In a cylindrical magnetic field $B$ is changing as $B = B_{0} + α t$ . The value of $\int \vec{E .} d \vec{l}$ for the path of BOA, as shown in the figure is $k (a b α)$ , then k is _________
(Line AOB is elliptical with semi major and minor axis $a$ & $b$ respectively and O is the centre of the magnetic field region. $\vec{E}$ is the induced electric field in the region)

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answer is 0.57.

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Detailed Solution

$ε_{A B} = \int_{- a}^{+ a} E co s θ d x = \int_{- a}^{+ a} (\frac{\sqrt{x^{2} + b^{2}}}{2} α) (\frac{b}{\sqrt{x^{2} + b^{2}}}) d x = \frac{b α}{2} (2 a)$

$ε_{B O A B} = A \frac{d B}{d t} = (\frac{π a b}{2}) (α)$

$ε_{B O A} = ε_{B O A B} - ε_{A B} = a b α (\frac{π}{2} - 1) = 0.57 a b α$

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