It has been known for a long time that a changing magnetic field induces and electric field and vice versa. An electromagnetic waves consists of electric fields and magnetic fields constantly changing and in a sense giving rise to each other. This means that electic and magnetic fields are interdependent.
Suppose we have a magnetic field which varies in time:
then
and similarly for an electric field
We can obtain an expression for
Similarly,
Maxwell's equation
becomes
Substituting
and
gives
A second relationship can be similarly be found to write the equation
as
or
Take the dot product of
and
to give
The left hand side can be written
so
but the dot product is even so
and
is perpendicular to
for an electromagnetic wave.