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 init

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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= B_0^2 L^2 R M in appropriate units, ignoring self-inductance of the loop and gravity, which of the following statements is/are correct:

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Correct answer: D. If v_0=3 K L, the complete loop enters inside the region of magnetic field at time t= ( 1 K ) ( 3 2 ).

- d dt = d dt ( B _0 y )= BV aligned & F = B ( i )( )(- j ) \ & ma =- B _0 [ ~B _0 ~V R ]( ) \ & a =- B _0^2 ^2 ~V mR aligned aligned & Also K= B_0^2 ^2 V R M \ & So [a=-k v] \ & d v d t =-k v \ & _ v_0 ^v d v d t = _0^t-k d t \ & n v v_0 =-k t aligned [ v = v _0 e ^ - kt ] [ v = v _0 e ^ - kt ] .( i ) dx dt = v _0 e ^ - kt ( x ) aligned & _0^ x dx = _0^ t v _0 e ^ - kt dt \ & = v _0 k (1- e ^ - kt ) aligned When x= = V _0 k (1- e ^ - kt ) Option (D) ( v _0=3 k ) aligned & = 3 k k (1- e ^ - kt ) \ & 1 3 =1- e ^ - kt \ & f 2 3 =2 e ^ - kt \ & - kt =8 n ( 2 3 ) \ & t = 1 k n ( 2 3 ) aligned Complete loop will enter at t = 1 k ( 2 3 ) Option (B) d dt =0, e =0, i =0, ~F =0 Ans. B,D)

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