Electromagnetic Induction · Question 4

A conducting square loop of side L, mass M and resistance R is moving in the XY plane with its edges parallel to the X and Y axes. The region y 0 has a uniform magnetic field, B=B0 k. The magnetic field is zero everywhere else. At time t=0, the loop starts to enter the magnetic field with an initial velocity v0 j m/s, as shown in the figure. Considering the quantity K= B02L2RM in appropriate units, ignoring self-inductance of the loop and gravity, which of the following statements is/are correct: [figure]

1. If v0=1.5KL, the loop will stop before it enters completely inside the region of magnetic field.
2. When the complete loop is inside the region of magnetic field, the net force acting on the loop is zero.
3. If v0= KL10, the loop comes to rest at t= ( 1K ) ( 52 ).
4. If v0=3KL, the complete loop enters inside the region of magnetic field at time t= ( 1K ) ( 32 ).

Answer

(2) When the complete loop is inside the region of magnetic field, the net force acting on the loop is zero., (4) If v0=3KL, the complete loop enters inside the region of magnetic field at time t= ( 1K ) ( 32 ).

Solution

While entering: =B0Lv, F=- B02L2vR, so dvdt=-Kv v=v0e-Kt and dvdx=-K v=v0-Kx.

The loop enters completely only if v0>KL (penetration xmax= v0K).

(A) v0=1.5KL>KL: it enters completely — false.

(B) Fully inside a uniform field: flux constant, no current, net force zero — true.

(C) v0= KL10: v=v0e-Kt never becomes zero at a finite time — false.

(D) x= v0K(1-e-Kt)=L with v0=3KL: 1-e-Kt= 13 t= 1K 32 — true.

Answer: (B), (D)