## 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.

read more Boundary conditions on electric and magnetic 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.

## Electromagnetic Nature of Light — A brief history of light.

Let us begin this lecture which has roughly two parts;

1. the history of light and its understanding through the centuries

and

2. the electromagnetic nature of light

A brief history of light

Various optical devices and optical phenomena have been known since close to 4000 years. The optical devices of ancient time includes mirrors, burning glasses, lenses and other magnifying devices.

Accordingly various properties and laws of light were understood and developed since these times. E.g.

a. light was understood to propagate rectilinearly and

b. light was understood to reflect and refract.

read more Electromagnetic Nature of Light — A brief history of light.

## The Maxwell’s equations, from nature to instruments.

The beauty of Maxwell’s equations can be seen in how it helps us understand nature as well as instruments, at the same time. Medical devices are simply an advanced understanding that began with understanding electromagnetic waves through Maxwell’s equations.

Each of the following 4 equations has a different name, by which we call’em, but together they are called as the Maxwell’s equations. Together they constitute what I am inspired to say; the golden equations of Physics. If we do some easy tricks they will be converted into whats called as the Wave Equations (of motion) ! Yes, they describe the wave behavior “fully”. — By waves I don’t mean sound waves, but any sort of waves that move at the speed of light. Sound waves are ordinary pressure oscillations, that travel much slower than even rockets.

The 4 equations therefore describe how electromagnetic waves are created and broadcast. Hence TV radio and satellite communication were understood because these 4 equations were understood.

First two are time-independent or static equations.

Gauss law of electrostatics
The first equation is known as Gauss law of electrostatics, it says “Electric fields (E) are a result of sources of electrostatic charges”.
Gauss law of magnetostatics
The second equation is known as Gauss law of static magnetic field ( or magnetostatic field ) it says “apparently there are no sources of magneto-static charge or single magnetic pole from which the magnetic field B is created”.
Then how are magnetic fields created? We needed to know further to find the answer. Lets look at the 3rd and 4th equations.

The last two equations are time-dependent, time varying or dynamic equations. Which is why sometimes we see electro and magneto statics and some times we see electrodynamics and magneto-dynamics or simply electrodynamics, in nomenclature of these fields of studies.

Ampere’s circuital law
The 3rd equation is called as Ampere’s circuital law. Its also whats known as Faraday’s law of electromagnetic induction and Lenz’s law. It says changing magnetic fields can produce electric fields. Not only electrostatic charge but changing magnetic fields as well produce electric fields. Although we don’t know yet how the magnetic fields were produced.
Modified Ampere Circuital law
The last equation is called as Modified Ampere Circuital law  or Ampere-Maxwell law.This provides the remaining links in the understanding of Electromagnetic field’s creation and motion. It says magnetic fields are produced by actual electric currents (I) that is; change of electrostatic charges over time produces magnetic fields. Magnetic fields are also produced by changing electrostatic fields, this type of pseudo current is called as Displacement current.

read more The Maxwell’s equations, from nature to instruments.