A conducting rod of length L rotates with a constant angular velocity ω about one of its ends (pivot at x = 0 ) in a magnetic field given by B = B 0 x k ^ , where B 0 is a constant and x is the di

Question

A conducting rod of length L rotates with a constant angular velocity ω about one of its ends (pivot at x = 0 ) in a magnetic field given by B = B 0 ​ x k ^ , where B 0 ​ is a constant and x is the distance from the pivot. The rod lies in the x y -plane. Match the quantities in List-I with their corresponding expressions in List-II. List - I List - II (P) EMF induced across the rod (1) 9 R B 0 2 ​ ω L 6 ​ (Q) Current in the rod if its ends are connected by an external resistor R (2) 3 B 0 ​ ω L 3 ​ (R) Torque required to maintain constant angular velocity if the ends are connected by R (3) 9 R B 0 2 ​ ω 2 L 6 ​ (S) Power dissipated in the external resistor R (4) 3 R B 0 ​ ω L 3 ​

TestHub · Accepted Answer

The velocity of a segment d x at distance x is v = ω x . The induced EMF d E = B v d x = ( B 0 ​ x ) ( ω x ) d x = B 0 ​ ω x 2 d x . Integrating from 0 to L , E = 3 B 0 ​ ω L 3 ​ . The current I = R E ​ = 3 R B 0 ​ ω L 3 ​ . The force on a segment d x is d F = I B d x = I ( B 0 ​ x ) d x . The torque d τ = x d F = x I B 0 ​ x d x . Integrating, τ = ∫ 0 L ​ 3 R B 0 ​ ω L 3 ​ B 0 ​ x 2 d x = 3 R B 0 2 ​ ω L 3 ​ 3 L 3 ​ = 9 R B 0 2 ​ ω L 6 ​ . Power dissipated is P = I 2 R = ( 3 R B 0 ​ ω L 3 ​ ) 2 R = 9 R B 0 2 ​ ω 2 L 6 ​ .

List - I	List - II
(P) EMF induced across the rod	(1) $\frac{B _{0}^{2} ω L ^{6}}{9 R}$
(Q) Current in the rod if its ends are connected by an external resistor $R$	(2) $\frac{B _{0} ω L ^{3}}{3}$
(R) Torque required to maintain constant angular velocity if the ends are connected by $R$	(3) $\frac{B _{0}^{2} ω ^{2} L ^{6}}{9 R}$
(S) Power dissipated in the external resistor $R$	(4) $\frac{B _{0} ω L ^{3}}{3 R}$

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