Two smooth parallel metal rails, 0.5 m…

Two smooth parallel metal rails, 0.5 m apart, are inclined at 30^° to the horizontal and joined at the bottom by a resistor R=2 Ω. A uniform magnetic…

One or more options correct

Two smooth parallel metal rails,

0.5\ \mathrm{m}

apart, are inclined at

30^\circ

to the horizontal and joined at the bottom by a resistor

R=2\ \Omega

. A uniform magnetic field

B=0.8\ \mathrm{T}

is perpendicular to the plane of the rails. A rod of mass

0.1\ \mathrm{kg}

and negligible resistance, kept perpendicular to the rails, is released from rest and slides down with good contact. Take

g=10\ \mathrm{m\,s^{-2}}

. Which of the following statements is/are correct?

A. The terminal speed of the rod is $6.25\ \mathrm{m\,s^{-1}}$ .
B. When its speed is $3.125\ \mathrm{m\,s^{-1}}$ , the acceleration of the rod is $2.5\ \mathrm{m\,s^{-2}}$ .
C. At terminal speed the power dissipated in the resistor is $3.125\ \mathrm{W}$ .
D. The speed of the rod after $1.25\ \mathrm{s}$ is $4.5\ \mathrm{m\,s^{-1}}$ .

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Answer: A, B, C

The motional emf is

Blv

, the current

Blv/R

, and the magnetic retarding force

B^2l^2v/R

acts up the incline. So

m\dot v=mg\sin\theta-\dfrac{B^2l^2}{R}v

. With

B^2l^2=0.16

, terminal speed is

v_T=\dfrac{mgR\sin\theta}{B^2l^2}=\dfrac{0.1\times10\times2\times0.5}{0.16}=6.25\ \mathrm{m\,s^{-1}}

(A true). At half of

v_T

the magnetic force is half of

mg\sin\theta

, so

a=\tfrac12g\sin\theta=2.5\ \mathrm{m\,s^{-2}}

(B true). At terminal speed the power dissipated equals the power of gravity,

mg\sin\theta\,v_T=0.5\times6.25=3.125\ \mathrm{W}

(C true). The time constant is

\tau=mR/B^2l^2=1.25\ \mathrm{s}

and

v(t)=v_T\left(1-e^{-t/\tau}\right)

, so

v(1.25)=6.25(1-e^{-1})\approx3.95\ \mathrm{m\,s^{-1}}

, not

4.5

(D false). Correct options: A, B and C.

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Original question written in the JEE Advanced 2017 Paper 1 pattern and checked twice for the correct answer. Not an official question paper. How we write and check questions · Corrections policy · Report a mistake