Section7: Maxwell's Equations

In this section we consider forms on R^4. The four coordinates are {x,y,z,t}.

Input := 

<<forms.d-defs4.m;

First we define three differential 2-forms F, G and s (for source):

Input := 

F = (Ex[x,y,z,t] d[x] + Ey[x,y,z,t] d[y] + 
Ez[x,y,z,t] d[z])^(c d[t]) +
(Bx[x,y,z,t] d[y]^d[z] + By[x,y,z,t] d[z]^d[x] + 
Bz[x,y,z,t] d[x]^d[y]);

Input := 


G = -1 (Bx[x,y,z,t] d[x] + 
By[x,y,z,t] d[y] + 
Bz[x,y,z,t] d[z])^(c d[t]) +
(Ex[x,y,z,t] d[y]^d[z] + 
Ey[x,y,z,t] d[z]^d[x] + 
Ez[x,y,z,t] d[x]^d[y]);

Input := 

s = (Jx[x,y,z,t] d[y]^d[z] + 
Jy[x,y,z,t] d[z]^d[x] + 
Jz[x,y,z,t] d[x]^d[y])^d[t] - 
rho[x,y,z,t] d[x]^d[y]^d[z];

Next we write down Maxwell's equations:

Input := 

MaxwellEquations = 
{D[Ex[x,y,z,t],x] + D[Ey[x,y,z,t],y] + D[Ez[x,y,z,t],z] ->
   4 Pi rho[x,y,z,t],
 D[Bx[x,y,z,t],x] + D[By[x,y,z,t],y] + D[Bz[x,y,z,t],z] -> 0,
 c D[Ez[x,y,z,t],y] - c D[Ey[x,y,z,t],z] + D[Bx[x,y,z,t],t] -> 0,
 - c D[Ex[x,y,z,t],z] + c D[Ez[x,y,z,t],x] - D[By[x,y,z,t],t] -> 0,
 c D[Ey[x,y,z,t],x] - c D[Ex[x,y,z,t],y] + D[Bz[x,y,z,t],t] -> 0,
 - c D[Bz[x,y,z,t],y] + c D[By[x,y,z,t],z] + D[Ex[x,y,z,t],t] ->
  4 Pi Jx[x,y,z,t],
 c D[Bx[x,y,z,t],z] - c D[Bz[x,y,z,t],x] - D[Ey[x,y,z,t],t] ->
  - 4 Pi Jy[x,y,z,t],
 - c D[By[x,y,z,t],x] + c D[Bx[x,y,z,t],y] + D[Ez[x,y,z,t],t] ->
  4 Pi Jz[x,y,z,t]
};

Then we consider the quantities dF and dG + 4 Pi s:

Input := 

Simplify[(Collect[
d[F],{d[x]^d[y]^d[z],d[t]^d[x]^d[y],d[t]^d[y]^d[z],
      d[t]^d[x]^d[z]}]) /. MaxwellEquations]

Output =

0

Input := 

Collect[d[G] + 4 Pi s,
        {d[x]^d[y]^d[z],d[t]^d[x]^d[y],d[t]^d[y]^d[z],
         d[t]^d[x]^d[z]}] /.MaxwellEquations

Output =

0

Now it can be seen that the 4 Maxwell's equations can be written in the compact form dF = 0 and dG + 4 Pi s = 0. Moreover, the equation of continuity comes out naturally from Poincare's lemma when we d the equation d[G] + 4 Pi s = 0.

Statistical Mechanics: Entropy, Order Parameters, and Complexity, now available at Oxford University Press (USA, Europe).