A flat circular loop of wire has radius 8 cm and resistance of 10-4 Ω. A uniform magnetic field is directed perpendicular to the plane of the loop, and its magnitude is increasing linearly with time, t, so that B = 13t (T) The electrical power, in watts, generated in the loop is to zero decimal places:

The power generated in a loop due to a changing magnetic field is given by Faraday's law of electromagnetic induction. The emf (ε) induced in the loop is given by:

ε = -N * dΦ/dt

where: N is the number of turns in the loop (which is 1 in this case, since it's a single loop), Φ is the magnetic flux through the loop, and dΦ/dt is the rate of change of the magnetic flux.

The magnetic flux (Φ) through a loop is given by:

Φ = B * A

where: B is the magnetic field, and A is the area of the loop.

Substituting B = 13t and A = πr^2 (where r is the radius of the loop) into the flux equation gives:

Φ = 13t * π * (0.08 m)^2

Differentiating this with respect to time gives:

dΦ/dt = 13 * π * (0.08 m)^2

Substituting this into the emf equation gives:

ε = -1 * 13 * π * (0.08 m)^2 = -0.2624 V

The negative sign indicates that the emf is in the opposite direction to the change in magnetic field, but we're only interested in the magnitude of the emf, so we can ignore the negative sign.

The power (P) generated in the loop is given by:

P = ε^2 / R

where: ε is the emf, and R is the resistance of the loop.

Substituting ε = 0.2624 V and R = 10^-4 Ω into the power equation gives:

P = (0.2624 V)^2 / 10^-4 Ω = 688.5 W

So, the power generated in the loop is approximately 688.5 W. To zero decimal places, this is 689 W.