A Derivation of the Lorentz Form of the Maxwell Equations, 6/14/98

 

I.  The Four Maxwell Equations

 


    1.  Coulomb’s Law:                                                              2.  No magnetic monopoles implies that:

         Coulomb’s Law states what Coulomb

discovered experimentally in the early 1800’s.

He found that the force between two electrically-

Charged spheres was proportional to the product

of the strength of their charges, and inversely

proportional to the square of the distance

between them. In other words, it was like illumination,

which is proportional to the brightness of the light,

and

 

                                                                                            

 

 


     3..  Lenz’ Law:                                                                    4.  Ampere’s Law:

 


                                                                       

 

 

 


Now the divergence of a vector field,is everywhere zero (Equation 2) if and only that field is the curl of another vector field. Thus Equation 2’ becomes:


 


                                                                 

 

In other words, the magnetic field, is generated by the curl of the “magnetic vector potential” field,

 


Shifting our attention to Lenz’ Law (Equation 3), we can replacewith Then Equation 3’ becomes:

 


But inasmuch as partial derivatives can be taken in any order (unlike total derivatives), Equation 3’ may be rewritten as:

                                                                             

But the curl of a vector field,  is everywhere zero (Equation 3) if and only if that field is the gradient of some other vector field j. Thus,  or

                                                                      

 

Now let’s look at Coulomb’s Law (Equation 1). Substituting the above expression fortransforms Equation 1 into:

                       Equation 1’                           

But since we can take partial derivatives in any order (as we did in Equation 3’), and since the calculation of the divergence involves only partial derivatives, we can rewrite Equation 1’ as                                   

 

                                                                       

Now the expression is the Laplacian operator, which, in Cartesian coordinates, is given by

 

                                                                            .

 

At this point, we invoke the “gauge condition” that  which allows us to replace  with

 

Equation 1’ then becomes

                                                                           .

 

This is now the first of our wave equations.

 

Turning to Equation 4, as usual, we replacewith


But the vector identity  allows us to rewrite Equation 4 as:

 


                                                                          .

Substituting  for  in Equation 4 gives us

                                                                         

 

Regrouping terms, Equation 4 becomes:

                                                                         

 

                                                                         

Invoking the gauge condition and remembering thatis the Laplacian operator, we have:

                                                                         , or

 

                                                                             .

 

Gathering up the equations for the electric potential, j, and for the magnetic vector potential, we have:

 

 

 

                                                            

 

                                                           

 

                                                              

 

                                                              

 

The Meaning of , the Magnetic Vector Potential.

 

.

 

 

 is a vector pointing in the same direction as the current vector Since it falls off as  it will generate vortices (curl), giving rise to afield.