A region in the form of an equilateral triangle (in x − y plane) of height L has a uniform magnetic field B pointing in the + z -direction. A conducting loop PQR , in the form of an equilateral triang

Question

A region in the form of an equilateral triangle (in $x - y$ plane) of height $L$ has a uniform magnetic field $B$ pointing in the $+ z$ -direction. A conducting loop $PQR$ , in the form of an equilateral triangle of the same height $L$ , is placed in the $x - y$ plane with its vertex $P$ 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 $v$ 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 $x = 0$ ?

Question diagram: A region in the form of an equilateral triangle (in x − y pl

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Answer:

A

Solution:

For, 0 to $L$ $ε = B ℓ_{eff} v = B \times \frac{x}{3} v$ For, $L$ to $2 L$
$∣ e m f ∣ = B (\frac{L}{3} - \frac{x _{0}}{3}) v - B \frac{2 x _{0}}{3} v = \frac{B vL}{3} - 3 Bvx_{0} = Bv [\frac{L}{3} - 3 (x - L)] = \frac{B v}{3} [L - 3 x + 3 L] = \frac{B v}{3} [4 L - 3 x] at x = \frac{4 L}{3} e m f = 0$

TestHub · Accepted Answer

For, 0 to L ε = B ℓ eff ​ v = B × 3 ​ x ​ v For, L to 2 L ​ ∣ e m f ∣ = B ( 3 ​ L ​ − 3 ​ x 0 ​ ​ ) v − B 3 ​ 2 x 0 ​ ​ v = 3 ​ B vL ​ − 3 ​ Bvx 0 ​ = Bv [ 3 ​ L ​ − 3 ​ ( x − L ) ] = 3 ​ B v ​ [ L − 3 x + 3 L ] = 3 ​ B v ​ [ 4 L − 3 x ] at x = 3 4 L ​ e m f = 0 ​

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