The system shown in Figure below is in the homogeneous magnetic field B = 0.1 T.

Question

The system shown in Figure below is in the homogeneous magnetic field B = 0.1 T. The distance between two conducting vertical line is l = 10 cm. A rod having a mass M = 10 g start moving in a gravitational field (initial velocity is zero), and it is always in electrical contact the battery rails. The emf of the battery is elementof = 12 V, and the resistance of the resistor is R = 4 Om, Two forces are exerted on the rod: one is the gravity fore (F = mg), and the second is the force due to the magnetic field (F_A = I Bl, its direction is given by the corresponding rule). At the beginning (v = 0), the current in the circuit is determined by the emf of the battery only. Under the action of these two forces, the rod starts moving, and once it acquires some velocity, the induction emf appears and changes the current in the circuit and the magnitude of the magnetic force. Eventually, the velocity of the rod reaches some magnitude at which the gravity force is equal to the magnetic force and, after that, there is no velocity change any more (because the acceleration of the rod is zero). Please, find this velocity of the rod. Find the magnitude of this velocity in the case when emf of the battery elementof = 0 V. What is the magnitude of this velocity if also the resistance R is zero?

Explanation / Answer

Given,

B = 0.1 T ; l = 10 cm = 0.1 m ; M = 10 g = 0.01 kg ; E = 12 V ; R = 4 Ohm

Since the system is in a magnetic field, the wire must be experiencing a magnetic force of magnitude

F(m) = B I L

where, I is the current, B is the magnetic field and L is the length

from Ohm's law, V = IR=> I = V/R

where, I is the current, V is the voltage and R is the resistance of the wire.

V = induced voltage - battery voltage

induced voltage = B L v

V = B L v - E

where, L is the length and v is the velocity with which the system moves

putting the values in the equation of force we get

F = B I L = L (V/R) B = L B(B L v - E)/R

Rearranging we get

F = (BL/R)(B L v - E)

The force above is getting balanced by gravity, so

Mg = (BL/R)(B L v - E)

Solving for v we get

v = M g R/(BL)^2 + E/BL

v = 0.01 x 9.8 x 4/(0.1 x 0.1)^2 + 12/0.1 x 0.1 = 5120 m/s

Hence, v = 5120 m/s

when, E = 0

v = M g R/(BL)^2 = 0.01 x 9.8 x 4/(0.1 x 0.1)^2 = 3920 m/s

Hence, v = 3920 m/s ; when E = 0

when, R = 0

v = E/BL = 12/0.1 x 0.1 = 1200 m/s

Hence, v = 1200 m/s ; when R = 0

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