​​Calculating Electromagnetic Force Density ​​​Mathematical Proof of Local Intrinsic Field Equilibrium (LIFE) in a Laser Beam
,propagatinginthez-directionwiththespeedoflight:c=
1
ϵ0μ0
"In classical electrodynamics, a laser beam propagating in a vacuum is assumed to be divergence-free. However, this assumption ignores the outward transverse radiation pressure (the gradient of the energy density).This Wolfram Mathematica calculation proves that a stable laser beam can only exist if this outward pressure is perfectly counterbalanced by an inward electromagnetic force density, generated by the Coulomb and Lorentz forces.Step 1: We define the Electric (E) and Magnetic (H) field intensities of a Gaussian laser beam propagating in the z-direction at the speed of light.Step 2: We substitute these into the force density equation (derived from Newton's Third Law: ∑ F = 0
[N/
3
m
]Step 3: The script calculates the resulting force densities in the x, y, and z directions. The output demonstrates that perfect mechanical equilibrium ∑ F = 0
[N/
3
m
] is achieved only when the wave propagates at exactly c.​​
​​
  LIFE (Local Intrinsic Field Equilibrium) Field EquationMinkowski-Maxwell Equation​​
f-
1
2
c
∂(E×H)
∂t
+
ϵ
0
E(∇·E)-
ϵ
0
E×(∇×E)+
μ
0
H(∇·H)-
μ
0
H×(∇×H)0​​
in which:      
and for the electromagnetic energy density :   
ε
0
 ∇. 
¯
E
=
ρ
E
3
 Electric Charge Density  [C/ m

               
 
μ
0
 ∇. 
¯
Η

 = 
ρ
M
3
 Magnetic Flux Density [Vs/ m

3
 or [Wb/ m
] 
​​
Electric Energy Density:           w
E
   =  
1
2
 
ε
0
2
 E
Magnetic Energy Density:         w
M
  =  
1
2
 
μ
0
2
 H
 
        
​​
In[]:=
Needs["DifferentialEquations`NDSolveProblems`"]
In[]:=
Needs["DifferentialEquations`NDSolveUtilities`"]
In[]:=
Needs["DifferentialEquations`InterpolatingFunctionAnatomy`"]
In[]:=
ϵ0=.
In[]:=
μ0=.
In[]:=
x=.
In[]:=
y=.
In[]:=
z=.
In[]:=
t=.
In[]:=
Get["VectorAnalysis`"]
General
:VectorAnalysis` is now obsolete. The legacy version being loaded may conflict with current functionality. See the Compatibility Guide for updating information.
In[]:=
InverseFunctions->True
Out[]=
InverseFunctionsTrue
In[]:=
Needs["DifferentialEquations`NDSolveProblems`"]
In[]:=
Needs["DifferentialEquations`NDSolveUtilities`"]
In[]:=
Needs["DifferentialEquations`InterpolatingFunctionAnatomy`"]
In[]:=
SetCoordinates[Cartesian[x,y,z]]
Out[]=
Cartesian[x,y,z]
In[]:=
{Coordinates[Cartesian],CoordinateRanges[Cartesian]}
Out[]=
{{x,y,z},{-∞<x<∞,-∞<y<∞,-∞<z<∞}}
In[]:=
K2=.
In[]:=
The Propagation of a Laser Beam
In[]:=
In[]:=
WhereMaxwellfails:
In[]:=
f[x,y]=g[x,y]
-K2
2
x
+
2
y

Out[]=
-K2
2
x
+
2
y

g[x,y]
In[]:=
f[x,y]=g[x,y]
-K2
2
x
+
2
y

Out[]=
-K2
2
x
+
2
y

g[x,y]
Theelectricfieldstrengthcomponentofthelaserbeam,expressedasev,​​propagatinginthez-direction.​​TheconstantK1isanarbitraryconstantthatallowsforanarbitraryphaseshift​​oftheelectricfield
ev={f[x,y]*g[t-(K1/z+1)*z*Sqrt[ϵ0]*Sqrt[μ0]],0,0}​​
-K2
2
x
+
2
y

gt-1+
K1
z
z
ϵ0
μ0
g[x,y],0,0
The magnetic field strength component of the laser beam, expressed as mv, propagating in the z - direction . The constant K1 is an arbitrary constant that allows for an arbitrary phase shift of the magnetic field​​
Gaussian intensity profile for a laser beam . De function g[x, y] represents an arbitrary function.