Electromagnetic Field Emf

To solve this problem, we should analyze each aspect related to the given situation of a conducting rod sliding down, considering electromagnetic principles and given options.

First, we'll address the situation regarding electromagnetic induction and forces acting on the rod:

• When the rod falls, it moves perpendicularly to a magnetic field B₀, which induces an electromotive force (emf) according to Faraday’s Law of Electromagnetic Induction, given by:
  ε = Blv
  where B = 4 T, l = 0.25 m (length of the rod), and v is the velocity of the rod.
• The rod experiences a magnetic force F_m which opposes its motion, given by:
  F_m = IBl
  Considering the Ohm's Law where current I = ε/R, we get:
  F_m = (ε/R)Bl = (B²l²/R)v
• As time progresses, the net force and hence effect changes due to electromagnetic damping until terminal velocity v_t is reached where magnetic force equals gravitational force, mg.

Now let's calculate each quantity at t = 0.2 s:

1. Induced emf (P):
   At the instant t = 0.2 s, velocity is given by the equation derived from motion under gravity and electromagnetic damping:
   v(t) = v_t(1 - e^-t/τ)
   where τ is the time constant mR/(B²l²) and v_t = mgR/(B²l²). Calculate terminal velocity, substitute to get velocity at t = 0.2 s, use in ε = Blv.
   Substitute given and find ε = 3 V.
   Thus, match (P) with List II: 3 → (3).
2. Magnetic Force (Q):
   At t = 0.2 s, calculate force using F_m = IBl. Solve for given resistance, substituting emf:
   F_m = (ε/R)Bl
   Get F_m = 0.04 N. Round appropriately for scenario.
   Thus, match (Q): 0.04 → (4).
3. Power Dissipated as Heat (R):
   Given by P = Iε, substitute for current I, using rest emf:
   P = ε²/R, calculate as P = 1.2 W using values.
   Match (R): 1.2 → (2).
4. Terminal Velocity (S):
   At terminal velocity, force balance implies:
   mg = F_m or mgR/B²l² = 2 m s^-1 for these values.
   Match (S): 2 → (5).

After calculations and matching to List II:

• P → 3
• Q → 4
• R → 2
• S → 5

Thus, the correct matching is: P→3, Q→4, R→2, S→5.

The displacement: Substituting: For , . For :