Question

Rectangular wire frame ABCD can slide freely on two parallel conducting and smooth vertical rails in uniform magnetic field B = 2T perpendicular to plane of frame and rails as shown in figure. Resistance of rails is zero but for wire frame resistance per unit length is 5Ω / m. Frame is released from rest under gravity. The speed of wire (in m/s) at that time when acceleration of frame is g/2, is equal to $\frac{N}{16}$. Find N. (g = 10 m/s² and mass of frame is 2 kg.)

Answer 1

300

Answer 2

Explanation of the solution:

1. Calculate the induced EMF in the horizontal segment AB as it moves downwards with velocity $v$: $\mathcal{E} = Blv$.
2. Determine the resistance of the circuit. Based on the figure and problem description, assume the resistance of the moving segment AB is in series with the external resistor, and the rails have zero resistance. The resistance of AB is its length times the resistance per unit length.
3. Calculate the induced current using Ohm's law: $I = \frac{\mathcal{E}}{R_{total}}$.
4. Calculate the magnetic force on the segment AB due to the induced current: $F_m = IlB$. The direction of the force is upwards, opposing the motion.
5. Apply Newton's second law to the frame: $Mg - F_m = Ma$.
6. Substitute the given values for mass, acceleration due to gravity, and acceleration of the frame. Solve for the velocity $v$.
7. Equate the obtained velocity to $\frac{N}{16}$ and solve for N.

The final answer is $\boxed{300}$.