San José State University
Thayer Watkins
Silicon Valley
& Tornado Alley

The Dynamics of Free
Fields in 2D 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/∂²t) = C²∇²E

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

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

∇²(r, θ) = (∂²/∂r²) + (1/r)((∂/∂r) + (1/r²)(∂²/∂θ²)

For a field that is a function of only the radial distance r the wave equation reduces to

(∂²E/∂t²) = C²[(∂E²/∂r²) + (1/r)((∂E/∂r)]

For a field that is initially

E(r, 0)=K/r for r≥R
E(r, 0)=0 for r<R

the RHS of the previous equation evaluates to C²[2K/r³ −K/r³]=C²K/r³ .


(∂²E/∂t²) = C²K/r³

(To be continued.)

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