Wave Dispersion EM radiation Maxwell’s Equations 1

2 - Simply stated, a dispersion relation is the function ω(k) for an harmonic wave. - A dispersion relation connects different properties of the wave such as its energy, frequency, wavelength and wavenumber. - From these relations the phase velocity and group velocity of the wave can be found and thereby refractive index of the medium can be determine. Wave Dispersion

t0t0 t1t1 t2t2 Non dispersive : All colours moving with same speed Dispersive: Red moving faster than blue 3

Normal dispersion of visible light Shorter (blue) wavelengths refracted more than long (red) wavelengths. Refractive index of blue light > red light.

A medium in which phase velocity is frequency dependent is known as a dispersive medium, and a dispersion relation expresses the variation of  as a function of k. Group velocity If a group contains number of components of frequencies which are nearly equal, then:

Phase and Group velocity P’ vgvg 6

Non dispersive waves v p = Constant Signal is propagated without distortion More generally v p is a function of (or k) 7

Usually, is positive, so that v g < v p Normal Dispersion When, is negative, so that v g > v p Anomalous Dispersion More in electromagnetic waves When, is constant, so that v g = v p Non-Dispersive medium (Ex: Free space)

Wave packet (without Dispersion) Wave packet (with Dispersion)

Non-dispersive Dispersive Wikipedia.org

Wave Packets

y(t) = Sin t 0 < t < 200 Superposition of waves and wave packet formation 12

y(t) = [Sin t + Sin (1.08 t)]/2 0 < t <

Suppose we have group of many frequency components lying within the narrow frequency range  …

y(t) = [Sin t + Sin(1.04 t) + Sin (1.08 t)]/3 0 < t <

y(t) = [Sin t + Sin(1.02 t) + Sin (1.04 t) + Sin(1.06 t) + Sin (1.08 t)]/5 0 < t <

y(t) = [Sin t + Sin(1.01 t) + Sin (1.02 t) + Sin(1.03 t) + Sin (1.04 t) + Sin (1.05 t) + Sin (1.06 t) + Sin (1.07 t) + Sin (1.08 t)]/9 0 < t <

y(t) = [Sin t + Sin(1.01 t) + Sin (1.02 t) + Sin(1.03 t) + Sin (1.04 t) + Sin (1.05 t) + Sin (1.06 t) + Sin (1.07 t) + Sin (1.08 t)]/9 0 < t <

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Electromagnetic Radiation

Let‘s first develop the understanding by taking the example of oscillating charge and/or dipole oscillator 21

For stationary charges the electric force field Coulomb’s law © 2005 Pearson Prentice Hall, Inc

Coulomb’s law What is the electric field produced at a point P by a charge q located at a distance r? where e r is an unit vector from P to the position of the charge

If a charge moves non-uniformly, it radiates © 2005 Pearson Prentice Hall, Inc

The electric field of a moving point charge

Electric field - P :unit vector directed from q to P at earlier time q

_ The correct formula for the electric field Important features 1.No information can propagate instantaneously 2. The electric field at the time t is determined by the position of the charge at an earlier time, when the charge was at r’, the retarded position. 3. First two terms falls off as 1/r’ 2 and hence are of no interest at large distances

Correct Expression (at large distances) This is electro-magnetic radiation or simply radiation. It is also to be noted that only accelerating charges produce radiation.

Electric Dipole Oscillator © SPK/SB

Radio-wave transmission

Car Antenna TV Antenna

“Let there be electricity and magnetism and there is light” J.C. Maxwell

Vector Analysis (Refresh)

- The gradient points in the direction of the greatest rate of increase of the function, - and its magnitude is the slope (rate of increase) of the graph in that direction. GRADIENT For a scalar function T of three variable T(x,y,z), the gradient of T is a vector quantity given by:

DIVERGENCE For a vector T the divergence of T is given by: It is a measure of how much the vector T diverges / spreads out from the point in question.

CURL For a vector T the Curl of T is given by: It is a measure of how much the vector T curls around the point in question.

DIVERGENCE THEOREM / Green’s Theorem / Gauss’s Theorem Integral of a derivative (in this case the divergence) over a volume is equal to the value of the function at the surface that bounds the volume.

Integral of a derivative (in this case the curl) over a patch of surface is equal to the value of the function at the boundary (perimeter of the patch). STOKES’ THEOREM

What we know from previous classes? 1)Oscillating magnetic field generates electric field (Faraday´s law) and vice-versa (modified Ampere´s Law). 2)Reciprocal production of electric and magnetic fields leads to the propogation of EM waves with the speed of light. Question: WAVES?????? How do we show that a wave is obtained? 40

Our Attempt: To derive the relevant wave equation 41

x y z EyEy BzBz Consider an oscillating electric field E y January 21, This will generate a magnetic field along the z- axis

C E y (x) E y (x+  x) x Y Z We know that Faraday´s law in the integral form in given as: where C is the rectangle in the XY plane of length l width  x, and S is the open surface spanning the contour C January 21, Faraday’s Law N: Number of turns B: External magnetic field A: Area of coil The induced electromotive force in any closed circuit is equal to the negative of the time rate of change of the magnetic flux through the circuit.

Using the Faraday´s law on the contour C, we get: this implies... Keep this in mind... 44

We know that the Ampere´s law with displacement current term can be written as: EyEy xx C/C/C/C/ x z Y B z (x) B z (x+  x) 45

Using the Ampere´s law, for the Contour C /, we get: this implies... 46

Using the eq. obtained earlier i.e., Note: Similar Equation can be derived for B z Form of wave equation 47

Solution of EM Wave equation

Electromagnetic waves for E field for B field

In general, electromagnetic waves Where  represents E or B or their components

# A plane wave satisfies wave equation in Cartesian coordinates # A spherical wave satisfies wave equation in spherical polar coordinates # A cylindrical wave satisfies wave equation in cylindrical coordinates

Solution of 3D wave equation In Cartesian coordinates Separation of variables

Substituting for  we obtain Variables are separated out Each variable-term independent And must be a constant

So we may write where we use

Solutions are then Total Solution is plane wave

Traveling 3D plane wave

spherical waves Spherical coordinates (r, θ, φ): radial distance r, polar angle θ (theta), and azimuthal angle φ (phi)

Spherical waves

Alternatively The wave equation becomes

Put Then  Hence

Therefore Wave equation transforms to 

Which follows that Separation of variables Solutions are Total solution is

outgoingwaves incomingwaves Final form of solution General solution spherical wave

Cylindrical waves Cylindrical Coordinate Surfaces(ρ, φ, z). The three orthogonal components, ρ (green), φ (red), and z (blue), each increasing at a constant rate. The point is at the intersection between the three coloured surfaces.

with angular and azimuthal symmetry, the Laplacian simplifies and the wave equation

The solutions are Bessel functions. For large r, they are approximated as

Maxwell’s equations

Use B in Divergence Theorem No magnetic monopoles II

Use E in Stokes’ Theorem From Faraday’s Law III

IV Use B in Stokes’ Theorem From Ampere’s Law

Charge conservation is a fundamental law of Physics which is written as a continuity equation

IV

Maxwell’s equations

Plane EM waves in vacuum

Wave vector k is perpendicular to E Wave vector k is perpendicular to B

B is perpendicular to E

B, k and E make a right handed Cartesian co-ordinate system

Plane EM waves in vacuum