Wave solutions of Maxwell's equations and Ampere-Maxwell law

An electromagnetic signal is generated by a Hertzian dipole located at a point P, which has the position vector r = ?(100m) ez. The signal is detected by a small wire loop located at the origin. Apart from the dipole and the loop, the nearby space is empty.

Experimentation reveals that the detected signal is induced by a changing magnetic field:
Bphys(t)= B0 sin (2?ft) ex, where B0 =0.1 ?T and f = 30MHz.

Show that the given physical magnetic field at the loop, Bphys(t), is consistent with a monochromatic plane wave solution of Maxwell's equations given by B =iB0 exp[i(kz ? ?t)] ex.

Solution Summary

We solve a problem in electromagnetism involving a Hertzian dipole.

1.
Explain how the wave equation arises out of Maxwell'sEquations. Relate your discussion to the following simulation of an electromagnetic wave: Electromagnetic wave
2.
Explain the effect of the speed of light on electromagnetic waves.
3.
Explain how you could find the frequency of

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.

Write out Maxwell'sEquations component-wise in cylindrical coordinates in their most general form. Remember that each field component can be a function of all three spatial dimensions as well as time. You should end up with eight equations. Assume that all source terms are present, or that there exists both a charge density a

1. Derive wave equation (about E, B ) by using Maxwell`s equations.
2. Derive wave equation (about vector potential, scalar potential) by using Maxwell`s equations.
There are no conditions (such as no charges or no current) given, which means I have to solve in a general form!
***I am currently using Reitz's book

Assume that both the E field an the B field are time-harmonic, so that each can be written as:
E = E(0)exp(i(k*r-wt))
B = B(0)exp(i(k*r-wt)) where * = dot
The time and spatial derivatives can then be written as
partial of E with respect to t = -iwE
Partial of B with respect to t = -iwB
divergence of E = ik*E

See attached file for complete formulation of the problem.
The problem deals with the electric and magnetic field behavior at an interface.
The first question asks to show that using Maxwell'sequations we can obtain the wave equation for the fields.
The second question asks as to draw the fields at the interface.
The th

2. (a) (i) Write down Maxwell'sequations for static electric and magnetic ﬁelds in the vacuum (note that you should include charge and current densities).
(ii) How did Maxwell modify Ampére's law to account for dynamic electric ﬁelds?
(b) In a region of space in which the relative permittivity is (attached equation),