmonic motion. Find the period of oscillations.

Answer 1

Let m be the mass per unit length of wire AB. At a height x above the wire CD, magnetic force per unit length on wire AB will be given by $\, \, \, \, \, \, \, \, \, \, \, \, \, \, \, F_m=\frac{\mu_0}{2\pi}\frac{i_1 i_2}{x}$ (upwards) $\, \, \, \, \, \, \, \, \, \, \, \, \, \, \, ...(i)$ Weight per unit length of wire AB is $\, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, F_g=mg \, \, \, \, \, \, \, $ (downwards) Here, m = mass per unit length of wire AB At x = d, wire is in equilibrium i.e. $\, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, F_m=F_g$ $or \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \frac{\mu_0}{2\pi}\frac{i_1 i_2}{d}=mg$ $or \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \frac{\mu_0}{2\pi}\frac{i_1 i_2}{d^2}=mg \, \, \, \, \, \, \, \, \, ...(ii)$ When AB is depressed, x decreases therefore,$F_m$ will increase, while $F_g$ remains the same. Let AB is displaced by dx downwards Differentiating Eq. (i) w.r.t. x, we get $\, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, dF_m=-\frac{\mu_0}{2\pi}\frac{i_1}{x^2}dx \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, ...(iii)$ i.e. restoring force, $F = df_m \propto -dx$ Hence, the motion of wire is simple harmonic. From Eqs. (ii) and (iii), we can write $\, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, dF_m=-\Bigg(\frac{mg}{d}\Bigg).dx \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, (x = d)$ $\therefore $ Acceleration of wire a $=-\Bigg(\frac{g}{d}\Bigg).dx$ Hence, period of oscillation $\, \, \, \, \, \, \, \, \, \, \, \, \, T=2\pi\sqrt{\Bigg|\frac{dx}{a}\Bigg|}=2\pi\sqrt{\frac{| displacement |}{| |acceleration |}}$ $or \, \, \, \, \, \, \, \, \, \, \, \, T=2\pi \sqrt{\frac{d}{g}}=2\pi \sqrt{\frac{0.01}{9.8}}$ $or \, \, \, \, \, \, \, \, \, \, \, \, T=0.2 s$