Final answer:
To induce a current of 0.392 A in the coil with the given parameters using Faraday's law, we calculate the rate of change of the magnetic field (dB/dt) that is required. The resulting mathematical expression includes the current, the total resistance of the coil, the number of turns, and the area of the coil.
Step-by-step explanation:
The question requires using Faraday's law of electromagnetic induction to determine the rate of change of magnetic field (dB/dt) necessary to induce a specified current (0.392 A) in a coil with given dimensions (5 cm by 8 cm), number of turns (147), and total resistance (12.7 Ω).
To solve this problem, we apply Faraday's law which states that the induced electromotive force (emf) in a coil is equal to the negative of the number of turns times the rate of change of the magnetic flux through the coil. This is represented as emf = -N(dΦ/dt), where emf = IR and Φ = B×A (magnetic flux is the product of magnetic field and the area of the coil).
We are given the current (I) and resistance (R), and we can calculate the area (A) as 5 cm × 8 cm = 0.05 m × 0.08 m. Since the field is perpendicular to the coil, the angle does not affect our calculation, and the magnetic flux simplifies to Φ = BA. We must therefore find dB/dt so that -N(dB/dt)A = IR. Rearranging and solving for dB/dt gives us dB/dt = -IR/(NA). Plugging in the values: dB/dt = -(0.392 A × 12.7 Ω)/(147 turns × 0.05 m × 0.08 m), we can calculate the required rate of change of the magnetic field.