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As shown in the figure, a rectangular loop of a conducting wire is moving away with a constant velocity $v$ in a perpendiculat direction from a very long straight conductor carrying a steady current $l$. When the
breadth of the rectangular loop is very small compared to its distance from the straight conductor, how does the emf. $E$ induced in the loop vary with time $t ?$
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breadth of the rectangular loop is very small compared to its distance from the straight conductor, how does the emf. $E$ induced in the loop vary with time $t ?$
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1734 Upvotes
Verified Answer
The correct answer is:
$E \propto \frac{1}{t^{2}}$
$\therefore$ Motional emf in the loop.
$$
\begin{aligned}
E &=E_{1}-E_{2}=vl l\left(B_{1}-B_{2}\right) \\
&=v\left(\frac{\mu_{0}}{2 \pi} \cdot \frac{I}{y}-\frac{\mu_{0} I}{2 \pi(y+b)}\right] \\
&=\frac{\mu_{0}}{2 \pi} \cdot I \cdot v l\left[\frac{y+b-y}{y(y+b)}\right]=\frac{\mu_{0}}{2 \pi} \cdot \frac{v I l}{y(y+b)} b
\end{aligned}
$$
$\because \quad b< < y$
$\therefore \quad E=\frac{\mu_{0}}{2 \pi} \cdot \frac{v l I}{y^{2}} b$
$\because v=$ constant
$$
$$
Thus, $\quad E=\frac{\mu_{0}}{2 \pi} \cdot \frac{v l l}{v^{2} t^{2}} b$
Hence, $\quad E \propto \frac{1}{t^{2}}$
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