calculating the induced emf in a coil?

Question

A 28 turn circular coil of radius 3.00 cm and resistance 1.00 is placed in a magnetic field directed perpendicular to the plane of the coil. The magnitude of the magnetic field varies in time according to the expression B = 0.0050t + 0.0380t^2, where t is in seconds and B is in tesla. Calculate the induced emf in the coil at t = 4.20 s.

Answer 1

You use one of Maxwell's equations to solve this:

(line)∫E.dl = -(surface)∫∂B/∂t*dA

Because of the geometry of your problem, these integrals are not difficult. Symmetry dictates that the electric field generated by the changing magnetic field is the uniform around the loops in the coil, so (line)∫E*dl is just E*2π*r which is a voltage V induced in each loop of the coil. There are 28 loops (basically they are all in series). so

V=-28*(surface)∫∂B/∂t*dA

Again, since the magnetic field is spacially uniform, the surface integral is merely the area of the coil times ∂B/∂t.
The area of the coil is π*r^2, so

V =- 28*π*r^2*∂B/∂t

You are given B(t) = 0.0050*t + 0.030*t^2; then

∂B/∂t = 0.0050 + 0.060*t. Putting this into the above equation for V you get

V = -28*π*r^2*(0.0050 + 0.060*t)

Plug in the values for r = 3.00cm and t = 4.2s to get V. Watch the units (convert cm to m to be consistent with tesla units which are weber/m^2).