# Maxwell's equations and the wave equation.

The question attached is from this page.

http://farside.ph.utexas.edu/teaching/em/lectures/node48.html

Please answer with vector notation. The question is only about (450) so I don't believe that you have to read through all of it to answer.

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#### Solution Preview

See the attached files.

We have shown in equation (443) that for any vector in the form

(1.1)

we have:

(1.2)

And fro equation (442)

(1.3)

Then we "assume" that the magnetic and electric fields have a plane wave form:

(1.4)

In the absence of charges and currents (vacuum). the first Maxwell equation gives using (1.3):

(1.5)

Which means that the electric field vector is perpendicular to the wave vector

The second Maxwell equation yields the same result for the magnetic field:

(1.6)

That is, both electric and magnetic fields are perpendicular to the wave vector k (which points in the direction of the electromagnetic wave's propagation).

The third Maxwell equation is:

(1.7)

We ...

#### Solution Summary

The solution shows that a plane-wave solution satisfies Maxwell's equations in vacuum and then goes on and derive the wave equations for the fields directly from the differential general Maxwell equations, therefore defining the speed of light.