Induced EMF in a Loop from Magnetic Bullet – Physics Solution

Physics Problem: Magnetic Induction in a Loop

Question:

A magnetized bullet is fired through the center of a loop of wire with radius r = 30 cm. At its peak, the average magnetic field in the plane of the loop and inside the loop is B_avg = 0.01 T. The magnetic field increases from B = 0 to B = 0.01 T and returns to B = 0 in a time interval Δt = 0.0065 sec.

Find:

(a) The average induced emf in the loop.
(b) The average current in the loop, given that its total resistance is R = 0.001 Ω.
(c) The total energy dissipated in the loop.

Solution:

Step 1: Area of the Loop

Radius: r = 30 cm = 0.30 m

A = πr² = π(0.30)² ≈ 0.2827 m²

Step 2: Use Faraday’s Law to Find Average Induced EMF

Faraday’s Law:

ε_avg = ΔΦ / Δt = A × ΔB / Δt

ε_avg = (0.2827 × 0.01) / 0.0065 ≈ 0.4357 V

Step 3: Use Ohm’s Law to Find Average Current

Ohm’s Law:

I_avg = ε / R = 0.4357 / 0.001 = 435.7 A

Step 4: Total Energy Dissipated in the Loop

Use:

E = I² × R × Δt = (435.7)² × 0.001 × 0.0065 ≈ 1.234 J

✅ Final Answers:
(a) Average induced emf: 0.436 V
(b) Average current: 435.7 A
(c) Total energy dissipated: 1.23 J

learnlyfly