Maxwell Equations

Dr. Anurag Srivastava

Web address: http://tiiciiitm.com/profanurag

Email: profanurag@gmail.com

Visit me: Room-110, Block-E, IIITM Campus

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Electrodynamics: Maxwell’s equations: differential and integralforms, significance of Maxwell’s equations, displacement current andcorrection in Ampere’s law, electromagnetic wave propagation,transverse nature of EM waves, wave propagation in bounded system,applications.

Quantum Physics: Dual nature of matter, de-Broglie Hypothesis,Heisenberg uncertainty principle and its applications, postulates ofquantum mechanics, wave function & its physical significance,probability density, Schrodinger’s wave equation, Eigen values &Eigen functions, Applications of Schrodinger equation.

Syllabus

Prerequisite… E & M

The History of

Electromagnetics

The history of electromagnetics shows that it

is a series of discoveries of many people

instead of by just one person.

Early Electromagnetics

Amber –

The ancient Greeks realized that it attracts chaff and

feather particles.

Loadstone –

A naturally magnetized mineral; magnetite.

Chinese used it properties to invent the compass.

Charles Coulomb

He was a civil engineer in the

French army, but had to quit

due to illness.

He discovered that electric

forces obey the same inverse

square law Newton

discovered.

Alessandro Volta

Learned that body tissue

could conduct electricity from

his friend Galvini.

Discovered that all metals

could conduct electricity.

Created the voltaic pile as

the first continuous electric

power source.

André-Marie Ampère

Early in life his father was

beheaded during the

French Revolution.

After Orsted discovered in

1820 that electricity induced

magnetism

he wrote a mathematical

paper to describe the

behavior in one week.

Michael Faraday, early life

Grew up in a poor family in

England. He received very

little formal education.

Read many books as an

apprentice bookbinder and

did experiments in the shop.

After attending a lecture by

Humphrey Davy he was hired

as a lab assistant.

Michael Faraday, later life

While Faraday made

discoveries Davy became

jealous and tried to take

credit.

Knew that electricity could

produce magnetism, but

could magnetism produce

Electricity?

James Clerk Maxwell (Dafty)

Born into a wealthy family in

Edinburgh. Was well educated and

inquisitive.

Went to Cambridge college, but moved

to Trinity for competitive reasons.

Was a professor at Trinity, Aberdeen,

and London.

Put Faraday’s Law in mathematical

form.

Discovered the famous 4 equations

that govern electromagnetics.

Devout Christian.

Quantities

In mathematics there are two types of

quantities:

Vector: Direction + Magnitude

Scalar: Only Magnitude

Gradient: An operator in vector calculus.

Two key concepts in vector calculus are divergence and curl, the latter of

which is sometimes called circulation. Basically, divergence has to do with

how a vector field changes its magnitude in the neighborhood of a point,

and curl has to do with how its direction changes.

Materials

Comprises of many atoms.

Atoms have neutron, proton and electrons.

Protons are positively charged.

Electrons are negatively charged.

Interactions among the particles through long range as well as short range

forces.

Electrons

Current

Flow of electrons per unit time is called current.

Two types:

Direct current- electrons flow in same direction

Alternating current- electrons flow in different direction

Circuit

Path for the flow of electrons

Three types

Series:

Parallel:

Hybrid

Voltage

Voltage is defined as the electromotive force

or the electric potential energy difference

between two points (often within the context

of an electrical circuit) per unit of charge.

Expressed in volts (V).

Coulomb's law

Force on a test charge Q1 due to a single point charge

Q2, is given by

Electric Field Intensity

Charge distribution

Total charge due to these distributions are given by:

Electric flux

Flux For a Closed Area

Then flux is given by:

Permittivity

Measure of a material's

ability to resist an electric

field.

Denoted by the symbol .

Gauss law of electric fields

The total of the electric flux out of a closed surface is

equal to the charge enclosed divided by the permittivity.

Magnetism Is a physical phenomenon produced by the motion of electric

charge, which results in attractive and repulsive forces between

objects.

Is a region around a magnetic material or a moving electric charge within

which the force of magnetism acts. Magnet produces magnetic force and

have magnetic field lines.

