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Physics-

General

Easy

Question

Shown in the figure is a circular loop of radius r and resistance R. A variable magnetic field of induction $B equals B subscript 0 end subscript e to the power of negative t end exponent$ is established inside the coil. If the key (K) is closed, the electrical power developed right after closing the switch is equal to

$\frac{B_{0}^{2} π r^{2}}{R}$
$\frac{B_{0} 10 r^{3}}{R}$
$\frac{B_{0}^{2} π^{2} r^{4} R}{5}$
$\frac{B_{0}^{2} π^{2} r^{4}}{R}$

The correct answer is: $fraction numerator B subscript 0 end subscript superscript 2 end superscript pi to the power of 2 end exponent r to the power of 4 end exponent over denominator R end fraction$

$P equals fraction numerator e to the power of 2 end exponent over denominator R end fraction semicolon e equals negative fraction numerator d over denominator d t end fraction left parenthesis B A right parenthesis equals A fraction numerator d over denominator d t end fraction left parenthesis B subscript o end subscript e to the power of negative t end exponent right parenthesis equals A B subscript o end subscript e to the power of negative t end exponent$
$rightwards double arrow P equals fraction numerator 1 over denominator R end fraction left parenthesis A B subscript o end subscript e to the power of negative t end exponent right parenthesis to the power of 2 end exponent equals fraction numerator A to the power of 2 end exponent B subscript o end subscript superscript 2 end superscript e to the power of negative 2 t end exponent over denominator R end fraction$
At the time of starting t = 0 so $P equals fraction numerator A to the power of 2 end exponent B subscript o end subscript superscript 2 end superscript over denominator R end fraction$
$rightwards double arrow P equals fraction numerator left parenthesis pi r to the power of 2 end exponent right parenthesis to the power of 2 end exponent B subscript o end subscript superscript 2 end superscript over denominator R end fraction equals fraction numerator B subscript o end subscript superscript 2 end superscript pi to the power of 2 end exponent r to the power of 4 end exponent over denominator R end fraction$

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