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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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Reflection and transmission of EM waves
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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Reflection and transmission of EM waves
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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EM waves in conductors
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.
EM waves in conductors
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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
EM waves in conductors
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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.
EM waves in conductors
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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:
EM waves in conductors
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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/