A homogeneous magnetic field B is perpendicular to a sufficiently long track of width ℓ which is horizontal. A frictionless conducting resistanceless rod of mass m straddles the two rail of the track as shown in the figure. Entire arrangement lies in horizontal plane. For the situation suggested in column-II match the appropriate entries in column-I. The rails are also resistancesless. Column-I Column-II (A) A is a battery of emf V and internal resistance R. The rod is initially at rest. (P) Energy is dissipated during the motion. (B) A is a charged capacitor. The system has no resistance. The rod is initially at rest. (Q) The rod moves with a constant velocity after a long time. (C) A is an inductor with initial current i0. It is having no resistance. (R) After a certain time interval rod will change its direction of motion. (D) A is a resistance. The rod is projected to the right with a velocity V0 (S) If a constant force is applied on the rod to the right, it can move with a constant velocity. (T) The rod stops after some time in absence of an external force.

Engineering

Physics

Self and Mutual Inductance

Question

A homogeneous magnetic field B is perpendicular to a sufficiently long track of width ℓ which is horizontal. A frictionless conducting resistanceless rod of mass m straddles the two rail of the track as shown in the figure. Entire arrangement lies in horizontal plane. For the situation suggested in column-II match the appropriate entries in column-I. The rails are also resistancesless.

Column-I	Column-II
(A) A is a battery of emf V and internal resistance R. The rod is initially at rest.	(P) Energy is dissipated during the motion.
(B) A is a charged capacitor. The system has no resistance. The rod is initially at rest.	(Q) The rod moves with a constant velocity after a long time.
(C) A is an inductor with initial current i₀. It is having no resistance.	(R) After a certain time interval rod will change its direction of motion.
(D) A is a resistance. The rod is projected to the right with a velocity V₀	(S) If a constant force is applied on the rod to the right, it can move with a constant velocity.
	(T) The rod stops after some time in absence of an external force.

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(A) $\frac{V - Bℓv}{R} = i$

$F = \frac{V - Bℓv}{R} \times Bℓ$

(B) i ℓ B = ma

$\Rightarrow \frac{ℓBdQ}{dt} = \frac{mdv}{dt} \Rightarrow v = \frac{BℓQ}{m}$

Ui > Uf because energy is dissipated in em radiation.
(C)   No energy dissipated as no resistance
R → Red executes SHM.
       F = iℓB to left ⇒ f_ext to right
       F = iℓB, $\frac{Ldi}{dt} = Bℓv \Rightarrow$ not possible to move with constant v

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