A rectangular conducting loop of length 4 cm and width 2 cm is in the $xy$-plane, as shown in the figure. It is being moved away from a thin and long conducting wire along the direction $\frac{\sqrt{3}}{2} \hat{x}+\frac{1}{2} \hat{y}$ with a constant speed $v$. The wire is carrying a steady current $\mathrm{I}=10 \mathrm{~A}$ in the positive $x$-direction. A current of $10 \mu \mathrm{~A}$ flows through the loop when it is at a distance $d=4 \mathrm{~cm}$ from the wire. If the resistance of the loop is $0.1 \Omega$, then the value of $v$ is ____ $\mathrm{ms}^{-1}$. [Given: The permeability of free space $\mu_{0}=4 \pi \times 10^{-7} \mathrm{NA}^{-2}$ ]

JEE Advanced · 2023 · 04 Jun 2023 · Paper 2 · Electromagnetic Induction

A rectangular conducting loop of length 4 cm and width 2 cm is in the

x y

-plane, as shown in the figure. It is being moved away from a thin and long conducting wire along the direction

\frac{3}{2} \overset{x}{^} + \frac{1}{2} \overset{y}{^}

with a constant speed

v

. The wire is carrying a steady current

I = 10 A

in the positive

x

-direction. A current of

10 μ A

flows through the loop when it is at a distance

d = 4 cm

from the wire. If the resistance of the loop is

0.1Ω

, then the value of

v

is ____

ms^{- 1}

.
[Given: The permeability of free space

μ_{0} = 4 π \times 1 0^{- 7} NA^{- 2}

]

Answer: 4

The magnetic field from a long wire at distance

r

B = \frac{μ _{0} I}{2 π r}

.

The induced emf is due to the change in flux as the loop moves. The component of velocity perpendicular to the wire is

v_{y} = v sin (30°) = v /2

.

The emf induced is:

ε = (B_{1} - B_{2}) b v_{y} = \frac{μ _{0} I}{2 π} (\frac{1}{d} - \frac{1}{d + a}) b v_{y}

where

a = 4

cm (length),

b = 2

cm (width),

d = 4

cm.

Current:

i = \frac{ε}{R}

10 \times 1 0^{- 6} = \frac{2 \times 1 0 ^{- 7} \times 10}{0.1} (\frac{1}{4} - \frac{1}{8}) \times 2 \times v_{y} \times 1 0^{- 2}

Solving:

v_{y} = 2

m/s.

Since

v_{y} = v /2

, we get

v = 4

m/s.

JEE Advanced · 2023 · 04 Jun 2023 · Paper 2 · Integer answer