Lecture 05

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PX384 - Electrodynamics 1

V. Electromagnetic waves


1. Characteristics of E.M. waves (revision) 1.1. Existence

The governing equations for E.M. potentials under the Lorenz gauge are:

and

In this course, we shall not consider the excitation of E.M. waves by E.M. sources (retarded potentials)*, but focus on the propagation of E.M. waves through media. In this section, we consider a vacuum, i.e. :

and

* Excitation of E.M. waves using retarded potentials will be covered in Advanced Electrodynamics

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Using the relations and it is clear that also the E.M. fields are governed by the same type of equation:

This result can also be found from Maxwell’s equations without the sources.

[Calculation V.1]


1.2. d’Alembert solutions

We consider the wave equation for the electric field. The same can be said for the wave equations for the magnetic field.

1)!Wave equation in each of the three components. 2)!Equation is a second-order partial differential equation in

time and space. 3)!Equation is linear: if f and g are solutions, then !f + µg is

also a solution. Hence, the general solution can be written as a superposition of basic solutions.

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This wave equation has two solutions known as d’Alembert solutions:

where is the light velocity vector with the direction of propagation.

We consider three geometrical cases: a)!1d E.M. wave b)!spherical E.M. wave c)! cylindrical E.M. wave


a) 1d E.M. wave

To understand what the two d’Alembert solutions mean, we consider only the x-component of the electric field and only the propagation in the z-direction:

•! E+(z-c0t) is a signal of shape f propagating in the positive z-direction at the speed of light co. •! E-(z+c0t) is a signal of shape f propagating in the negative z-direction at the speed of light co.

f(z-cot)

z 0 cot -cot

f(z) f(z+cot)

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b) spherical E.M. wave

The Laplacian in spherical geometry, using spherical symmetry, is . Hence, the E.M. wave equation becomes:

which has the d’Alembert solutions

The E.M. wave has a spherical wave front. For a central source, i.e. a lamp, only the outgoing wave exists ( ). The singularity at r=0 is unphysical and is due to the neglect of the E.M. sources. The 1/r factor is geometrical and reflects the conservation of wave energy as the wave front grows (surface sphere ~ r2).

r t0 t1 t2

wave energy density


b) cylindrical E.M. wave

The Laplacian in cylindrical geometry, using azimuthal and z-symmetry, is . Hence, the E.M. wave equation becomes:

which has the d’Alembert solutions in function of Hankel functions. However, for large r, this can be approximated as

The wave has a cylindrical wave front. The same other comment apply as for the spherical wave.

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1.3. Plane wave solutions

We have not said anything about the form of the function f. It can be taken to be a plane wave, i.e.

and are the wave vector and angular frequency, resp. and are related to the wave length ! and wave period P as

We can see that this represents a plane wave front by considering that at fixed time:

The latter is the equation for a plane with normal vector

They are related to each other through the dispersion relation [Calculation V.2]


The general E.M. wave solution can be written as a superposition of plane waves:

This is actually the 4d inverse Fourier transform.

[Calculation V.3]

We recover the d’Alembert form of the solution by splitting the frequency integral into two parts and redefine " as –" in the second term.

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1.4. Phase and group velocity

Phase velocity is the velocity at which a wave front travels and is in the direction of the normal to the wave plane.

Group velocity is the velocity at which the centre of a wave train travels and is linked with the propagation of wave energy.

space vph

vg [Calculation V.4]


For an E.M. wave in vacuum:

phase velocity:

group velocity:

The group speed does not depend on the wave vector. Therefore, E.M. waves in vacuum are non-dispersive.

Using

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1.5. Wave polarisation

The components of the electric and magnetic field vectors of the E.M. wave are related through the source-less Maxwell’s equations. For a solution written as a superposition of plane waves, note that:

For an individual wave component Maxwell’s eqs. for the wave amplitude vectors are:


We learn the following: •! Wave amplitude vectors are orientated perpendicular to the direction of propagation ! transverse wave.

•!Electric and magnetic amplitude vectors are perpendicular to each other.

•!The ratio of the magnitudes of the electric over the magnetic amplitude vector is equal to the speed of light.

•! forms a triad (right-handed coordinate system).

[Calculation V.5]

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With the component of parallel to equal to zero, that leaves choices for the two perpendicular components.

