derivation of wave equation from Maxwell's equations
Maxwell was the first to note that Ampère's Law does not satisfy conservation of charge
(his corrected form is given in Maxwell's equation). This can be shown using the equation of conservation of electric charge:
Now consider Faraday's Law in differential form:
Taking the curl
of both sides:
The right-hand side may be simplified by noting that
Recalling Ampère's Law,
Therefore
The left hand side may be simplified by the following Vector Identity:
Hence
Applying the same analysis to Ampére's Law then substituting in Faraday's Law leads to the result
Making the substitution
we note that these equations take the form of a transverse wave travelling at constant speed
. Maxwell evaluated the constants
and
according to their known values at the time and concluded that
was approximately equal to 310,740,000
ms
, a value within 3% of today's results!
Contributors to this entry (in most recent order):
As of this snapshot date, this entry was owned by invisiblerhino.