As an example of the statement that Maxwell's equations
completely define
electromagnetic phenomena, it will be shown that Coulomb's law
may be derived
from Gauss' Law
for electrostatics. Consider a point charge
of charge
; we
can obtain an expression for the Electric Field
at a point in space due to this
charge by surrounding it with a "virtual" sphere of radius
, and then using
the Gauss' law in integral form:
The surface integral on the the right-hand-size of the equation can be written in spherical polar coordinates over the "virtual" sphere, considering the point charge at its centre. Under the assumption that the electric field is spherically symmetric, its value over the sphere surface is constant. Then, we can write
hence,
or
The usual form can then be recovered from the Lorentz force law
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