Electromagnetic theory

Helmholtz theorem in electrodynamics, Gauge transformation.

Electromagnetic theory, lecture — IV

Topics covered in this lecture

a. Helmholtz theorem — in electrodynamics

b. Gauge transformation — of scalar and vector potential in electrodynamics

c. Coulomb and Lorentz gauge

All electromagnetic theory lectures of this series, will be found here (https://mdashf.org/category/electromagnetic-theory/)

In our previous lecture — lecture — III, we discussed in quite detail, the problem of electrostatics and magneto-statics.

We understood how deeply the Helmholtz theorems formulate the entire question of these two branches of electromagnetic phenomena.

But static problems are not sufficient for any rigorous treatment of the electromagnetic theory.

We promised in that lecture to study how Helmholtz theorems lend their magical power to understand the most general nature of electromagnetic phenomena.

In this lecture we will study precisely the applicability of Helmholtz theorems to the problem of electrodynamics and we will see how it leads to a great deal of success in advancing the ability to solve electromagnetic problems of a great variety. 

Boundary conditions on Electric and magnetic fields.

Electromagnetic theory, Lecture — II. 

Boundary conditions on Electric and magnetic fields in Maxwell’s equations

Topics covered

A. Summary of Maxwell’s equations — in free space and in material media

B. Integral forms of Maxwell’s equations — by application of vector calculus

C. Derivation of boundary conditions — on electric and magnetic fields

In the last lecture we formulated the Maxwell’s equations, for free space as well as any material medium in their differential form.

Remember that we say free space to mean that the sources of charge densities and sources of current densities that experience our field vectors, viz. $latex vec{E}$ and $latex vec{B}$ — which are produced by other source densities of charges and currents, are non-existent.

That is there is no hindrance or onlookers our $latex vec{E}$ and $latex vec{B}$ fields meet on their way when they go on a sojourn, in that space. I also hear they call it by the name vacuum. As far as I know I testify, there is no difference between vacuum and free space.

Vacuum simply means for our purpose and many others, there is no glimpse of matter in the space of consideration. It is therefore the simplest of situation to harp on, before we can target our intelligence for achieving more complicated scenario, and yes there certainly are such situations and they take most of our coveted attention in asking us to solve them. 

And sooner than later we would be on our toes trying to grasp the burden the more complicated situations would unleash our way. For the time being we focus on free space which means the sources are zero.

Again by sources we mean, not the sources that produce our vector field $latex vec{E}$ and $latex vec{B}$ but the ones that interact with them, in the path of our fields.