Friday, May 30, 2014

Whacko and the Free Photons

I don't have Guass's Equations but, for a charge/dipole in motion,

B=iEx

the negative sign is the result of E pointing towards the negative charge in circular motion.  And the complex imaginary i is the result of rotation π2 anti-clockwise.  And x is in the plane perpendicular to direction x

In case of photons in motion,

B=iEx

These equations prove one thing,  I am a complete WHACKO.  Because,

B=iEx

Bt=i2Ext

Bt=i2Exxxt

Since substituting for B again,

 itEx=i2Ex2xt

Multiplying both sides by i rotates both B and E again by π2 anti-clockwise, into the direction of x, ie xx,

 tEx=2Ex2xt

since xt=c

2Etttx=2Ex2c

2Et21c=2Ex2c

2Et2=2Ex2c2

Which is the wave equation for photons at speed c.  This wave equation was derived from one relationship between B and E,  this suggest that a dipole with one end spinning is all that is required for wave propagation.   In this case of a photon, the positive end is in circular motion behind the negative charge, both moving against a E-field.  Furthermore, for the case of charge/dipole

B=iEx

Bt=i2Ext

Bt=i2Exxxt

Substituting for B again,

 it(Ex)=i2Ex2xt

 itEx=i2Ex2xt

Multiplying both sides by i rotates both B and E again by π2 anti-clockwise, into the direction of x, ie xx and since xt=v

2Etttx=2Ex2v

2Et21v=2Ex2v

2Et2=2Ex2v2

This is the wave equation for electromagnetic wave at speed v.  In this case, the negative end of the dipole is in circular motion behind the positive charge,  both moving in the direction of the E field.  The initial negative sign cancels when B is substituted again into the derivation.  The negative sign, however indicates that 

B=iEx

obeys Lenz's Law, that work is done against the change taking place; that electromagnetic wave established between two point requires continued energy input.  The wave disappears when power is lost.  In the case of photons

B=iEx

runs away.  Photons continues to propagate after source power has been turned off.  Photons are free from the source once it leaves the source.