A conducting square loop of side L , mass M and resistance R is moving in the X Y plane with its edges parallel to the X and Y axes. The region y ≥ 0 has a uniform magnetic field, B = B 0 k . The ma

Question

A conducting square loop of side L , mass M and resistance R is moving in the X Y plane with its edges parallel to the X and Y axes. The region y ≥ 0 has a uniform magnetic field, B = B 0 ​ k . The magnetic field is zero everywhere else. At time t = 0 , the loop starts to enter the magnetic field with an initial velocity v 0 ​  ^ ​ m / s , as shown in the figure. Considering the quantity K = RM B 0 2 ​ L 2 ​ in appropriate units, ignoring self-inductance of the loop and gravity, which of the following statements is/are correct:

TestHub · Accepted Answer

⇒ dt − d ϕ ​ = dt d ​ ( B 0 ​ × ℓ × y ) = BV ℓ ​ F = B ( i ^ ) ( ℓ ) ( − j ^ ​ ) ma = − B 0 ​ [ R B 0 ​ V ℓ ​ ] ( ℓ ) a = − mR B 0 2 ​ ℓ 2 V ​ ​ ​ Also K = RM B 0 2 ​ ℓ 2 V ​ So [ a = − k v ] d t d v ​ = − k v ∫ v 0 ​ v ​ d t d v ​ = ∫ 0 t ​ − k d t ℓ n v 0 ​ v ​ = − k t ​ [ v = v 0 ​ e − kt ] [ v = v 0 ​ e − kt ] … . ( i ) dt dx ​ = v 0 ​ e − kt ( x ≤ ℓ ) ​ ∫ 0 x ​ dx = ∫ 0 t ​ v 0 ​ e − kt dt = k v 0 ​ ​ ( 1 − e − kt ) ​ When x = ℓ ℓ = k V 0 ​ ​ ( 1 − e − kt ) Option (D) ( v 0 ​ = 3 k ℓ ) ​ ℓ = k 3 k ℓ ​ ( 1 − e − kt ) 3 1 ​ = 1 − e − kt f 3 2 ​ = 2 e − kt − kt = 8 n ( 3 2 ​ ) t = k 1 ​ ℓ n ( 3 2 ​ ) ​ Complete loop will enter at t = k 1 ​ ln ( 3 2 ​ ) Option (B) dt d ϕ ​ = 0 , e ​ ​ = 0 , i = 0 , F = 0 Ans. B,D)

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