JEE Advanced · 2021 · 03 Oct 2021 · Paper 1 · Electromagnetic Induction

A long straight wire carries a current,

I = 2

ampere. A semi-circular conducting rod is placed beside it on two conducting parallel rails of negligible resistance. Both the rails are parallel to the wire. The wire, the rod and the rails lie in the same horizontal plane, as shown in the figure. Two ends of the semi-circular rod are at distances 1 cm and 4 cm from the wire. At time

t = 0

, the rod starts moving on the rails with a speed

v = 3.0

m/s (see the figure).

A resistor

R = 1.4Ω

and a capacitor

C_{0} = 5.0 μ

F are connected in series between the rails. At time

t = 0, C_{0}

is uncharged. Which of the following statement(s) is(are) correct?

[

μ_{0} = 4 π \times 1 0^{- 7}

SI units. Take

ln 2 = 0.7

]

(A)Maximum current through $R$ is $1.2 \times 1 0^{- 6}$ ampere✓ Correct
(B)Maximum current through $R$ is $3.8 \times 1 0^{- 6}$ ampere
(C)Maximum charge on capacitor $C_{0}$ is $8.4 \times 1 0^{- 12}$ coulomb✓ Correct
(D)Maximum charge on capacitor $C_{0}$ is $2.4 \times 1 0^{- 12}$ coulomb

Answer: (A), (C)

Solution

EMF induced in the semi-circular rod moving through the magnetic field of the wire:

E = \int_{0.01}^{0.04} \frac{μ _{0} I}{2 π r} v d r = \frac{μ _{0} I v}{2 π} ln (\frac{4}{1}) = \frac{μ _{0} I v}{2 π} \cdot 2 ln 2 = \frac{μ _{0} I v l n 2}{π}

Substituting values:

E = \frac{4 π \times 1 0 ^{- 7} \times 2 \times 3 \times 0.7}{π} = 24 \times 7 \times 1 0^{- 8} = 168 \times 1 0^{- 8}

V.

Maximum current (at t = 0, when capacitor is uncharged):

i_{m a x} = \frac{E}{R} = \frac{168 \times 1 0 ^{- 8}}{1.4} = 120 \times 1 0^{- 8} = 1.2 \times 1 0^{- 6}

A. ✓ (A)

Maximum charge on capacitor (when fully charged):

Q_{m a x} = C_{0} E = 5 \times 1 0^{- 6} \times 168 \times 1 0^{- 8} = 840 \times 1 0^{- 14} = 8.4 \times 1 0^{- 12}

C. ✓ (C)

JEE Advanced · 2021 · 03 Oct 2021 · Paper 1 · One or more correct options

Practise more: all JEE Advanced PYQs on Electromagnetic Induction →

JEE Advanced · 2021 · 03 Oct 2021 · Paper 1 · Electromagnetic Induction

Solution

More JEE Advanced questions on Electromagnetic Induction