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 √(3) 2 hat x +1/2 hat 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 0.1 Ω , then the value of v is \ \ \ \ m s -1 . [Given: The permeability of free space μ 0 =4 π × 10 -7 N A -2 ]

Physics · Electromagnetic Induction

JEE Advanced 2023 — Paper 2 — Question 22

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 $\frac{\sqrt{3}}{2} \hat{x}+\frac{1}{2} \hat{\mathrm{y}}$ with a constant speed v . The wire is carrying a steady current $I=10 \mathrm{~A}$ in the positive x-direction. A current of $10 \mu \mathrm{~A}$ flows through the loop when it is at a distance $\mathrm{d}=4 \mathrm{~cm}$ from the wire. If the resistance of the loop is $0.1 \Omega$ , then the value of $v$ is $\_\_\_\_$ $\mathrm{m} \mathrm{s}^{-1}$ . [Given: The permeability of free space $\mu_{0}=4 \pi \times 10^{-7} \mathrm{~N} \mathrm{~A}^{-2}$ ]

Answer: 4

Numerical answer — enter this value.

Step-by-step solution

Emf induced in the loop is $\varepsilon=\frac{\mu_{0} \mathrm{I}}{2 \pi \mathrm{~d}} \cdot \frac{\mathrm{v}}{2} \mathrm{~b}-\frac{\mu_{0} \mathrm{I}}{2 \pi(\mathrm{~d}+\ell)} \cdot \frac{\mathrm{v}}{2} \cdot \mathrm{~b}$ ( $\ell=4 \mathrm{~cm}, \mathrm{~b}=2 \mathrm{~cm}, \mathrm{~d}=4 \mathrm{~cm}$ ) $\Rightarrow \varepsilon=\frac{\mu_{0} \mathrm{Ivb}}{4 \pi}\left(\frac{1}{\mathrm{~d}}-\frac{1}{\mathrm{~d}+\ell}\right)$ $\Rightarrow \mathrm{I}_{0}=\frac{\varepsilon}{\mathrm{R}}=\frac{\mu_{0} \operatorname{Ivb} \ell}{4 \pi \mathrm{~d}(\mathrm{~d}+\ell) \mathrm{R}}$

\mathrm{I}=10 \mathrm{~A}

Substituting all the values, $\mathrm{v}=4 \mathrm{~m} / \mathrm{s}$

Answer key and solution verified before publishing.

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Exam: JEE Advanced 2023
Paper: Paper 2
Subject: Physics
Chapter: Electromagnetic Induction
Topic: Motional EMF