A rod of length ℓ = 1 m and resistance R = 10 Ω is being pulled with a constant velocity of 4 m/s towards right. At t = 0 the rod is 1 m away from the left end. The magnetic field is B = ( 1 − 3 t ) t

Question

A rod of length ℓ = 1 m and resistance R = 10 Ω is being pulled with a constant velocity of 4 m/s towards right. At t = 0 the rod is 1 m away from the left end. The magnetic field is B = ( 1 − 3 t ) tesla. Find the work done (in J) by the external agent to move the rod with constant velocity for first 5 sec. The rest of circuit is resistance less. Neglect friction.

TestHub · Accepted Answer

At time t , B = ( 1 − 3 t ) T and the rod is at x = ( 1 + 4 t ) m from the left end. ϕ = B ℓ x = ( 1 − 3 t ) ( 1 + 4 t ) = 1 + t − 12 t 2 ε = d t d ϕ ​ = 1 − 24 t ; i = R ε ​ = 10 1 − 24 t ​ F e x t ​ = B i ℓ = 10 ( 1 − 3 t ) ( 1 − 24 t ) ​ = 10 1 − 27 t + 72 t 2 ​ W = ∫ 0 5 ​ F e x t ​ v d t = 10 4 ​ [ t + 24 t 3 − 13.5 t 2 ] 0 5 ​ W = 0.4 ( 5 + 3000 − 337.5 ) = 1067 J

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