Chapter 23: Electromagnetic waves. What will we learn in this chapter?

Transcription

1 Chapter 23: Electromagnetic waves Arecibo 305m radio telescope What will we learn in this chapter? Contents: Speed of EM waves The EM spectrum Transmitted energy Nature of light Reflection and Refraction Total internal reflection Polarization Huygens principle Scattering of light Convention: In this lecture EM = electromagnetic. solar spectrum

2 Brief introduction Recall Faraday s law:!! A changing magnetic field B (flux) induced an emf in a circuit: E = Φ B t!! An induced emf produces an electric field E. This suggests that time-dependent electric and magnetic fields are intrinsically related, thus we can use the term electromagnetic field. We can assume that energy travels in waves, like ripples in a pond. This is summarized in Maxwell s equations: E = ρ/ɛ 0 B E = t B =0 B E = µ 0 ɛ 0 t + µ 0 j Speed of an electromagnetic wave Setup: Assume we have an E-field in the z-direction and a B-field in the y-direction producing an electromagnetic wave traveling in the x-direction. Rudimentary EM wave: The yellow plane corresponds to the wave front traveling with a speed c. In front of the wave front, there is no field. Behind the wave front, E and B fields are perpendicular to each other.

3 Speed of an electromagnetic wave contd. Conditions for the rudimentary wave to be an EM wave: At any point in time, the magnitudes of E and B have to satisfy: The wave front moves with a speed c given by c = 1 ɛ0 µ 0 where ɛ 0 = C 2 /Nm 2 µ 0 =4π 10 7 N/A 2 Note: E = cb Note that c = m/s is the speed of light in vacuum! By combining two of Maxwell s equations, we obtain the wave equation (harmonic motion) with the aforementioned speed. Characteristics of EM waves in vacuum Conditions a EM wave has to fulfill: The wave is transverse, i.e., both E and B are perpendicular to the direction of propagation and to each other. There is a definite ratio between the magnitudes: E = cb The wave travels in vacuum with a constant speed c. The wave does not need a medium to propagate. The fields oscillate. Modified right-hand rule for EM waves: Point the right thumb in the travel direction of the wave. Index fingers points in E direction. Middle finger points in B direction.

4 The electromagnetic spectrum long wavelength short wavelength EM waves cover a broad spectrum of wavelengths λ, but all propagate with the same speed. Hence c = λf. The visible light only covers a narrow range of the spectrum. Although we cannot see outside the visible range, there are machines and animals that can. The Milky Way in different wavelengths what we see

5 Nature and invisible spectrum ranges Pit vipers: Can see in the infra-red spectrum. Detect heat generated by prey and hunt blindly. Bees: Can see in the ultraviolet. Some plants advertise in this wavelengths. Shell parakeets: Females determine the nicest mate in the UV range. UV visible visible pit viper UV Harmonic waves Harmonic EM waves are analogous to transverse mechanical waves in stretched strings. In an EM wave, the E and B field are sinusoidal in time. This is similar to the rudimentary wave in that the fields are uniform in the propagation front. We call such a wave a plane wave. EM waves can be described by wave functions: ( t y(x, t) =A sin 2π T x ) = A sin(ωt kx) λ where y is the transverse displacement at time t of a point with coordinate x on a string. A is the amplitude, ω =2πf and k =2π/λ is the wave number.

6 Harmonic waves contd. We can now represent the EM fields as follows: ( t E = E max sin 2π T x ) = E max sin(ωt kx) λ ( t B = B max sin 2π T x ) = B max sin(ωt kx) λ The amplitudes must be related via E max = cb max Note: A wave whose E-field always lies along the same line is said to be linearly polarized. In the example the wave is linearly polarized in the y direction. Energy in electromagnetic waves Energy can be associated with electromagnetic waves. Examples: Note: Sun (heats up planet) Microwave oven (resonance with water, heats food) X-rays Energy density of electric and magnetic fields: The energy density u (energy per volume) in a vacuum with electric and magnetic fields present is u = 1 2 ɛ 0E µ 0 B 2 Use B = c 1 E = ɛ 0 µ 0 E and replace the B field by an E field. It follows that u = ɛ 0 E 2. The energy density of both fields is the same. Since EM waves travel, it follows that they transfer energy.

