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Maxwell’s Equations and some applications

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 Maxwell’s work.

Some important Equations in Electromagnetism • Coulomb's law • Gauss’s law

(1)

(2)

• Biot-Savart law (3)

• Ampere’s law • Faraday’s law

(4)

(5)

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 ampere’s law and introduced the concept of displacement current. • Let’s see what Maxwell did. If we take the divergence of the Ampere’s law in differential form, i.e.

• •

Take divergence => Which is problematic as the divergence of the current should be equal to the decrease in the density of the charge inside a closed surface.

What Maxwell did • So he added the following term to the Ampere’s law

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

• So by introducing this term he also made the equation more symmetric with faraday’s equation.

The Maxwell’s Equations • Gauss’s law • Ampere’s law with displacement current

• Faraday’s law

• And

No Monopoles!

Maxwell Equations in Vacuum • We now consider what Maxwell concluded from his equations in free space (6) (7)

(8)

(9)

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

Maxwell Equations in Vacuum •

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

We can get a similar equation for the B.

•

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 matter •

Maxwell equations inside matter

•

Where

•

Jf is the free current density

•

D is the electrical displacement

•

And

ρf is the density of free charge in the medium.

, M being the magnetization

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.

μ and ε

do not vary

from point to point, Maxwell’s equation reduce to

Maxwell Equations in matter

•

Which differ from the vacuum analog only in the replacement of μ0 and ε0 by μ and ε

•

By using Maxwell’s equations in the above form we can study the behavior of electromagnetic waves or light in linear media, i.e., their reflection, transmission, absorption, etc.

Reflection and transmission of EM waves •

Maxwell’s conclusion of the wave nature of EM fields makes it easy 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.

Reflection and transmission of EM waves •

We consider sinusoidal wave forms.

•

It gives rise to a reflected wave

•

And a transmitted wave

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.

EM waves in conductors •

If we are dealing with conductors than we cannot set

ρf and Jf

equal to zero in the Maxwell’s equations. So the equations read

Where the free current density Jf has been placed equal to σE in (iv). • Now the continuity equation for the free charge is •

EM waves in conductors • Now with ohm’s law and Gauss’s 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 τ=σ/ε. This also reflects the fact that any free charge placed on a conductor flows to the surface. This time constant in a way gives a definition of a good and a bad conductor. For a good conductor τ should be very small where as the opposite will be the case for a bad conductor.

EM waves in conductors •

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 • 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 and imaginary parts

EM waves in conductors • 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 •

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:

Wave guides •

A waveguide is a structure used to guide waves like the EM waves. Lets see how we can study these waves by using the Maxwell’s Equation. So far we have considered waves of infinite extent; now lets consider EM waves confined in a hollow pipe. We will assume that the waveguide is a perfect conductor so that E and B are zero inside the material. So that

(i) and (ii) are BC at the inner wall. Free charge will be induced on the surface as to impose these constraints.

Wave guides •

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

•

E and B should satisfy the Maxwell’s Equations in the interior of the wave guide

Wave guides •

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

Wave guides •

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.

Importance of Maxwell’s 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. • Maxwell’s 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.

Importance of Maxwell’s work •

Maxwell’s 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.

References • Lectures on Physics by R. P. Feynman. • Introduction to Electrodynamics by D. J. Griffiths • http://www.bookrags.com/

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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 Maxwell’s work.

Some important Equations in Electromagnetism • Coulomb's law • Gauss’s law

(1)

(2)

• Biot-Savart law (3)

• Ampere’s law • Faraday’s law

(4)

(5)

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 ampere’s law and introduced the concept of displacement current. • Let’s see what Maxwell did. If we take the divergence of the Ampere’s law in differential form, i.e.

• •

Take divergence => Which is problematic as the divergence of the current should be equal to the decrease in the density of the charge inside a closed surface.

What Maxwell did • So he added the following term to the Ampere’s law

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

• So by introducing this term he also made the equation more symmetric with faraday’s equation.

The Maxwell’s Equations • Gauss’s law • Ampere’s law with displacement current

• Faraday’s law

• And

No Monopoles!

Maxwell Equations in Vacuum • We now consider what Maxwell concluded from his equations in free space (6) (7)

(8)

(9)

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

Maxwell Equations in Vacuum •

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

We can get a similar equation for the B.

•

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 matter •

Maxwell equations inside matter

•

Where

•

Jf is the free current density

•

D is the electrical displacement

•

And

ρf is the density of free charge in the medium.

, M being the magnetization

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.

μ and ε

do not vary

from point to point, Maxwell’s equation reduce to

Maxwell Equations in matter

•

Which differ from the vacuum analog only in the replacement of μ0 and ε0 by μ and ε

•

By using Maxwell’s equations in the above form we can study the behavior of electromagnetic waves or light in linear media, i.e., their reflection, transmission, absorption, etc.

Reflection and transmission of EM waves •

Maxwell’s conclusion of the wave nature of EM fields makes it easy 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.

Reflection and transmission of EM waves •

We consider sinusoidal wave forms.

•

It gives rise to a reflected wave

•

And a transmitted wave

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.

EM waves in conductors •

If we are dealing with conductors than we cannot set

ρf and Jf

equal to zero in the Maxwell’s equations. So the equations read

Where the free current density Jf has been placed equal to σE in (iv). • Now the continuity equation for the free charge is •

EM waves in conductors • Now with ohm’s law and Gauss’s 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 τ=σ/ε. This also reflects the fact that any free charge placed on a conductor flows to the surface. This time constant in a way gives a definition of a good and a bad conductor. For a good conductor τ should be very small where as the opposite will be the case for a bad conductor.

EM waves in conductors •

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 • 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 and imaginary parts

EM waves in conductors • 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 •

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:

Wave guides •

A waveguide is a structure used to guide waves like the EM waves. Lets see how we can study these waves by using the Maxwell’s Equation. So far we have considered waves of infinite extent; now lets consider EM waves confined in a hollow pipe. We will assume that the waveguide is a perfect conductor so that E and B are zero inside the material. So that

(i) and (ii) are BC at the inner wall. Free charge will be induced on the surface as to impose these constraints.

Wave guides •

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

•

E and B should satisfy the Maxwell’s Equations in the interior of the wave guide

Wave guides •

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

Wave guides •

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.

Importance of Maxwell’s 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. • Maxwell’s 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.

Importance of Maxwell’s work •

Maxwell’s 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.

References • Lectures on Physics by R. P. Feynman. • Introduction to Electrodynamics by D. J. Griffiths • http://www.bookrags.com/

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