Different types of polarisation:

•! Linearly:

•! Circularly polarised:

Right circularly polarised Left circularly polarised

Linearly polarised


1.5. Electromagnetic spectrum

Orange

100 km

10 km

1 km

100 m

10 m

1 m

100 mm

10 m

m

1 mm

100 µm

10 µm

1 µm

100 nm

10 nm

1 nm

100 pm

10 pm

1 pm

100 fm

10 fm

1 fm

wave length

radio waves micro waves

infrared radiation

gamma rays X rays

UV radiation

Red

Yellow

Green

Blue

Violet

700 nm

400 nm

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1.6. Conservation of E.M. wave energy

Electromagnetic energy density is given by (with EM sources):

The rate of change of u [Jm-3] has two contributions:

[Calculation V.6]

Poynting flux [Jm-2s-1] is the rate of E.M. energy flow per unit area across surface perpendicular to in which and ly. is the work done by E.M.; translates into heat through Ohm’s law:


Integral form of E.M. energy conservation:

Using divergence theorem:

E.M. energy within a volume V is gained or lost by an energy flux through the surface A, or is lost due to work done which heats the volume.

S1

A1

V

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Electromagnetic wave energy density for a plane wave component in vacuum of the form

Using it is clear that the wave obeys the energy conservation law

Then,


2. E.M. wave guides An electromagnetic wave guide is a conduit for EM waves, i.e. a tube or box with perfectly conducting walls.

At the perfectly conducting walls we have the conditions:

No electric fields tangential to the wall, and no magnetic field lines entering the wall. You can still have a normal electric field due to charges on the wall, or a tangential magnetic field due to surface currents.

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2.1. Transverse electric and magnetic wave

The simplest manner to satisfy the boundary conditions is to satisfy them everywhere.

Two parallel perfectly conducting infinite planes at y = 0,a. z

x

y

a

The boundary conditions impose:

A plane wave solution to the EM wave equation with Ex=Ez=0 is


The magnetic field components are calculated using Faraday’s Law:

Hence,

This EM wave is called the Transverse Electric and Magnetic wave (TEM) and propagates along transmission lines at any frequency. However, if we consider a wave guide with boundaries in multiple dimensions (e.g. a box), a TEM cannot propagate.

Of course we still find that . They are also perpendicular to the direction of wave propagation:

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2.2. Transverse electric or magnetic wave We consider a more general wave solution with an E.M. that is polarised in the x-direction.

The EM wave equation turns into an ODE for y:

We define the ‘wavenumber’ normal to the boundaries as

The form of the solution depends on the sign of "2 and thus on the frequency of the wave.


We wish that the EM wave can propagate (is oscillatory) in all directions.

- oscillatory solution along the wave guide:

- oscillatory solution between two walls:

The wave can only propagate with a frequency above a certain value, which is called the cut-off:

Below the cut-off frequency the wave solution is evanescent, with exponentially decaying or growing solutions.

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It has to satisfy the boundary conditions at y=0,a:

We have a wave which propagates in the x-z plane, but is a standing wave in the y-direction with discrete wavelengths ! = 2#/$ = 2a/n

The oscillatory solution is (valid above the cut-off):

To have a non-zero solution, we need

and the wave solution is (B=E0):


The condition that the wave propagates in the x-z plane imposes a condition on n and ":

If we fix the frequency, that means that not all standing modes (n) are possible. Only the low harmonics that satisfy above condition.

Example: a = 1.5 cm

% f = "/2# = 15 GHz

Only the fundamental harmonic n=1 can propagate!

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Using again Faraday’s Law, we calculate the magnetic components of the wave:

The component of the magnetic field parallel to the direction of propagation is non-zero!

Because only the electric field is purely transverse, this type of wave is called a Transverse Electric wave (TE).

Starting with a wave with a transverse magnetic field in the x-direction, the same analysis leads to the Transverse Magnetic wave (TM).


Physical picture of a TE or TM between two parallel plates.

y=0

y=a

wave plane

The EM wave reflects at the two plates. The incident and reflected wave fronts interfer with each other. When half the wavelength (or any integer multiple of that) in the y-direction fits exactly between the two plates, then the interference is constructive.

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Phase and group speeds of TE and TM waves. We calculate the speeds for propagation along the plates.

For large n, the phase speed can be larger than the speed of light when the wave becomes more and more purely a standing wave in the y-direction.

The propagation of wave energy along the plates is below the speed of light because of the zig-zag path of the wave. Also, the group speed depends on the wavelength, and is thus dispersive.


