Final answer:
To induce an average emf of 10,000 V in a coil rotated one-fourth of a revolution within 4.17 ms, the needed magnetic field strength is calculated to be 0.477 T, which is feasible with today's magnet technology.
Step-by-step explanation:
To find the magnetic field strength needed to induce an average emf of 10,000 V when a coil is spun, we can use Faraday's law of electromagnetic induction. The average emf (E) generated in a coil of N turns and area A experiencing a change in magnetic flux (ΔΨ) over a time Δt is given by E = -N(ΔΨ/Δt). Since the coil is rotated one-fourth of a revolution (Π/2 radians) with respect to the magnetic field, the change in flux is equal to BA(cos0 - cos(Π/2)), which simplifies to BA since cos(Π/2) is zero.
The area of the coil (A) is Πr^2, where r is the radius of the coil. With a radius of 0.250 m, the area becomes Π(0.250 m)^2. The number of turns (N) is 500, and the time (Δt) is 4.17 ms or 4.17×10^-3 s. Substituting these values into Faraday's law and solving for the magnetic field (B), we find the required magnetic field strength.
Thus, using the formula E = N(Πr^2)B(Π/2) / Δt, and rearranging to solve for B, we find that the magnetic field strength required is 0.477 T. This value is reasonable and can be obtained with current magnet technology.