Introduction to Classical Electrodynamics।।(part-1)।।Inverse Square Law।।Electricity and Magnetism।।

This is the first part of Classical Electrodynamics.. But before that we should have some clear concepts about some topics.. Today we will do that...

First we need to learn some Mathematical Concept,

1. Inverse Square Law:-

This is a symmetry we often find in the Newtonian Gravitational Equation, Coulomb's law of Electrostatic force etc..Carl Frederick Gauss was the first to explain the symmetry..

                     Carl Fredrick Gauss 

According to Gauss, If any source of Force or Energy exists in a three-dimensional space, then we can simply say that

Where ∮is closed surface integral, F ⃗ is force vector ds ⃗ is infinitesimally small area vector..

which basically means that Force or Energy from the source evenly permeates through 3-dimensional space... so, we can conclude that (F∝1 / 4πR²) or
(F∝1 / R²).. actually Newton's formula of gravitation is F = ((k / 4πR²) M • m) generally (G) gravitational constant is eqivalent to (k/4πR²) similarly Culomb's law of attraction is F = ((k / 4πR²) • Qq) This inverse square symmetry is also evident in Kepler's light source.

2. FIELD:-

In simple words Field are the effects on surrounding space due to the existence of energy or force..Or for rigorous definition Fields are just numbers which measure the effects on surrounding space by the source of Force..
So, we can say that effects on the surrounding space due to the presence of electrical charge is called Electric Field and the effects on space for the existence of a magnet is called Magnetic Field..
For mathematical convenience Fields are omnipresent..Even if there is no Charge or Magnetic source in space the fields are still present but their values are zero..

3. CHARGE:-

It's difficult to say what actually charge is rather than it's effect.. what we can say about charge is that it's a property of some object, particles etc which can perform actions at a distance on other charges..

In 1748 Benjamin Franklin Categorized Charges into two categories Positive and Negative..

                     Benjamin Franklin 

In 1785 Charles-Augustin de Coulomb expressed this phenomena of Electrostatic attraction and repulsion in a equation,

We know that this is another example of Inverse Square law..so we can modify the equation like this... 

Where Fₑ is Electrostatic Force; q1,q2 are the charges; R is the distance between the two charges; ε is the permitivity of the space (i.e constant) and R̂ is the direction of the action...

             Charles-Augustin de Coulomb

But though the equation can tell us about the consequences of the charges but it's unable to say how does this action at a distance takes place..For that we have to take another bold assumption, FIELD..
The equation of the Electric Field Vector is

4. Magnets:-

We know lots about the permanent magnets from our daily experiences.. But we will talk about electromagnets (i.e magnetism produced by the flow of electric charges) here.. The permanent magnets are just the manifestation of this electromagnets at Quantum level..we will get to that later..
In 1819 Hans Christian Ørsted noticed compass needle deflects near electric current..

                      Hans Christian Ørsted

Next year Jean Baptiste Biot and Felix Savart found some patterns in the incident..

             Jean Baptiste Biot and Felix Savart 

They found that this induced magnetic field strength at a point is directly proportional to the length of the conductor, rate of the flow of electric charge, and the sine value off the angles between the the direction of the electric current and the point..

Some time later Pierre-Simon Laplace generalized this idea in an equation..

B ⃗   is the Magnetic Field Vector I is the rate of flow of electric charges; δl is the infinitesimally small length of the conductor; θ is the angle between the direction of the flow and the point R is the distance between the conductor and the point. μ is the permeability of the space..

                     Pierre-Simon Laplace

now it's the time to to define the relationship between electromagnets and permanent magnets there are various levels to determine the magnetic property of a material...

First atomic and sub atomic Level:-

we all know that atoms are made up of positively charged protons negatively charged electrons and neutral particle neutrons.. The protons and the neutrons are clamped together in the the atom so the protons directly do not determine the magnetic behaviour of the atom..
It's the electrons, which move around an atom and determine its property.. Firstly electrons are not actually particles neither waves.. they are actually Quantum probability waves and the most probable places to find an electrons around the nuclei are called orbitals..The orbitals(Azimuthal Quantum Number(ℓ)) with non zero angular momentum such as p,d,f orbitals which are not spherically symmetric act like little bar magnets..

But in the most cases this magnetic behaviours got canceled out by the opposite directions of the orbital magnetic fields..

Secondly every single electron has a unique property called spin angular momentum which exerts a magnetic field making the whole electron like a small bar magnet but due to Pauli's Exclusion Principle which states that two electrons can't occupy the same Quantum states.. So, if two electrons which have same Principal Quantum Number(n), Azimuthal Quantum Number(ℓ), Magnetic Quantum number (mℓ) must have different Spin Quantum Numbers(s)or in other words two electrons with same first 3 Quantum numbers must have opposite spin(i.e opposite magnetic moment)..

                         Wolfgang Pauli 

so, in most cases at this level this magnetic behaviour also cancel one another.. but in some cases while filling the orbitals, the electrons at first fill all the Magnetic Quantum Numbers(mℓ) of a degenerated(i.e. same) orbital(Azimuthal Quantum Number(ℓ)) and the magnetic moment of all the electrons of the orbital point in a same direction due to Hund's law..That's why the atoms in the middle of the Periodic Table have net magnetic moment..

Second Molecular and Crystal Level:-

The atom is magnetic that necessary to not make the whole material magnetic.. For a material to be magnetic at first the atoms must have to be magnetic then all the the atoms in a collection of atom(domain) must point in a same direction so the magnetic moment of individual atoms do not cancel Each Other out and then the collection of domains..

 

When these requirements are satisfied a material becomes a Ferromagnetic.. So, if the the atoms are magnetic the material may not be magnetic at all that's why Atom of Chromium is magnetic but the material is not magnetic but Iron is magnetic..Truth is that this is so rare that only 4 materials (Iron, Nickel, Cobalt, Gadolinium) out of 118 elements exerts magnetic properties at room temperature...

To be continued...

Written by Shravan Ghosh