3. E.M. waves in dielectrics

An E.M. wave in a dielectric is governed by the wave eq.

where

We define the refractive index of a medium n [1]:

Because c & co , n ' 1 The relative permeability is often close to 1, then

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The solution of the wave equation is:

where

Materials usually show a smooth decrease in #r with increasing $. Superimposed on this are anomalies due to some internal natural resonances at the microscopic level. Energy can be absorbed from the E.M. wave.

Examples: - water has vibrational modes that absorbs microwaves. - Earth’s atmosphere is opaque in the IR due to many excitations of rotations of air molecules


Opacity of the Earth’s atmosphere

100 pm

1 nm

10 nm

100 nm

1 µm

10 µm

100 µm

1 mm

10 mm

100 mm

1 m

10 m

100 m

1 km

0%

50%

100%

wave length

Atm

osph

eric

O

paci

ty

Gamma rays, X rays and UV blocked by upper atmosphere

Visible window

Most IR absorbed by atmospheric gases

Radio window

Long wave length radio waves blocked

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This absorption effect is modelled by attributing to the medium a complex relative permittivity:

Then,

which turns the wave solution into

and

The factor represent exponential decay over a typical length-scale called the skin depth % [m]:


Examples:

- Glass: In visible light: n2 << 1 which implies % >> 1: weak damping In UV, n2 grows which implies that % shortens: growing absorption. This is due to excitation of electrons in the glass molecules (Si).

No point in sunbathing behind glass!

- Atmosphere: In IR between 10 µm and 1 mm wavelength, n2 is large due to molecular rotations.

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4. E.M. waves in plasmas

We consider a uniform, static plasma without external electric or magnetic fields applied. This plasma is perturbed and we study the types of waves the plasma can support.

For simplicity, we neglect the effect of random thermal motions and collisions of the charged particles and the ions are assumed to be fixed.

The equation describing the electrons as a ‘fluid’ is given by

where is the electron fluid bulk velocity and is a function of space and time. is the plasma electron number density.

4.1. Generalised Ohm’s Law for E.M. wave


We now write all the variables to be consisting of two terms: an equilibrium term (subscript 0) and a perturbation (prime).

With assumptions of the plasma equilibrium imply

We consider the perturbation quantities to be small so that combinations of them are neglected, i.e. linearisation:

Multiplying by –e and using the definitions of the electric current density and electron plasma frequency, we find:

[Calculation V.7]

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is part of the generalised Ohm’s Law for E.M. waves. For a wave of frequency ", we can write this as

where is a conductivity but is not due to collisions (note it is imaginary!). Collisions would add a conductivity as:

If we turn to the energy conservation Eq. and ignore the Poynting flux for the moment ( ). We have u ~ E’2. Thus,

The first factor is oscillatory (wave-like) and the second factor is exponential decay and represents Ohmic dissipation.


4.2. Two types of wave solutions

Taking the divergence of the generalised Ohm’s Law leads to an equation for oscillations in charge density:

using and

Equation for oscillations in electric charge density

electrostatic oscillations (seen earlier)

E.M. waves in plasma

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4.3. E.M. wave equation in a plasma

We focus on the case that (’=0 and derive the wave equation from Maxwell’s equations in the usual manner, but taking into account the generalised Ohm’s Law.

[Calculation V.8]

The third term reflects the plasma. If we let the plasma density tend to zero, then the electron plasma frequency tends to zero and we recover the E.M. wave equation in vacuum.


The solution to the wave equation is again in terms of a superposition of plane waves:

where, using , we find the dispersion relation for E.M. waves in a plasma:

4.3. E.M. wave dispersion relation in a plasma

Compared with vacuum, the term is added. We can rewrite the dispersion relation also as

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From we see that there are two regimes for the propagation of E.M. waves:

a)

k is real. Hence, and the wave solution is of the form

The wave is travelling with for each plane wave component of frequency " a wavelength


b)

k is purely imaginary. Hence, and the wave solution is of the form

The wave is evanescent, being damped for each plane wave component of frequency " over a typical length-scale, which is the skin-depth of E.M. waves in plasma:

For " = 0 : : strongest damping For " % "pe : : limit of no damping

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An E.M. wave of frequency " entering a plasma with an electron plasma frequency larger than " will damp over a typical length-scale of )(")

We call where the behaviour of the E.M. wave changes between travelling and damping, and where the wavelength or skin-depth become infinite, the cut-off frequency.


