maxwell’s eqautions and some applications

8/6/2019 Maxwells Eqautions and some applications

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Maxwells Equations and some

applications


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Summary of the lecture

Important equations of Electromagnetism.

What Maxwell did.

The Maxwell Equations.

Maxwell Equations in Vacuum.

Maxwell Equations in matter.

Reflection and transmission of EM waves

EM waves in conductors.

Wave guides

Importance of Maxwells work.


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Some important Equations in

Electromagnetism

Coulomb's law Gausss law

Biot-Savart law

Amperes law

Faradays law

(1)

(3)

(5)

(4)

(2)


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What Maxwell did Maxwell considered the known laws of electricity and

magnetism and showed that these laws imply the existence of

electromagnetic waves.

He made an important modification to the amperes law and

introduced the concept of displacement current.

Lets see what Maxwell did. If we take the divergence of the

Amperes law in differential form, i.e.

Take divergence =>

Which is problematic as the divergence of the current shouldbe equal to the decrease in the density of the charge inside a

closed surface.


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So he added the following term to the Amperes law

Now if we take the divergence of the above equationwe get the continuity equation.

So by introducing this term he also made the

equation more symmetric with faradays equation.

What Maxwell did


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Gausss law

Amperes law with

displacement current

Faradays law

And

The Maxwells Equations

No Monopoles!


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Maxwell Equations in Vacuum

We now consider what Maxwell concluded from his

equations in free space

Now if we take the derivative of equation (9) and use (8), we get

(6)

(8)

(9)

(7)


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Then by using the vector identityand equation (6), we get

Now interestingly this is a wave equation of a wave traveling

with velocity v.

Which is precisely the speed of light c!

Maxwell Equations in Vacuum

We can geta similar

equation for

the B.


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Maxwell Equations in matter Maxwell equations inside matter

Where fis the density of free charge in the medium. Jf is the free current density

D is the electrical displacement

And , M being the magnetization


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Maxwell Equations in matter Maxwell equations inside matter where there are no free

charges and no free currents are

If the medium is linear,

Also if the medium is homogenous i.e. and do not vary

from point to point, Maxwells equation reduce to


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Which differ from the vacuum analog only in the replacement of

0 and 0 by and

By using Maxwells equations in the above form we can studythe behavior of electromagnetic waves or light in linear media,

i.e., their reflection, transmission, absorption, etc.

Maxwell Equations in matter


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Maxwells conclusion of the wave nature of EM fields makes iteasy to imagine the behavior of waves on boundaries.

Consider EM waves approaching the boundary of two media

normally. A plane wave traveling in the z-direction and

polarized in the x direction approaches the interface from the

left.


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We consider sinusoidal wave forms.

It gives rise to a reflected wave

And a transmitted wave


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Reflection and transmission of EM waves We can use the boundary conditions for the electric and magnetic

fields to get the exact nature of the transmitted and reflected wave.

For example for z=0, EI and ER must sum up to ET,

and for B, (iv) gives

We can that solve for EOR, EOT and EOI and calculate the reflection and

transmission coefficient by using formula for the intensity I.


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EM waves in conductors

If we are dealing with conductors than we cannot set f and Jfequal to zero in the Maxwells equations. So the equations read

Where the free current density Jf

has been placed equal to Ein (iv).

Now the continuity equation for the free charge is


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Now with ohms law and Gausss law above equation becomes

Solving the above equation

Which means that if we have free charge with in the conductor

than that charge will dissipate in time =/. This also reflectsthe fact that any free charge placed on a conductor flows to thesurface. This time constant in a way gives a definition of a goodand a bad conductor. For a good conductor should be verysmall where as the opposite will be the case for a bad conductor.


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When the accumulated charge has disappeared the equations

read;

Taking the curl (iii) and (iv) we obtain the wave equations for E

and B.



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These equations admit plane wave solutions,

Plugging these solutions in the wave equations we see that the

wave number in this case is complex

Taking the square root and writing the wave number as real andimaginary parts



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Where, the real and imaginary parts are

The imaginary part of k results in an attenuation of the wave, i.e.

decreasing amplitude with z.



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The distance it takes to reduce the amplitude by a factor of

1/e( about one third) is called the skin depth

It is a measure of how far a wave penetrates into a conductor.

Meanwhile the real part of k determines the wavelength ,

propagation speed, and the index of refraction, in the usual

way:



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We are interested in monochromatic waves that propagatedown the tube so that E and B are of the form;

E and B should satisfy the Maxwells Equations in the interior

of the wave guide

Wave guides


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We can write E and B as

Where each component is a function of x and y. Putting this in

Maxwell equations (iii) and (iv), we get

Wave guides


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Equation (ii), (iii), (v) and (vi) can be solved for E x, Ey, Bx and By

Wave guides


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Inserting these equations in the remaining Maxwell equations (i) and (ii) we get

uncoupled equations for Ez and Bz

The boundary conditions can be used to solve the above equations.

If Ez=0 we call these trasnsverse electric TE waves and if Bz=0 they are called

transverse magnetic TM waves. And if both are zero we call them TEM waves.

Wave guides


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Importance of Maxwells work

Maxwell realized that the value of c which he calculated was the

same as the value of the speed of light available at that time

through experiment. So he concluded that light is an

electromagnetic wave which travels with speed c.

Maxwell and other scientists realized that visible light was a

tiny portion of the electromagnetic spectrum and that other

portions remained to be explored. Based on Maxwell's

equations, in 1888 German physicist Heinrich Rudolf Hertz,

demonstrated the existence of radio waves at frequencies and

Rontgen later discovered X-rays.

Maxwells equation also showed that light does not need a

medium (called ether) to travel as sound waves. This was later

experimentally confirmed by Michelson and Morley.


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Maxwells work laid the foundation of quantum theory, when

Planck explained the black body radiation by proposing that

atoms absorb and emit electromagnetic radiation in forms of

bundles of energy called quanta.

Maxwell's equations remain a powerful tool used by scientists

to understand and predict the behavior of electromagnetic

fields and waves in many engineering applications, including

the design of electrical transmission lines, electromagnetic

antenna (e.g., radio, television, microwave), radio telescopes,

and other instruments used to measure portions of the

electromagnetic spectrum.

Importance of Maxwells work


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References

Lectures on Physics by R. P. Feynman.

Introduction to Electrodynamics by D. J. Griffiths

http://www.bookrags.com/

maxwell’s eqautions and some applications

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