San José State University

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Thayer Watkins
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The Dynamics of Free
Fields in 3D Space

It was shown in a previous study that Maxwell's equations imply that electromagnetic fields in the absence of charges obey the wave equation

(∂²E/∂t²) = C²∇²E

where C is the speed of light in the medium in which the field exists.

In spherical coordinates (r, θ, φ) the Laplacian ∇² is given by

∇²(r, θ, φ) = (1/r²)(∂/∂r)(r²(∂/∂r) + (1/(r²sin(θ))(∂/∂θ)(sin(θ)(∂/∂θ)
+ (1/(r²sin²(θ))(∂²/∂φ²)

For a spherically symmetric field E in which all the derivatives with respect to the angle coordinates are zero the wave equation reduces to

(∂²E/∂t²) = C²[(1/r²)(∂/∂r)(r²(∂E/∂r) ]
or, equivalently
r²(∂²E/∂t²) = C²[2r(∂E/∂r) + r²(∂²E/∂r²) ]

Now consider the equation

(∂²(rE)/∂t²) = C²(∂²(rE)/∂r²)

When this is expanded it is

r(∂²E/∂t²) = C²[2(∂E/∂r) + r(∂²E/∂r²)]

And, when this equation is multiplied by r the result is

r²(∂²E/∂t²) = C²[2r(∂E/∂r) + r²(∂²E/∂r²) ]
the same as the above
wave equation

The equation

(∂²(rE)/∂t²) = C²(∂²(rE)/∂r²)

is a one dimensional wave equation in rE. The d'Alembert solution therefore applies; i.e.,

rE(r, t) = ½(r-Ct)E(r-Ct, 0) + ½(r+Ct)E(r+Ct, 0)

In this solution one half of the field shoots away toward the horizon at the speed of light. The other half moves through the center of the field at the speed of light. It would seem that this other solution is not physically relevant. The physically relevant solution is then

rE(r, t) = (r-Ct)E(r-Ct, 0)

Consider a field generated by a spherical charge of radius R.

E(r, 0) = K/r² for r≥R
E(r, 0) = 0 for r<R

If the charge were to disappear the hole in the field would expand from a radius of R at t=0 at the speed of light, as depicted below.

  

Thus free fields not generated by a charged particle disappear toward the horizon at the speed of light. There can be no static free-fields. The goal of Quantum Field Theory of replacing charged particles with autonomous free fields is thus grossly misguided.


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