Magnetic Field

Magnets

Magnets have two poles.

North pole

South pole

Opposite pole attract each

other

Similar pole repel each

other.

Permeability

Is the degree of magnetization that a material

obtains in response to an applied magnetic field

Tells, how easily an external magnetic field can

induce an internal field in the material

Where, B = induced magnetic field

H = externally applied magnetic field

Gauss law of Magnetism

The net magnetic flux out of any closed surface

is zero.

For any closed surface ,the magnetic flux

directed inward toward the south pole will equal

the flux outward from the north pole.

Electromagnetism

Moving charge create a

magnetic field in the

direction perpendicular to

the current.

Direction of magnetic field

is given by right hand rule

.

Thumb- direction of current

Fingers – direction of

magnetic field.

Faraday’s Law

Any change in the magnetic

field of a coil of wire will cause an

EMF to be induced in the coil.

This EMF induced is called

induced EMF and if

the conductor circuit is closed,

the current will also circulate

through the circuit and this current

is called induced current.

dl =

Ampere Law

Magnetic field created by an

electric current is proportional to

the current with constant of

proportionality equal to the

permeability of free space. d

l

Maxwell Equations

Differential form of

Maxwell’s equations

Integral form of

Maxwell’s equations

Physical Significance Of

Maxwell Equations Gauss law of electric fields:

It tells us that electric field origins from electric charge.

Gauss law of magnetic fields:

tells us that magnetic monopoles do not exist.

Faradays law:

Any change in magnetic flux across some closed path generates

e.m.f.

Ampere’s law:

Electric current generates magnetic field

Integral form of Maxwell Equation

Integral form of Maxwell Equation

Integral form of Maxwell Equation

Integral form of Maxwell Equation

Integral form of Maxwell Equation

Integral form of Maxwell Equation

Fundamentals of Electrical EnginPHYring

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Differential form of Maxwell Equation

Difference between differential and

integral form of Maxwell Equation

Fundamentals of Electrical EnginPHYring

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Difference between differential and

integral form of Maxwell Equation

• The equations are entirely equivalent, as can be proven using Gauss' and

Stokes' theorems.

• The integral forms are most useful when dealing with macroscopic problems

with high degrees of symmetry (e.g. spherical or axial symmetry; or,

following on from comments below, a line/surface integrals where the field is

either parallel or perpendicular to the line/surface element).

• The differential forms are strictly local - they deal with charge and

current densities and fields at a point in space and time. The differential

forms are far easier to manipulate when dealing with electromagnetic waves;

they make it far easier to show that Maxwell's equations can be written in a

covariant form, compatible with special relativity; and far easier to put into a

computer to do numerical electromagnetism calculations.

Fundamentals of Electrical EnginPHYring

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Maxwell’s first equation in differential form

Fundamentals of Electrical EnginPHYring

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Integral form of

Maxwell’s 1st

equation

It is called the differential form of Maxwell’s 1st equation.

The Second Maxwell’s equation (Gauss’s law for

magnetism)

Fundamentals of Electrical EnginPHYring

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The Gauss’s law for magnetism states that net flux of the magnetic field through a

closed surface is zero because monopoles of a magnet do not exist.

Fundamentals of Electrical EnginPHYring

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The Third Maxwell’s equation (Faraday’s law of

electromagnetic induction )According to Faraday’s law of electromagnetic induction

It is the differential form of Maxwell’s third

equation.

Fundamentals of Electrical EnginPHYring

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The Fourth Maxwell’s equation ( Ampere’s law)The magnitude of the magnetic field at any point is directly proportional to the

strength of the current and inversely proportional to the distance of the point from

the straight conductors is called Ampere’s law.

Third Maxwell’s equation says that a changing magnetic field

produces an electric field. But there is no clue in fourth

Maxwell’s equation whether a changing electric field produces

a magnetic field? To overcome this deficiency, Maxwell’s

argued that if a changing magnetic flux can produce an electric

field then by symmetry there must exist a relation in which a

changing electric field must produce a changing magnetic flux.