Advanced Electromagnetism Notes

Lecture #4 Radiation from charges and currents

i. Introduction

The notes aim to derive a solution for Maxwell's equations using vector and scalar potentials. Then to express the same potentials as functions of current densities. This method alows the user to more easily obtain the electric and magnetic fields instead of directly solving Maxwell's equations. The option of presenting the E and H fields is still explored, presenting the fields in terms of the sources that create them. The second part of the notes provides insight into far-fields in terms of sources, far-field conditions and derives the radiation integrals.

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ii. On notation

The notation in this note may prove to be confusing at first due to the lack of coding skills present in the author. This means that vector quantities in images containing equations will be denoted with an arrow above the letter, while the same vector quantities in the text will be denoted by the letter being in bold text. (I am sorry for the inconvenience, I will try to standardize the notation in further notes!). Also, of importance is that the differential equations given in the note are derived and shown in their phasor notation (in the complex domain). This is done for convenience and easier calculations.

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1. Fields in terms of potentials

This is the first out of four parts in this lecture. Each part follows a simple process that will be given as a list of steps and then each step will be presented in detail. Part one has the following process:

1. Separate Maxwell's equations by electric and magnetic sources.
2. Derive the vector potential quantities A and F.
3. Solve for B_A & E_A or D_F and H_F by using the potentialls for a general medium.
4. Derive Helmholtz equations for A and F in a simple medium by using the Lorenz gauge.
5. Derive equations for E_A and H_F as functions of A and F, respectively.

Maxwell's equations in phasor notation {1} are the base where the reader starts the derivation. It is also important to note that a magnetic current density J_m and a magnetic charge density ρ_m are included to make the equations simmetric. These sources of magetic fields do not exist physically, but prove to be usefull when trying to calculate the fields in certain scenarios due to a law of symmerty that will be explored in future notes.

To derive the vector and scalar potentials Maxwell's equations are split into two sets. The first one having only electric sources (J_e & ρ_e) {2} and the second one having only magnetic sources (J_m & ρ_m) {3}.

The subscript A denotes only electic sources and the subscript F denotes only magnetic sources. In future text the note will show the methodology of deriving the vector and scalar potentials from the set of equations with electric sources {2}, as derivation of the potentials from equations {3} is analogous.

From equation {2} is can be seen that the divergence of the B_A field is zero. This means that the magnetic flux density can be represented as the curl of an arbitraty vector A. After that the expression for B_A can be substituted in the curl equation of the electric field intensity E_A as shown in the image below:

The arbitrary vector A in equation {5} is called the vector magnetic potential and the derived scalar ϕ_e is called the scalar electric potential. Equations {4} & {5} are valid for a general (inhomogeneous, unisotropic, non-linear) medium/materials.

If simple (homogeneous, isotropic, linear) medium/materials is assumed, the equation for H_A from {2} can be used to derive equaiton {6} for A:

In the first row the vector identity ∇×∇×A=(∇∇⋅A-∇²A) is used. The second row utilizes the fact that the vector field A is fully defined only when its curl and divergence are known, and that it is an arbitrary non-unique field. Therefore, a condition called the Lorenz' gauge (often misspelled as Lorentz') is used allowing us to define the divergence as ∇⋅A=jωεμφ_e setting the third term in the second row to zero and thus coming to equation {6}. This is possible as due to the non-uniqueness of the field which means that no matter how the divergence is defined it will not have an effect on the physical field B_A.

From the third equation in set {2} an analogous euqation can be derived for ϕ_e as seen below:

Equations {6} and {7} are called inhomogeneous Helmholtz equations.

If the Lorenz' gauge defined as ∇⋅A=jωεμφ_e is substituted in {5} an equation for the electric intensity E_A that is a function of A only. It is important to remember that because the Lorentz' gauge is used on an equation derived in a simple medium, {8} is only valid for a simple medium/material.

A summary of all equations and their analogous counterparts for magnetic can be found below. The potentials derived from Maxwell's equations with magnetic sources are F the vector electric potential and φ_m the scalar magnetic potential.

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2. Potentials in terms of currents

The second part of the lecture aims to derive equations for potentials in terms of currents/sources. The process for this part is shown in the following list:

1. Find a solution to the Helmholtz' PDE equations by superposition of point source solutions.
2. Dirac's delta function is used to derive J_e(r) as a sum (integral) of point sources J_e(r').
3. Solutions for A(r) and F(r) are obtained by utilizing the 3D Green's function G.
4. Derive solutions for B_A,H_A,E_A and D_F,H_F,E_F. Find the total fields E and H by superposition.