7 Transferred energy Traveling EM waves transport energy in space. Intensity of a wave: Transferred power per unit area. Study a wave in x direction which moves by x = c t. Consider an area A on the wave front. The energy in that region is the volume V = A x times the energy density: U = u V =(ɛ 0 E 2 )(Ac t) The energy flow per area A and time t thus is S = 1 U A t = ɛ 0cE 2 This can be rewritten U( t) =u V Transferred energy contd. We can also derive the following alternative forms: The units of S are power per unit area [J/(sm 2 )]. Note: S = ɛ 0 ce 2 = 1 µ 0 EB = cu So far we have only treated instantaneous values of the fields. For a harmonic wave the average value of E 2 is 1/2 the amplitude. It follows: S av = 1 ɛ0 Emax 2 = 1 2 µ 0 2 ɛ 0cEmax 2 = 1 E max B max 2µ 0 Similarly to previous calculations we obtain for a sinusoidal wave S av = u av = u av c ɛ0 µ 0 which is also known as the Intensity I = S av.

8 Radiation pressure The fact that EM waves transport energy follows from the fact that energy is required to establish the E and B fields. It can also be shown that EM waves carry momentum p, with a corresponding momentum density p V = ɛ 0E 2 = EB c µ 0 c 2 = S c 2 For harmonic waves we can look at the average momentum: Note: p av V = ɛ 0Emax 2 = E maxb max c µ 0 c 2 = S av c 2 and the momentum flow p in a time interval t trough A is 1 p A t = 1 2 ɛ 0E 2 = S av = I c c The EM momentum is a property of the field and not associated with any moving particle masses. Radiation pressure contd. The momentum transfer is responsible for radiation pressure: A propagating wave excerpts a force on a surface. The average pressure is I/c. If the wave is totally reflected then the pressure is 2I/c. Examples: The radiation pressure of sunlight on an absorbing surface is approximately 4.7x10-6 Pa (this can be measured!). Theoretically, one could use the pressure of the sun s EM waves to power spacecrafts so called sun sails. Some projects are currently being explored. Tails of comets always face away from the sun due to the sun s radiation pressure.

9 Nature of light Is it a particle? Is it a wave? Is it a ray? It s all of them! Rays: Reflection, refraction, geometrical optics, Wave: Diffraction Particle: The emission/absorption of light by atoms suggests that light also has a particle character (quanta of energy). We call these photons. EM radiation (light) sources Electric charges in accelerated motion: X-ray machines Thermal radiation due to molecular motion: Flames & coals in a camp fire Hot metallic objects appear red hot Light bulbs (the old kind) Electrical discharge in ionized gases: Sodium lamps (crucial for astronomers) Neon signs Lasers: Monochromatic light sources

10 Wave fronts Definition (wave front):! Locus of all adjacent points at which the!!!!!!! phase of vibration of the wave is the same. Examples: ripples in a pond, sound waves. The wave crests are separated from each other by one wavelength. For EM waves the displaced quantity is the electric or magnetic field: point source far away from the source Wave fronts contd. It is often convenient to represent the light wave as light rays instead of wave fronts. In fact, early descriptions of light assumed particles traveling along straight paths. A ray is an imaginary line along the direction of travel of the wave. Rays can change direction when different mediums come together. This will be discussed next within the ray model of light propagation.

11 Reflection and refraction Common scenario: Different images of the hat can be seen. Reflection and refraction contd. Analysis: Light strikes the smooth glassy surface and changes medium. Part of the wave is refracted (transmitted). Part of the wave is reflected. Simplified problem: It is enough to study the behavior of one representative ray of light. One can define angles between incident/reflected/refracted rays with respect to the Normal.

12 Specular vs diffuse If the interface is rough, both transmitted and reflected light are scattered in various directions and there is no single angle of transmission or reflection. Specular reflection: On a smooth surface There is a definite angle Diffuse reflection: Rough surface Scattered reflection No definite angle Index of refraction Material-dependent quantity which plays a central role in geometric optics. Index of refraction: The index of refraction of an optical material (n) is the ratio of the speed of light in vacuum (c) to the speed of light in the material (v): n = c v For vacuum n = 1 by definition. Note: In general, light travels slower in a material hence n > 1. As we will see later, n can be determined geometrically as well.

13 Principles of geometric optics The incident, reflected, and refracted rays, and the normal to the surface, all lie in the same plane. The angle of reflection θ r is is equal to the incidence angle θ a for all wavelengths and for any pair of substances: θ a = θ r For monochromatic light passing from medium a to medium b the angle of refraction is given by sin θ a sin θ b = n b n a (Snell s law) Snell s law: some considerations A ray passes from material a to material b nb > na The refracted beam bends toward the normal. nb < na The refracted beam bends away from the normal If the angle of incidence is zero, the rays is not bent.

