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)