in Electromagnetics Numerical Method Introduction to Electromagnetics I Lecturer: Charusluk Viphavakit, PhD

Size: px

Start display at page:

Download "in Electromagnetics Numerical Method Introduction to Electromagnetics I Lecturer: Charusluk Viphavakit, PhD"

Osborn Charles
5 years ago
Views:

2141418 Numerical Method in Electromagnetics

1 Numerical Method in Electromagnetics Introduction to Electromagnetics I Lecturer: Charusluk Viphavakit, PhD ISE, Chulalongkorn University, 2 nd / charusluk.v@chula.ac.th Website:

2 Light Light is electromagnetic radiation within a certain region of the electromagnetic spectrum. Called Photons Carry energy from one place to the other Feel the heat of the sun Electromagnetic wave Propagates from one place to another See a light from far distances Mathematics for Nano-Engineering:

3 Light Phenomenon Particles Wave Reflection Refraction Interference Diffraction Polarization Photoelectric effect Mathematics for Nano-Engineering:

4 What is Electromagnetics?? Electromagnetics involves the macroscopic behavior of electric charges in vacuum and matter. Positive and negative electric charges are sources of the electric fields and moving electric charges yielding a current is the source of magnetic fields. This behavior can be accurately characterized by the Lorentz force law and Maxwell s equations Mathematics for Nano-Engineering:

5 Electromagnetics fields 4 different quantities The magnetic flux density B with the unit T The magnetic field intensity H with the unit A/m The electric field intensity E with the unit V/m The electric flux density D with the unit C/m 2 B = μh D = εe μ is the permeability (μ = μ 0 μ r ) ε is the permittivity (ε = ε 0 ε r ) The permittivity of free space e 0 (= x F/m) The permeability of free space m 0 (= 4p x 10-7 H/m) Mathematics for Nano-Engineering:

6 Electromagnetics wave Field propagation Direction The magnetic field, H, will be perpendicular to the electric field, E, and both are normal to the direction of propagation Mathematics for Nano-Engineering:

7 Electromagnetics wave Electromagnetic wave oscillates in space with a certain period, λ Electromagnetic wave oscillates in time with a certain frequency, ν d t The wavelength of a electromagnetic wave and its frequency are link through the speed of light. ν = c λ Mathematics for Nano-Engineering:

8 Electromagnetics spectrum Mathematics for Nano-Engineering:

9 Electromagnetics spectrum Type Wavelength (m) 1 Frequency (Hz) 1 Gamma rays < 6x10-12 > 5x10 19 X-rays 6x x x x10 19 UV 8x x x x10 16 Visible Infrared 7.6x x x10 14 Microwaves x10 11 Radio waves > 0.3 m < Mathematics for Nano-Engineering: <a href=" Spectrum - Wavelength, Frequency, And Energy, Wavelength Regions</a>

10 Wave Wave can oscillate in space and time ψ(x, t) where ψ is a wave function. Wave function is a complex-valued probability amplitude. f(x) ψ(x, t) v t = 0 t > 0 f(x x ) x For the electromagnetic wave: E = f(x vt) Mathematics for Nano-Engineering:

11 Harmonic wave A monochromatic wave which has one frequency represented by sine or cosine function. sine cosine x x ψ x, t = A sin k( x vt) ψ x, t = A cos k( x vt) For electromagnetic waves: where E = E 0 cos k( x vt) E 0 is wave amplitude (Energy carried by the wave) k is wave number; k = 2π λ v is velocity; v = νλ Mathematics for Nano-Engineering:

12 Harmonic wave ψ x, t = A cos k( x vt) in which k = 2π λ (spatial frequency) in which ω = 2πν (angular frequency) ψ x, t = A cos kx ωt Wave is a function of x vt or kx ωt so ψ x, t = ψ φ Mathematics for Nano-Engineering:

13 Phase of the wave φ(x, t) = kx ωt When light travels a distance x over a time t, it gains a phase φ. Phase is a property of waves having peaks and valleys with a zero-crossing between them. 2π φ The phase denotes a specific point on a wave Mathematics for Nano-Engineering:

