Trying to understand electromagnetic waves

[Agustin.R]

I'm having some trouble understanding visually the propagation of an electromagnetic wave. I'm self-studying electrodynamics so I've never had someone explain this properly to me.

I understand an electromagnetic wave is a propagating disturbance in an electromagnetic field. Originally, I thought this is a change in the radiating particle's field, so I was puzzled why wouldn't the wave fall off like r^2 like the electric field does. So I'm thinking that the electromagnetic wave is a separate, self-propagating electromagnetic field, distinct from the particle's field (or at least the field at the retarded time in which the particle radiated). Is this the meaning of "a changing electric field generating a magnetic field, and a changing magnetic field generating an electric one"?

Additionally, I don't understand what is the "Eo" value in the wave equation for a monochromatic plane wave. I know this is the amplitude, but I don't understand how I would calculate it.

Finally, most books I see have only a treatment of plane waves. Could anyone recommend me a book with a good treatment of spherical ones?

Thank you in advance for your help.

 

[Bill_K]

Agustin, This can lead to complicated math in a hurry, so I'll talk about about a simplified situation. Suppose you have two point charges located at z = +a and z = -a, and suppose the charges are equal and opposite and their magnitude varies sinusoidally: Q = ± Q₀ e^iωt. In between them there is a thin resistance-free wire that carries the current back and forth.

Let the field point be a distance r from the origin, and distances r_a, r_b from the charges. Look at the scalar potential, which depends only on the charges, not the current. It's like 1/r:

φ = Q₀ [e^iωt/r_a - e^iωt/r_b]

But that's wrong: we ignored the effect of retardation. Changes in the field propagate outward at the speed of light, and φ reflects the value of each charge at the retarded time. So φ is really:

φ = Q₀ [e^iω(t-r_a/c)/r_a - e^{iω(t- r_b/c})/r_b]

Let ω = ck, and for utter simplicity take a field point somewhere on the z-axis, so that r_a = r + a, r_b = r - a.

φ = Q₀ e^{ikct -ikr}[e^-ika/(r+a) - e^+ika/(r-a)]

Expand this, assuming the charges are close together: a << r and ka << 1. The leading terms are:

φ = Q₀ e^{ikct -ikr}[-2ika/r - 2a/r²]

(I hope I got that right.) Anyway the point is, when k ≠ 0 (time-varying) the potential goes like 1/r, and when k=0 (static) the 1/r term is missing and the potential goes like 1/r².