A rectangular loop of sides 12 cm and 5 cm, with its sides parallel to the x-axis and y-axis respectively, moves with a velocity of 5 cm/s in the positive x-axis direction, in a space containing a variable magnetic field in the positive z-direction. The f

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Solution

Q.
A rectangular loop of sides 12 cm and 5 cm, with its sides parallel to the x-axis and y-axis respectively, moves with a velocity of 5 cm/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/cm along the negative x-direction and it is decreasing with time at the rate of $10^{- 3}$ T/s. If the resistance of the loop is 6 $mΩ$ , the power dissipated by the loop as heat is ______ $\times 10^{- 9}$ W. [2024]

Ans.
(216)

$B_{0} is the magnetic field at origin$

$\frac{d B}{d x} = - \frac{10^{- 3}}{10^{- 2}} \Rightarrow \int_{B_{0}}^{B} d B = - \int_{0}^{x} 10^{- 1} d x$

$B - B_{0} = - 10^{- 1} x \Rightarrow B = (B_{0} - \frac{x}{10})$

$Motional emf in A B = 0$

$Motional emf in C D = 0$

$Motional emf in A D = ε_{1} = B_{0} l v$

$Magnetic field on rod B C,$

$B = (B_{0} - \frac{(- 12 \times 10^{- 2})}{10})$

$Motional emf in B C = ε_{2} = (B_{0} + \frac{12 \times 10^{- 2}}{10}) l \times v$

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

$For time variation$

$(ε_{eq})' = A \frac{d B}{d t} = 60 \times 10^{- 7} V$

$(ε_{eq})_{net} = ε_{eq} + (ε_{eq})' = 360 \times 10^{- 7} V$

$Power = \frac{{(ε_{eq})}_{n e t}^{2}}{R} = 216 \times 10^{- 9} W$

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