The dispersion relation of E.M. wave in a plasma:

"=c ok

For k = 0 : " = "pe

For : " * cok For high-frequency E.M. waves, the wave resembles E.M. wave in vacuum.

For " > "pe : travelling EM wave For " < "pe : damped EM wave

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4.4. Phase and group velocities

Phase velocity:

If , we retrieve the case of E.M. waves in vacuum with vph = co and vg = co.

[Calculation V.9]

Group velocity:


4.5. Example: the solar corona

solar eclipse 1999

The average number density of the solar corona is ne = 1014 m-3. The electron plasma frequency is then equal to

= 5.64 108 rad s-1

or = 9 107 Hz = 90 MHz

E.M. waves emitted from the solar photosphere will travel through the corona undamped if + > +pe . For visible light & is in the range 460 GHz to 750 GHz, so there is no damping. The cut-off frequency + = +pe lies in the radio domain. In fact, = 90 MHz corresponds to BBC Radio 2 FM transmission frequency! Radio signals with frequencies below that are damped.

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However, a constant density corona is simplistic. The density of the corona decreases with distance z from the sun as

The density scale-height H is about 50 106 m, which is much larger than the wavelength of a typical radio signal, i.e. ! ~ 1 m. We can thus assume that locally for the wave the density is quasi-constant, so we can apply our theory. The plasma frequency is a function of height:

The heigher in the corona, the lower and the lower the frequency of the escaping E.M. waves can be. Or, if we observe an E.M. wave, the lowest height z* it could have originated from is where .


z

damping

travelling

z*

To receive an E.M. signal from a height we need to have a receiver with a frequency of minimum (H=5 107 m, ne0 = 1015 m-3, ): From base of corona: z* = 0: ! > 300 MHz From 1 solar radius above: z*=6.96 108 m:

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Exercises Ex V.1: Assume a plasma of electrons and ions without an magnetic field, where random thermal motions and collisions of particles are neglected. However, the protons are no longer assumed fixed. a) Derive the relation between the time derivate of the electric current and the electric field . b) Derive the differential equation for the electric charge density. Discuss the difference between the equation where ions were assumed fixed. c) Find the wave equation of electromagnetic waves in a plasma. d) Derive and draw the dispersion relation for the E.M. waves. Also, calculate the phase and group speed. Ex V.2: Using spherical polar coordinates, verify that E = E0 f(r,cot)/r represents a spherical electromagnetic wave emanating from the origin. Calculate the wave energy density and show that the total wave energy is conserved as the wave spreads out. a) The Sun radiates electromagnetic waves corresponding to a black body at temperature of 6000 K. Use Stefan’s Law P = -ST4A to calculate the total E.M. power output P, where -S = 5.67 10,8 Wm,2K,4 is Stefan’s constant and A is the area through which the E.M. wave travels. The solar radius is 696000 km.


b) Calculate the solar power per square meter, known as the solar constant, received at Earth, which is 150 million km away from the Sun. Also, calculate the amplitudes of the electric and magnetic fields. c) Repeat the calculation for a star similar to the Sun which is located 10 light years from Earth. d) Repeat the calculation for a 5 W laser focused on a 1 µm spot. Compare and discuss the values you found in parts b,c and d. Ex. V.3: Massive stars at the end of their Hydrogen burning phase can become neutron stars. Neutron stars are very dense and small (size of a city) and spin very fast around their axis (periods between a few seconds to milliseconds). Also, the magnetic field has become so extremely powerful, that light escapes only near the magnetic poles in form a beam. A neutron star is a cosmic equivalent of a light house, called a pulsar. We see on Earth periodic pulses of light every time one of the two beams points our way. The first discovered pulsar is in the Crab nebula, which is about 6500 light years away. This pulsar emits short pulses of microwave radiation. When they reach the earth, the frequency components of the pulses are dispersed so that higher frequencies around +1 =115 MHz arrive first, after a travel time T , while lower frequencies around +2 =110 MHz appear about 'T=1.5 s later. The observed dispersion is believed to be

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due to the intervening interstellar plasma, which gives us an opportunity to estimate the electron density of this plasma. a) Write the group speeds for E.M. waves in plasma for those two frequencies, in function of the electron plasma frequency. b) Determine the electron plasma frequency from those two group speeds, taking into account the difference in travel time. c) Determine the average electron density of the interstellar medium.