Lectures & Presentations

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. 

Maxwell’s equations

Electromagnetic theory, Lecture — I.

Maxwell’s equations

This lecture, the web version of the first lecture given in the electromagnetic theory paper of the physics honors degree class, was delivered on 21st December 2017. All electromagnetic theory lectures of this series, will be found here. 

Also read part-2 of the linked lecture. That describes the subject matter of this lecture, in a good deal of depth.

Topics covered

A. Maxwell’s equations — basic form

B. Displacement current — Correction to Ampere’s law

C. Maxwell’s equations — in material media

Maxwell’s equations

Maxwell’s equations the basic forms

The Maxwell’s equations without the corrections to the Ampere’s law can be written as the following;

Electrostatics is when the electric charge and electric current densities, that produce these field, known therefore also, as the sources of the field, do not explicitly depend on time, that is, are constants. These sources or distributions depict the behavior of the field, and their independence from time means the fields do not vary in time, but vary only under spatial transformation.

Note that we are not talking about sources in the Maxwell’s equations above, but the ones that actually produce the E and B fields of the equations. The sources present in the equations above would alter these static fields though.

Accordingly the Maxwell equations would change their behavior in dynamic — i.e. time varying conditions, than they exhibit in the static conditions.

Equation (ii) has no names, but sometimes given a name, Gauss law — of magneto-statics.

Equation (iii) is known as Faraday’s law — of electromagnetic induction.

Equation (iv) is known as Ampere’s law.

Inconsistency in Maxwell’s equation
The Maxwell’s equations in this form are not the most general form of the eponymous set of equations. Also they are fraught with some degree of inconsistency.

Lets gaze deeper.

Microcanonical ensemble

Microcanonical ensemble
Lecture IV; This lecture, the 4th in the series of statistical mechanics lectures, a paper for the physics honors degree class, was delivered on the 10th of January this year (2018).

You can find the previous lectures here ( Lecture — I, II ) and here ( Lecture — III ).

Topics covered in this lecture

a. Recapitulation of some previous ideas and  — important remarks

b. Microcanonical ensemble — definition and properties 

c. Some basic parameters and formalism 

Recapitulation and remarks
In our previous lecture we defined the phase space density or distribution function rho (q, p; t) for a classical statistical system with an aim to connect it to a thermodynamic system.

We saw that an ensemble system would be stationary if rho does not have any explicit time dependence, …

Remarks
The type of general ensemble we defined as mental copies of actual system occupying each possible microstate can be called a Gibbsian Ensemble.
The above condition of statistical density as a stationary or time-independent variable would represent conditions of equilibrium.
We defined ensemble average of a physically measurable quantity

Phase space, ensemble and Liouville’s theorem

Phase space, Ensemble and Liouville’s theorem.

Topics covered in this lecture

a. Ensemble and average — thermodynamic systems

b. Phase space — a classical system

c. Liouville’s theorem

Ensemble and average in thermodynamic systems

For a given “macrostate” (N, V, E) a statistical system, at any instant of time, t, is likely to be found in any one of an extremely large number of distinct “microstates”.

When time passes, the system evolves into different microstates. In due course of time the system exhibits an average behavior of all microstates it passes through. 

We can equivalently depict this behavior by envisaging a large number of mental copies of the system, with the same macrostate as the original system, but all the possible microstates, in which the system can exist, all at once. Such a collection of hypothetical or mental copies of the given system is known as an ensemble. 

Thus the average behavior of the ensemble is expected to be identical with the time-averaged behavior of the actual physical system. In fact this is one of the fundamental requirements for statistical mechanics to be valid. No matter which mathematical avenue we prefer to meander through we must in the end reach our unique destination of physical validity. 

To understand the deeper aspects of this ensemble theory we need to define what is known as “phase space” of a statistical system.