In the figure, a conducting rod of length l = 1 meter and mass m = 1 kg moves with an initial velocity, u = 5 m s - 1 . On a fixed horizontal frame containing inductor L = 2 H and resistance R = 1 Ω .

Question

In the figure, a conducting rod of length $l = 1$ meter and mass $m = 1 kg$ moves with an initial velocity, u $= 5 m s^{- 1}$ . On a fixed horizontal frame containing inductor $L = 2 H$ and resistance $R = 1 Ω$ . PQ and MN are smooth, conducting wires. There is a uniform magnetic field of strength $B = 1 T$ . Initially, there is no current in the inductor. Find the total charge in coulomb, flown through the inductor by the time velocity of the rod becomes $v_{f} = 1 m s^{- 1}$ and the rod has travelled a distance $x = 3$ meter.

TestHub · Accepted Answer

Let i 1 and i 2 be the current through L and R at any time t ∴ i = i 1 + i 2 ⇒ B l v R = i 2 and B l v = L d i 1 d t Force on conducting rod = m d v d t = - i l B = - i 1 + B l v R l B ⇒ m d v = - l B i 1 d t - B 2 l 2 R v d t ⇒ m ∫ d v = - l B ∫ i 1 d t - B 2 l 2 R ∫ v d t ⇒ m v f - u = - l B Q - B 2 l 2 R x ( v 1 = velocity, when it has moved a distance 'x') ⇒ Q = - B 2 l 2 R x - m v f - u B l = 1 C

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