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 - 1 in the positive x axis direction, in a space containing

Question

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 - 1 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 - 1 along the negative x direction and it is decreasing with time at the rate of 10 - 3 T s - 1 . If the resistance of the loop is 6 m Ω , the power dissipated by the loop as heat is ______ × 10 - 9 W .

TestHub · Accepted Answer

Let B 0 is the magnetic field at origin. Then, d B d x = - 10 - 3 10 - 2 ⇒ ∫ B 0 B d B = - ∫ 0 x 10 - 1 d x ⇒ B - B 0 = - 10 - 1 x ⇒ B = B 0 - 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 0 - - 12 × 10 - 2 10 Motional emf in B C = ε 2 = B 0 + 12 × 10 - 2 10 ℓ × v Net equivalent emf ε eq = ε 2 - ε 1 = 300 × 10 - 7 V For time variation ε e q ' = A d B d t = 60 × 10 - 7 V ε eq net = ε eq + ε eq ' = 360 × 10 - 7 V Power r = ε eq net 2 R = 216 × 10 - 9 W

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