A rectangular conducting loop of length 4 ~cm and width 2 ~cm is … - Vidyadip

MCQ

A rectangular conducting loop of length $4 \mathrm{~cm}$ and width $2 \mathrm{~cm}$ 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 $\frac{\sqrt{3}}{2} \hat{x}+\frac{1}{2} \hat{y}$ 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 $\mu_0=4 \pi \times 10^{-7} \mathrm{NA}^{-2}$ ]

✓
$4$
B
$5$
C
$7$
D
$10$

✓

Answer

Correct option: A.

$4$

a
$\mathrm{R}=0.1 \Omega$

$\varepsilon=\left(\mathrm{B}_1-\mathrm{B}_2\right) \mathrm{bv}_y$

(image)

$\mathrm{i}=\frac{\varepsilon}{\mathrm{R}}=\frac{\mu_0 \mathrm{I}}{2 \pi \mathrm{R}}\left(\frac{1}{\mathrm{~d}}-\frac{1}{\mathrm{~d}+\mathrm{a}}\right) \mathrm{bv}_y$

$\Rightarrow 10^{-5}=\frac{2 \times 10^{-7} \times 10}{0.1}\left[\frac{1}{4}-\frac{1}{8}\right] \times 2 . \mathrm{v}_y$

$\therefore \mathrm{v}_{\mathrm{y}}=2$

$\tan \theta=\frac{\mathrm{v}_z}{\mathrm{v}_z}=\frac{1}{\sqrt{3}}$

(image)

$\therefore \mathrm{v}_{\mathrm{x}}=2 \sqrt{3}$

$\therefore \mathrm{v}=\sqrt{\mathrm{v}_x^2+\mathrm{v}_z^2}=4$

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