Consider a long thin conducting wire carrying a uniform current I. A particle having mass " M " and charge " q " is released at a distance " a " from the wire with a speed v₀ along the direction of current in the wire. The particle gets attracted to the wire due to magnetic force. The particle turns round when it is at distance x from the wire. The value of x is [ ₀ is vacuum permeability]

Step-by-step solution

aligned & A B & V =-v_x i +v_y j & B = ₀ I 2 r (- k ) & F = q ( v B )= ₀ Iq 2 r [- v _ x j - v _ y i ] & a _ x =- ₀ Iq 2 ~m v_y r & a _ y =- ₀ Iq 2 m v_x r aligned aligned & v _ x dv _ x dr =- ₀ Iq 2 ~m v _ y r & v _ x dv _ x v _ y =- ₀ Iq 2 ~m dr r & ₀^ v ₀ v _ x dv _ x v ₀^2- v _ x ^2 =- ₀ Iq ^ x ₁ 2 ~m _ a ^ dr r 2 & Let, z ^2= v ₀^2- v _ x ^2 & 2 zdz =-2 v _ x dv _ x aligned aligned & zdz =- v _ x dv _ x & v _ x dv _x v ₀^2- v _ x ^2 = - zdz z =- dz aligned then integral becomes aligned & - _ v ₀ ^0 dz =- ₀ Iq