Question

Consider the situation of the previous problem. (a) Calculate the force needed to keep the sliding wire moving with a constant velocity v. (b) If the force needed just after t = 0 is F₀, find the time at which the force needed will be F₀/2.

Answer 1

Emf induced in the circuit, e = Bvl
Current in the circuit, $i=\frac{e}{R}=\frac{Bvl}{2r(l+vt)}$
(a) Force F needed to keep the sliding wire moving with a constant velocity v will be equal in magnitude to the magnetic force on it. The direction of force F will be along the direction of motion of the sliding wire.
Thus, the magnitude of force F is given by
$$\begin{gathered} F=ilB=\frac{Bvl}{2r(l+vt)}\times lB \\ =\frac{B^2{l}^{2}v}{2r(l+vt)} \end{gathered}$$
(b) The magnitude of force F at t = 0 is given by
$$\begin{gathered} F_0=ilB=lB\left(\frac{lBv}{2rl}\right) \\ =\frac{l{B}^{2}v}{2r}...\left(1\right) \end{gathered}$$
Let at time t = T, the value of the force be F₀/2.
Now,

$\frac{F_0}{2}=\frac{l^2{B}^{2}v}{2r(l+vT)}$

On substituting the value of F₀ from (1), we get
$$\begin{gathered} \frac{l{B}^{2}v}{4r}=\frac{l^2{B}^{2}v}{2r(l+vT)} \\ \Rightarrow 2l=l+vT \\ \Rightarrow T=\frac{l}{v} \end{gathered}$$