Cornell electromagnetic radiation (1981) Robert C. Richardson - Cornell Laboratory of Atomic

https://www.youtube.com/watch?v=j2gOh39IyPM

Consider a point source with a sphere of arbitrary radius enclosing it. Both electric fields and gravitational fields obey the same basic differential equation:

∇⋅F=ρ$$\mathrm{\nabla }\cdot F=\rho$$

for some field F$F$ and some source ρ$\rho$. The divergence theorem tells us that fields obeying this equation also obey

∫VρdV=∮∂VF⋅dS$${\int }_V\rho dV={\oint }_{\mathrm{\partial }V}F\cdot dS$$

This is a mathematical consequence of obeying the first equation. This integral theorem must hold for all volumes enclosing our point source. That is to say, these integrals are constant--the left hand integral tells us the total charge or mass, and we have only one point source, so as long as the volume contains the point source, the integral must have a constant value regardless of size.

On the other hand, we've chosen a sphere for our surface of integration on the right-hand side. By symmetry, we conclude that this integral reduces to 4πR2|F(R)|$4\pi R^2|F(R)|$. For this to be constant, F(R)$F(R)$must fall off as 1/R2$1/R^2$.

In different numbers of dimensions, the dropoff will have a different form. For instance, in 1d there's no dropoff at all (cf. the electric field from an infinite sheet of charge).

Uploaded on Jul 30, 2009

Sparks fly—literally—as CU physicist Bob Richardson lectures on the propagation of electromagnetic radiation (1981)

Robert C. Richardson - Cornell Laboratory of Atomic and ...

www.lassp.cornell.edu/lassp_data/rcr.html

Cornell University

Vita. Ph.D., Duke University, 1966. Nobel Prize in Physics, 1996. Research Areas Low temperature, properties of liquid 3He and 4He, superfluid flow in bulk and ...

Robert C. Richardson - Cornell Laboratory of Atomic and ...

www.lassp.cornell.edu/lassp_data/rcr.html

Cornell University

Vita. Ph.D., Duke University, 1966. Nobel Prize in Physics, 1996. Research Areas Low temperature, properties of liquid 3He and 4He, superfluid flow in bulk and .