basic physics

Maxwell Boltzmann distribution for a classical ideal gas

i. We assume a dilute gas which is enclosed by a thermally insulated container on all sides.

Dilute gas in a thermally insulated container: Dilute means concentration of gas molecules is low. Insulated implies there is no reasonable flow of heat energy across the walls of the container.

ii. Each molecule is assumed to be a hard sphere which moves randomly in all directions such that its velocity vary from – infinity to + infinity.

Hard sphere: Remember a hard sphere is a classical analogy of a rigid sphere whose surfaces do not deform when an external object comes into contact. This essentially means the incoming object is scattered elastically that is without loss of kinetic energy, only momenta magnitude and directions are changed in accordance with the conservation of linear momentum.

iii. When molecules collide they do not lose energy or time. They bounce off each other so that ‘energy’ and ‘momenta’ are conserved.

We would like to obtain an expression for velocity distribution function. That is we would like to know the fraction of molecules having velocity between v to v+dv for all possible values of velocity.

For this we assume an ensemble of molecules in equilibrium. The ratio of number of molecules in a velocity range to the total number of molecules N gives the probability of finding a molecule in that velocity range.

v. The “phase space” of the ensemble of molecules is defined by a 6 N dimensional space, which constitutes of 3 N spatial components and 3 N velocity components of the N molecules in equilibrium. For a more advanced concept of phase space check the following statistical mechanics lecture.

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. 

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.