 A changing magnetic flux through a circular area generates a loop electric field which accelerates the electrons in a Betatron. Green is transformed into red. Faraday's law and the electromagnetic wave, +B→-E or -B→+E
E and B are sine and cosine waves because they are ninety degrees out of phase. B is the cosine since it has a sign change in its derivative.
Lenz's law comes from the sign change in the derivative. d(cos)/dt = -sin or d(-cos)/dt = sin. Faraday's law is applied twice per wavelength so there is no net sign change per wavelength since, -1*-1 = 1. This sign change does not occur in Ampere's law, noting d(sin)/dt = cos or d(-sin)/dt = -cos, does not have a sign change.
Our derivation of Faraday's law starts with the idea that the rate of change of B is 4*pi*B times the frequency of the wave.
d(B)/dt = 4*pi*B*frequency, kg/(A*s3) Teslas/second
d(B)/dt = 4*pi*B *c/(2*pi*r), frequency = c/wavelength = c/(2*pi*r). Frequency measures how many loops something, moving at c, does in the ring 2*pi*r per second.
d(B)/dt = 4*pi*-E/(2*pi*r), B*c = -E, in an electromagnetic wave. See appendix 2.
2*-E = r*d(B)/dt, group terms, volts/meter = kg*m/(A*s3) = kg*m/s2 * 1/(A*s) = force/charge
2*pi*r*-E = pi*r2*d(B)/dt, multiply by (pi*r)
2*pi*r*-E = d(B*pi*r2)/dt, or -E*ds = d( B)/dt, Faraday's law. volts = kg*m2/(A*s3) = amps*resistance = energy/charge = watts/amps. Toroidal red -E times the circumference of the loop equals the rate of change of the poloidal magnetic flux of green B times the area of the loop. Faraday's law.
When we divide the voltage on both sides of Faraday's law by the resistance of a loop or coil of wire then we get Ohm's law: volts/resistance = amps.
2*pi*r*-E /resistance = (d(B*pi*r2)/dt) /resistance, amps or E*ds / resistance = d( B)/dt / resistance, amps. The toroidal amps in the loop equals the poloidal flux of amps through the area of the loop. This is the reversing current seen on a galvanometer, when hooked to a coil of wire, while a magnet is inserted and removed from the coil of wire. This classic experiment is strong direct evidence for Faraday's law. E*ds = -d( B)/dt, integral form of Faraday's law. Hyperphysics or Wiki E*ds = 2*pi*r*-E, the line integral of the electric field equals the circumference of the loop times E.
d( B)/dt = d(B*pi*r2)/dt, the rate of change of the magnetic flux equals the rate of change of B times the area of the loop.