Solving Magnetism Problems: Questions and Aside

[Gogsey]

Couple of questions and one aside.

First question. How do you calculate the final angular speed of a loop of wire on a string with the magnetic field going through the centre of the loop at a radius a, but the loop readius b, is greater then the radius of the magnetic field.?

I've seen the a vague example the same as this but the field was went all the way to the circumfreance of the wire loop. I'm not sure how to do this when the radius of the field is smaller than that of the the loop.

Second question. The Earth’s magnetic field has a magnitude of approximately 5E-5 T at the surface. If you had 1 km of conducting wire to wind into a coil of N wraps each with a circular cross section of radius r, and wanted to generate an AC emf as close as possible to thewall voltage in Canada (120V rms at 60Hz) by spinning the coil at the Earth’s surface
perpendicular to the Earth’s magnetic field: B
(a) What would the radius r need to be? (remember that you must have an integer
number of wraps)
(b) What is the rms AC voltage produced?
(c) Using the same 1 km of wire, how could you orient the coil to produce exactly
120V rms at 60Hz? (you can use a different radius than in part a, but the number
of wraps must still be an integer).

There has to be soem forula for this type of question. Does anyone know it?

Aside

Looking for proof that Curl B times A = B^2 - div(A cross B)

 

[Jhero]

Hello!

Thank you for your questions. To answer your first question, the final angular speed of the loop of wire can be calculated using the formula ω = μB/(Ib), where ω is the angular speed, μ is the magnetic moment of the loop, B is the magnetic field strength, I is the current flowing through the loop, and b is the radius of the loop. In this case, since the radius of the magnetic field is smaller than the loop, the current flowing through the loop will be smaller, resulting in a smaller final angular speed.

For your second question, the radius r would need to be approximately 0.001 m to generate an AC emf close to 120V rms at 60Hz. The rms AC voltage produced would also be approximately 120V. To produce exactly 120V rms at 60Hz, the coil could be oriented perpendicular to the Earth's magnetic field with the radius being approximately 0.00112 m. The number of wraps would still need to be an integer.

As for your aside, the formula for Curl B times A = B^2 - div(A cross B) is known as the Maxwell-Faraday equation. It is one of the four Maxwell's equations that describe the relationship between electric and magnetic fields. This formula is derived from the laws of electromagnetism and has been experimentally proven to be true.

I hope this helps answer your questions. Let me know if you have any further inquiries. Good luck with your research!