14 Phase of the wave φ(x, t) = kx ωt k is referred to the wave number or the number of changes of the phase by 2π over a unit distance. Wavelength, λ Spatial frequency k = 2π λ x ω is referred to the angular frequency or the number of changes of the phase by 2π over a unit time. Period, τ Temporal frequency ω = 2πν ν = 1/τ t Mathematics for Nano-Engineering:

15 Wave number Number of wave per unit distance wave propagation To include the direction of propagation, wave vector is introduced y k = k a r where a r = a x, a y, a z and a r = 1 z k x = ka x k y = ka y k z = ka z x k = 2π λ Mathematics for Nano-Engineering:

16 Phase difference The difference in phase between two waves having the same frequency and referenced to the same point in time. Two waves oscillate with the same frequency and no phase difference are said to be in phase. In-phase waves The direction of propagation is drawn normal to the wave front. Line of equal phase is a position of identical phase which is a wave front (contour of maximum field or a line at the location of the peak) Mathematics for Nano-Engineering:

17 Phase difference The difference in phase between two waves having the same frequency and referenced to the same point in time. Two waves oscillate with the same frequency and difference phases are said to be out of phase with each other. The amount of out-of-phase waves can be expressed in degrees from 0 to 360, or in radians from 0 to 2π. Out-of-phase waves If the phase difference is 180 degrees (π radians), then the two waves are said to be in antiphase Mathematics for Nano-Engineering:

18 Phase velocity, v p How fast the wave propagating!!? The rate at which the phase of the wave propagates in space. Any given phase of the wave will appear to travel at the phase velocity. λ The phase velocity is the wavelength / period: x v p = ω k Mathematics for Nano-Engineering:

19 Group velocity, v g A signal velocity! The velocity at which the overall shape of the wave s amplitude (envelope of the wave) propagates through space. The group velocity is the derivative of the angular frequency with respect to the wave number. v g = ω k Mathematics for Nano-Engineering:

v p and v g The phase velocity of a wave is the velocity which its phase travels in space, while the group velocity is the velocity which its amplitude (envelope) travels in space.

20 v p and v g The phase velocity of a wave is the velocity which its phase travels in space, while the group velocity is the velocity which its amplitude (envelope) travels in space. ω red = 3π 5 rad/s ω green = 0.9ω red and v green = 9 8 v red v orange is the phase velocity of the orange wave which is the average of the speed of red and green waves. v p = The red and green waves travel with speeds indicated by the red and green dots. The phase and group velocities of their sum are (represented as the orange wave) the movement of the blue and black dots respectively Mathematics for Nano-Engineering: Source:

21 Wave using complex numbers Wave can be represented in complex numbers. y (imaginary) P Let x-coordinate be the real part and the y-coordinate be the imaginary part of a complex number. x (real) P = x + iy P (x, y) Mathematics for Nano-Engineering:

22 Wave using complex numbers: Euler s Formula Wave can be represented in complex exponential forms derived from Euler s theorem. e iφ = cos φ + i sin φ So, the point P = A cos φ + ia sin φ can also be written as; The wave function is dependent of amplitude (magnitude) and phase Mathematics for Nano-Engineering:

23 Wave using complex numbers The wave function is a function of space and time that returns complex number. E = E 0 cos kx ωt + ie 0 sin kx ωt i E = E 0 e kx ωt where E 0 is E at x = 0 and t = 0 The wave function is a linear function which means the two solutions can be added to each other to obtain another solution; For harmonic wave, the wave function can also be represented as; E = E 0 cos kx ωt where the argument of the cosine function is the phase of the wave, φ Mathematics for Nano-Engineering:

24 Amplitude and phase shift E = E 0 cos kx ωt + φ or E = E 0 ei kx ωt+φ where E 0 is amplitude of the wave which is the peak magnitude of the oscillation. φ is phase shift of the wave. Absolute phase = 0 E 0 2π kx At a given moment in time, a positive phase shift means the wave is shifted in the negative x-axis direction. A phase shift of 2π radians shifts it exactly one wavelength Mathematics for Nano-Engineering:

25 Plane wave If the peak locations across the space are aligned in planes, the wave is referred to as plane-wave. For plane-wave, the surface of equal phase is a plane with a wavelength apart. The phase of the wave changes uniformly in a plane normal to the direction of propagation. φ = k r ωt +φ where is vector dot product and r is positive vector which defines a point in 3-D space (x, y, z). Wave front (Peak locations or contours of maximum fields) A plane wave function can be generally written as; i k E r, t = E 0 e റr ωt+φ Numerical Method in Electromagnetics 40

26 Cylindrical wave If the surfaces of constant phase are cylinders, then the wave is referred to as Cylindrical wave. The phase of the cylindrical is calculated y using cylindrical coordinates. φ = k r ωt +φ x θ ρ z φ = ρ A cylindrical wave function can be approximately* written as; E r, t = E 0 ρ ei k ρ k x 2 + k z 2 + k y y ωt +φ where ρ = x 2 + z 2 x 2 +k 2 z +k y y ωt+φ *This solution is an approximation of the Hankle function for large values of ρ Numerical Method in Electromagnetics 42

27 Spherical wave If the surfaces of equal phase are spheres, then the wave is referred to as Spherical wave. x y θ φ r z The phase of the spherical is calculated using spherical coordinates. φ = k r ωt +φ A spherical wave function can be generally written as; E r, t = E 0 r ei r k x 2 +k 2 y +k 2 z ωt+φ Numerical Method in Electromagnetics 44

28 Spherical wave: Point source Usually, spherical wave represents light emitting from a very small source (point-source.) Direction of propagation x y θ φ r z For light emitting from a very small light source (point source); φ = k r ωt +φ A point source wave function located at r 0 can be generally written as; Wave front (Peak locations) E r, t = E 0 r ei k(r r 0) ωt+φ Numerical Method in Electromagnetics 46

29 Exercise Colour Frequency (Hz) Wavelength (nm) Red 4.615* Orange 5.084* Yellow 5.263* Green 5.882* Blue 6.315* Violet 7.500* Write the colors in the table above in terms of angular frequencies and wavenumbers Numerical Method in Electromagnetics 48

30 Exercise 2. What is the distance needed to make a blue light gain a phase of 800π? 3. Write the phase of a green light plane wave propagating in the x-y plane with an angle 30 to the x axis. 4. Write the equation for a violet light plane wave with an amplitude 0.9 (V/m) and angles of 45 to the x-z plane and 60 to the y axis. 5. Write the wave function of red light cylindrical wave. 6. Write the wave function of a yellow light spherical wave. 7. For the following spherical wave, what is the light color and where is the point source located? E r, t = E 0 r ei t r Numerical Method in Electromagnetics 49

Exercise ω red = 3π 5 rad/s with v red ω green = 0.8ω red and v green = 8 9 v red v orange is the phase velocity of the orange wave. 8. What is the phase velocity, v orange, of the orange wave? 9. Assume these waves are highly dispersive* in which v red = Cλ 1/2.

31 Exercise ω red = 3π 5 rad/s with v red ω green = 0.8ω red and v green = 8 9 v red v orange is the phase velocity of the orange wave. 8. What is the phase velocity, v orange, of the orange wave? 9. Assume these waves are highly dispersive* in which v red = Cλ 1/2. What is the group velocity of the envelope, v g, compare to the v red? 10. If the waves are in non-dispersive medium, what is v g? *Dispersive medium is a medium in which waves of different frequencies travel at different velocities. This occurs because the index of refraction of the medium is frequency dependent Numerical Method in Electromagnetics 50 Source:

MCQs E M WAVES. Physics Without Fear.

MCQs E M WAVES. Physics Without Fear. MCQs E M WAVES Physics Without Fear Electromagnetic Waves At A Glance Ampere s law B. dl = μ 0 I relates magnetic fields due to current sources. Maxwell argued that this law is incomplete as it does not