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VI. Electromagnetic Waves

All the important physics in electromagnetism can be expressed in

four Maxwell’s Equations, the Lorentz force and conservation of

charge with interesting consequences.

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VI–1 Maxwell’s Equations

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Main Topics

• Generalized Faraday’s Law.

• Generalized Ampère’s Law.

• Maxwell’s Equations.

• Production of Electromagnetic Waves.

• Electromagnetic Waves Qualitatively.

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Generalized Faraday’s Law I

• From electrostatics we know that:

• From Faraday’s law we know that if flux in a circuit changes the induced EMF is equal to work done by electric field to transport unit charge one closed loop around the circuit.

)()()()( BAABldEB

A

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Generalized Faraday’s Law II

• We can obtain this fact from:

• When we substitute for the EMF from the Faraday’s law we get general formula:

• The integration must be performed in the positive counterclockwise direction!

ldEqldF

dt

dldE m

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Generalized Faraday’s Law III

• The line integration can be performed along any closed path in any material or of course even in vacuum.

• We consider change of flux through a surface surrounded by our integration path.

• When flux changes the electric field is not conservative any more. Otherwise the integral along a closed path would be zero.

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Generalized Faraday’s Law IV

• Let’s have a magnetic field (with a non-zero component) perpendicular to the plane of drawing and flux through some fixed loop increases. Then:

• This means that the electric intensity must be clockwise, which is in accord with Lenz law.

• Note that we have information only about the change of the magnetic induction not about the direction of the original field!

0

dt

dldE m

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Generalized Ampère’s Law I

• We have learned that line integral of magnetic induction over any closed path depends on the total current which is enclosed by this path.

• But is it valid generally?• If we are charging a capacitor, experiment

shows that magnetic field can be measured near to the capacitor as if a current went through it. But charge doesn’t pass through the capacitor!

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Generalized Ampère’s Law II

• If a theory is in a contradiction with the experiment it must be improved or changed!

• We have to accept that whatever happens in the capacitor when it is being charged or discharged behaves like a current because it produces magnetic field. It must be a new type of current, since it is definitely not movement of charges.

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Generalized Ampère’s Law III

• It is, of course, the electric field what changes in the charged (or discharged) capacitor. So we define a new type of current which we call a displacement or shift current, which we attribute to the change-in-time of the electric field, more accurately of the electric flux.

• Enclosing means the same as in the previous case of Faraday’s law:

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Generalized Ampère’s Law IV

• Up to now, when we used Ampere’s law we usually used a circular closed path and have taken into account the total current which passed the circular surface bounded by it.

• Generally, we can consider surface of any shape, bounded by the enclosed path. The currents, which would enter the surface but would not pass through our path would have to leave the surface somewhere so they would not be counted in.

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Generalized Ampère’s Law V

• The fact that some of these surfaces can also pass between the plates of the capacitor means that what passes there through them must be equivalent to the electric current. And now since we are interested in an electric field passing through surfaces, we are, in fact interested in electric flux.

• Again we integrate in the positive sense.

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Generalized Ampère’s Law VI

• The existence of displacement current actually means an important symmetry between electric and magnetic fields since not only changes of magnetic fields produce electric fields but also the changes of electric fields produce magnetic fields!

• Thanks to this symmetry electromagnetic waves and also WE do exist!

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Generalized Ampère’s Law VII

• The displacement current for a parallel plate capacitor can be calculated simply from the definition of its capacity and the current:

Q = CV = (0A/d)(Ed) = 0AE

I = dQ/dt = d(0AE)/dt = 0 de/dt

• This conclusion is valid generally so the Ampere’s law has a new term:

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Generalized Ampère’s Law VIII

• If we consider the relation:

• We can finally write:

dt

dIldB e

enc 000

dt

d

cIldB e

enc 20

1

200

1

c

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Generalized Ampère’s Law IX

• For instance if we are charging a (parallel plates) capacitor from a power source V0 , through a resistor R. The current decreases exponentially from an initial value I0 = V0/R and :

)]exp(1[)( 0

RC

t

d

VtE

A

I

AdR

dV

RCd

V

dt

dE

0

0

0

00

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Generalized Ampère’s Law X

• The magnetic field B outside the capacitor can be obtained from the Amperes law:

Which is exactly the induction we get close to a straight wire feeding the capacitor.

dt

dEA

dt

drrB e

00002)(

r

IrB

2

)( 00

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Maxwell’s Equations I

• Now we are ready to write the Maxwell’s equations.

• There are several types of them and several levels of generality. So exactly we shall deal with Maxwell’s equations in integral form valid for vacuum.

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Maxwell’s Equations II

dt

dIldB

AdB

dt

dldE

QAdE

e

m

000

0

0

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Maxwell’s Equations III

• The first equation is the Gauss law of electrostatics. It means that:• Sources of electric field – charges exist.

• If charges are present, the electric field lines begin in positive charges (or infinity) and end in negative charges (or infinity).

• The field of a single charge decreases as 1/r2.

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Maxwell’s Equation V

• The second equation is the Faraday’s law of electromagnetic induction. It means that:• The electric field can be produced also by time

changes of magnetic field. This field is not conservative and its field lines are closed.

• In the absence of time-varying magnetic field the magnetic field is conservative and a scalar potential exists.

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Maxwell’s Equation IV

• The third equation is the Gauss law of magnetism. It means that:• Separate sources of magnetic field – magnetic

monopoles do not exist.

• The magnetic field lines are closed.

• The field of an elementary current decreases as 1/r2.

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Maxwell’s Equation VI

• The fourth ME is the generalized Ampères law. It means that:• The magnetic field is produced either by

electric currents or equivalently by time changes of electric field.

• The field lines are closed lines.

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Maxwell’s Equations VII

• ME plus the Lorentz force carry all the information on all electromagnetism.

• They have many important consequences.• We are actually dealing with one electro-magnetic

field but when electric and magnetic fields don’t change in time the first two equations are not connected with the other two and electricity and magnetism can be treated separately.

• The most important outcome is the existence of electromagnetic waves.

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Planar Electromagnetic Waves

• Important solution of ME is a planar wave. If it moves in positive x direction with a speed c, the field can be described as:

E = Ey =E0sin(kx - t)

B = Ez =B0sin(kx - t)• E and B are in phase• Vectors , , form right (hand) turning system• wave number: k = 2/• angle frequency: = 2f• speed of the wave: c = f = /k

c

E

B

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Production of Electromagnetic Waves

• Since changes of electric field produces magnetic field and vice versa these fields once generated can continue to exist and spread into the space.

• This can be illustrated using a simple dipole antenna and an AC generator.

• Planar waves will exist only far from the antenna where the dipole field disappears.

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Homework

• No homework assigned today!

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Things to read and learn

• This lecture covers

Chapter 29 – 7; 32 – 1, 2, 3, 4

• Advance reading

Chapter 32 – 5, 6, 7, 8, 9

• Try to understand the physical background and ideas. Physics is not just inserting numbers into formulas!