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A rectangular loop of sides 12cm and 5cm, with its sides parallel to the x-axis and y-axis respectively moves with a velocity of 5cm/s in the positive x axis direction, in a space containing a variable magnetic field in the positive z direction. The field has a gradient of $10^{- 3} T / c m$ along the negative x direction. If the resistance of the loop is $6 m Ω$ , the power dissipated by the loop as heat is______

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$216 \times 10^{- 9} W$

$150 \times 10^{- 9} W$

$900 \times 10^{- 9} W$

$225 \times 10^{- 9} W$

answer is A.

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Detailed Solution

Question Image
$B_{0}$ is the magnetic field at origin
$\frac{d B}{d x} = - \frac{10^{- 3}}{10^{- 2}}$

$\int_{B_{0}}^{B} d B = \int_{0}^{x} 10^{- 1} d x$

$B - B_{0} = - 10^{- 1} x$

$B = (B_{0} - \frac{x}{10})$

Motional emf in AB = 0
Motional emf in CD = 0
Motional emf in AD = $ε_{1} = B_{0} l v$
Magnetic field on rod BC B $= (B_{0} - \frac{(- 12 \times 10^{- 2})}{10})$
Motional emf in BC $= ε_{2} = (B_{0} + \frac{12 \times 10^{- 2}}{10}) l \times v$

$ε_{e q} = ε_{2} - ε_{1} = 300 \times 10^{- 7} V$

power $= \frac{{(ε_{e q})}_{}^{2}}{R} = 150 \times 10^{- 9} W$

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