Faraday's law: A coil of 160 turns and area 0.20 m² is placed with its axis parallel to a magnetic field of initial magnitude 0.40 T. The magnetic field changes uniformly from 0.40 T in the +x direction to 0.40 T in the -x direction in 2.0 s. If the resistance of the coil is 16 Ω, at what rate is power generated in the coil?

Question

Quizplus · Accepted Answer

Faraday's law states that the EMF induced in a coil is proportional to the rate of change of magnetic flux through the coil.

EMF = - N dΦ/dt

where N is the number of turns in the coil, Φ is the magnetic flux through the coil, and t is time.

We can find the magnetic flux through the coil by multiplying the magnetic field by the area of the coil:

Φ = BA

where B is the magnetic field and A is the area of the coil.

The change in magnetic field is:

ΔB = -0.80 T

The time taken for the magnetic field to change is:

Δt = 2.0 s

The average magnetic field during the change is:

B_avg = (0.40 T + 0.00 T)/2 = 0.20 T

The initial magnetic flux through the coil is:

Φ_1 = B_avg A = (0.20 T)(0.20 m^2) = 0.040 Wb

The final magnetic flux through the coil is:

Φ_2 = (-B_avg) A = (-0.20 T)(0.20 m^2) = -0.040 Wb

The change in magnetic flux through the coil is:

ΔΦ = Φ_2 - Φ_1 = -0.080 Wb

The rate of change of magnetic flux is:

dΦ/dt = (ΔΦ)/(Δt) = (-0.080 Wb)/(2.0 s) = -0.040 Wb/s

The EMF induced in the coil is:

EMF = - N dΦ/dt = (-160)(-0.040 Wb/s) = 6.4 V

The current in the coil is:

I = EMF/R = (6.4 V)/(16 Ω) = 0.40 A

The power generated in the coil is:

P = I^2 R = (0.40 A)^2 (16 Ω) = 2.56 W

The rate at which power is generated is:

dP/dt = I^2 (dR/dt) + 2IR (dI/dt)

We are told that the resistance of the coil is constant, so dR/dt = 0. We are also told that the magnetic field changes at a constant rate, so dI/dt = 0.

Therefore, the rate at which power is generated is simply:

dP/dt = I^2 (dR/dt) + 2IR (dI/dt) = 2IR (dI/dt) = 2(0.40 A)(16 Ω)(0) = 0 W

The power generated in the coil is constant over time, so the answer is:

dP/dt = 2.56 W / 2.0 s = 1.28 W/s = 1.28 W/s × 3600 s/h = 4608 W/h = 4.6 kW/h

However, the question asks for the rate at which power is generated, which is given by:

dP/dt = 2.56 W / 2.0 s = 1.28 W/s

Therefore, the best choice is B (10 W).

Faraday's Law: a Coil of 160 Turns and Area 0