Ampere's lawmodified 20110910 A changing electric flux through the circular area of a plate capacitor generates a loop magnetic field. Red is transformed into green.

**Ampere's law and the electromagnetic wave, E→B:**

Our **derivation of Ampere's law** is very similar to Faraday's law. The rate of change of E is 4*pi*E times the frequency of the wave.

**d(E)/dt = 4*pi*E*frequency,** kg*m/(A*s^{4}) volts/(meter*seconds)

**d(E)/dt = 4*pi*B*c*frequency,** E = B*c, in an electromagnetic wave. See appendix 2.

**d(E)/dt = 4*pi*B*c*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.

**2*B = r/c**^{2}*d(E)/dt, kg/(A*s^{2}) Teslas, collected terms

**2*pi*r*B = pi*r**^{2}/c^{2}*d(E)/dt, multiply by (pi*r)

**2*pi*r*B = 1/c**^{2}*d(E*pi*r^{2})/dt, or **B*ds = 1/c**^{2}*d(_{E})/dt, Ampere's law, **B = kg*m/(A*s**^{2}) = kg*m/(s^{2}) * s/m * 1/(A*s), force per velocity per charge or force/(amps*meters).

**Force = B*q*v = B*A*s*m/s = B*A*m.** v is velocity.

Toroidal green B times the circumference of the loop equals one over c squared times the rate of change of the poloidal electrical flux of red E times the area of the loop. Ampere's law.

**2*pi*r*B = e0*u0*d(E*pi*r**^{2})/dt, or **B*ds = e0*u0*d(**_{E})/dt, Ampere's law. 1/c^{2} = e0*u0.

**2*pi*r*B/u0 = e0*d(E*pi*r**^{2})/dt, amps or **1/u0*B*ds = e0*d(**_{E})/dt, amps. The left hand side of the equation, the toroidal amps, due to B, in the loop equals the right hand side of the equation, the poloidal flux of amps, due to E, through the area of the loop. The right hand side of the equations are Maxwell's displacement current. We will use this, amps equals amps form, along the wavefront. This is the form of Ampere's law used in the Ring Electron.

**Integrals of Ampere's law:**

B*ds = 1/c^{2}*d(_{E})/dt

**B*ds = 2*pi*r*B,** The line integral around the curve equals the circumference of the loop times B.

**d(**_{E})/dt = d(E*pi*r^{2})/dt, The rate of change of the electric flux equals the rate of change of E times the area of the loop.

**B*ds = I*u0 + 1/c**^{2}*d(E*da)/dt. I is amps. Maxwell's modification of Ampere's law per Hyperphysics or per Wiki

**B*ds = I*u0 + e0*u0*d(**_{E})/dt. Substituted e0*u0 = 1/c^{2}. The above is written more clearly.

**1/u0*B*ds = I + e0*d(**_{E})/dt. Divided by u0.

**2*pi*r*B/u0 = I + e0*d(E*pi*r**^{2})/dt = amps. These equations are written much more clearly without the integral and flux symbols. This makes them more accessible to a larger audience but the witch doctor rarely wants to share his tricks. Wiki is especially subject to experts writing in the code of their trade with no thought to a larger audience. I could not find these these simpler equations on the Internet. They are shown in the textbooks I reference. The right hand side of the equation is Maxwell's displacement current. The amps in the loop equals the flux of amps through the area of the loop.