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WBJEEA region in the form of an equilateral triangle (in \(x-y\) plane) of...
A region in the form of an equilateral triangle (in \(x-y\) plane) of height \(L\) has a uniform magnetic field \(\vec{B}\) pointing in the \(+z\)-direction. A conducting loop \(\mathrm{PQR}\), in the form of an equilateral triangle of the same height \(L\), is placed in the \(x-y\) plane with its vertex \(\mathrm{P}\) at \(x=0\) in the orientation shown in the figure. At \(t=0\), the loop starts entering the region of the magnetic field with a uniform velocity \(\vec{v}\) along the \(+x\)-direction. The plane of the loop and its orientation remain unchanged throughout its motion.
Which of the following graph best depicts the variation of the induced emf \((E)\) in the loop as a function of the distance \((x)\) starting from \(x=0\) ?
Which of the following graph best depicts the variation of the induced emf \((E)\) in the loop as a function of the distance \((x)\) starting from \(x=0\) ?
Solution:
For, 0 to \(\mathrm{L}\)
\(\varepsilon=\mathrm{B} \ell_{\text {eff }} \mathrm{v}=\mathrm{B} \times \frac{\mathrm{x}}{\sqrt{3}} \mathrm{v}\)
For, \(\mathrm{L}\) to \(2 \mathrm{~L}\)
\(\begin{aligned} & |e m f|=B\left(\frac{L}{\sqrt{3}}-\frac{x_0}{\sqrt{3}}\right) v-B \frac{2 x_0}{\sqrt{3}} v \\ & =\frac{B v L}{\sqrt{3}}-\sqrt{3} \mathrm{Bvx}_0 \\ & =\operatorname{Bv}\left[\frac{L}{\sqrt{3}}-\sqrt{3}(x-L)\right] \\ & =\frac{B v}{\sqrt{3}}[L-3 \mathrm{x}+3 \mathrm{~L}] \\ & =\frac{B v}{\sqrt{3}}[4 L-3 \mathrm{x}] \\ & \text { at } x=\frac{4 L}{3} \\ & e m f=0 \\ & \end{aligned}\)
\(\varepsilon=\mathrm{B} \ell_{\text {eff }} \mathrm{v}=\mathrm{B} \times \frac{\mathrm{x}}{\sqrt{3}} \mathrm{v}\)
For, \(\mathrm{L}\) to \(2 \mathrm{~L}\)
\(\begin{aligned} & |e m f|=B\left(\frac{L}{\sqrt{3}}-\frac{x_0}{\sqrt{3}}\right) v-B \frac{2 x_0}{\sqrt{3}} v \\ & =\frac{B v L}{\sqrt{3}}-\sqrt{3} \mathrm{Bvx}_0 \\ & =\operatorname{Bv}\left[\frac{L}{\sqrt{3}}-\sqrt{3}(x-L)\right] \\ & =\frac{B v}{\sqrt{3}}[L-3 \mathrm{x}+3 \mathrm{~L}] \\ & =\frac{B v}{\sqrt{3}}[4 L-3 \mathrm{x}] \\ & \text { at } x=\frac{4 L}{3} \\ & e m f=0 \\ & \end{aligned}\)
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