Recent Posts - page 2

  • Helmholtz theorem in electrodynamics, Gauge transformation.

    The northern light or aurora is an electromagnetic phenomenon, produced due to motion of charged cosmic particles entering earth's magnetic field.

    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. 

  • Helmholtz theorem. Scalar and vector potentials

    This image shows the fabulous colors cities wear at night. And it all comes from electromagnetic waves. Our topic today is an attempt at understanding the deeper formal frame work of electromagnetic waves.

    This lecture develops the formalism of electrodynamics in a very methodical way.

    It covers the following topics in detail.

    A. Formalism of electrodynamics — fundamental theorem

    B. Application of Helmholtz theorem — to electrostatics

    C. Application of Helmholtz theorem — to magnetostatics

  • Boundary conditions on electric and magnetic fields.

    Boundary conditions on electric and magnetic field: When electric or magnetic fields go across the boundary of material media their values might or might not change. There are 4 possibilities and we derive them in this article. These values depend upon the surface current charge densities and the volume charge densities present on the surface of the media. Accordingly the tangential and normal components of electric and magnetic fields have 4 different possibilities. 2 of these change and are named as discontinuous while the other two don't change and are therefore called as continuous. Photo Credit: mdashf.org

    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

    Maxwell's equations: Ampere - Maxwell equation: this image shows an Amperian loop, a battery and a capacitor circuit. The image describes how Maxwell fixed one inconsistency in the form of one of the Maxwell's equations, which was known by the name Ampere’s law. When he removed the inconsistency it was then known as Ampere - Maxwell equation ( of Maxwell’s equations ) or simply modified ampere’s law. Photo Credit: mdashf.org

    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