14 Snell s law contd. The path of the refracted ray is always reversible. The ray follows the same path when going from b to a as when going from a to b. The path of the reflected ray is also always reversible. The intensities of the reflected and refracted rays depend on the angle of incidence, the indices of refraction and the polarization of the incoming ray. Reflection is smallest when the angle of incidence is zero. Index of refraction for yellow sodium light λ = 589nm

15 Dispersion The index of refraction depends on the wavelength of the incident light. This is called dispersion. This is how rainbows form and why prisms produce mini rainbows. Wavelengths at interfaces When light passes from one material to another, the frequency f does not change. Reason: The boundary cannot create/destroy waves. The number arriving per unit time must be the same as the number leaving per unit time. In any material v = λf. Because f is fixed and v = c/n it follows that in a medium where λ 0 λ = λ 0 n is the vacuum wavelength.

16 Total internal reflection Rays are emitted from a point P in a material a and strike the surface of a material b with nb < na. Because na/nb > 1 there must be a critical angle for which θ b = 90. θ a The critical angle for which light travels along the interface is given by sin θ crit = n b n a For larger angles than θ crit the sine is not properly defined, the light is trapped in the lower material and reflected back. Total internal reflection: If nb < na a ray is reflected back into the material if the angle of incidence is greater than. θ crit = arcsin(n b /n a ) Total internal reflection contd. For a glass-air surface θ crit = arcsin(1/1.52) = The fact that this angle is smaller than 45º makes it possible to use triangular prisms as a totally reflecting surface. This has advantages over metallic mirrors since the latter oxidize and do not reflect 100%. This technology is commonly used in optical instruments: Porro prism binoculars

17 Total internal reflection contd. Optical fibers: When light enters a transparent rod which is not curved too strongly it is trapped (optical fibers). Technological applications: Data transfer Endoscopes International radio transmission: Thanks to total reflection on the atmosphere we can tune into Deutsche Welle in the USA. Or just listen online Texas DW Polarization (mechanical analogy) Definition: When a transverse wave has only displacement in the!-axis we say the wave is!-polrized. Examples: linearly-polarized in the y-direction linearly-polarized in the z-direction The wave will oscillate in the x-y or x-z plane, respectively. Polarizing filters: Permit motion only in a given direction.

18 Polarizing filters for EM waves Construction: Depends on the wavelength range to be polarized. Examples Microwaves: A grid of closely-spaced, parallel conducting wires. If E wires the field passes. unpolarized light Optical: Polaroid filters use materials that exhibit dichroism (anisotropic absorption of polarizing components). 80% of the intensity is transfered parallel to the polarizing axis, less than 1% in other directions. horizontal polarization absorbed filter linearly-polarized light Polarized EM waves Ordinary light sources: Light is emitted by many out of sync molecules. Thus light is unpolarized. Radio transmitter antennas: Usually linearly polarized (vertically in the US, horizontally in the UK). Material science: Can be used to detect stress in materials. Later

19 Polarizing light The intensity of the transmitted light is the same for all filter orientations. Why? The incident E-field can be described as parallel and perpendicular to the polarization axis. When we measure the intensity (power/area) only 50% get trough since the incident light is randomly polarized and thus the components of the field are on average equal. Place a second polarizer before photocell angle between polarizer and analyzer linear polarized light can be divided into components maximal intensity for zero angle Once the light is linearly polarized it can be cleanly divided into components. Only the parallel component E = E cos φ is transmitted. The intensity of EM waves is proportional to the square of the amplitude. It follows

20 Polarizing filters contd. Light transmitted by polarizing filters (Malus law): When linear-polarized light strikes a polarizing filter with its axis at an angle φ to the direction of the polarization, the intensity of the transmitted light is I = I max cos 2 φ with Imax the maximum intensity at φ =0. The effects of polarizing filters are nicely illustrated by the pair of sunglasses. No transmission for 90º. The reason sunglasses have polarizing filters is to eliminate glare from asphalt surfaces. φ = 90 φ =0 Polarization by reflection Unpolarized light can be partially polarized by reflection on a surface between two optical materials if it strikes the surface at an angle. Parallel to surface is refracted. θ p Refracted light is slightly polarized in the plane of incidence.

21 Polarization by reflection: Brewster s law When lights strikes a surface at a particular angle θ p, the field perpendicular to the plane of incidence is reflected and thus polarized in that direction (parallel to the surface). θ p When light at an angle hits the surface, none of the E-field parallel to the plane of incidence is reflected; it is transmitted 100% in the refracted beam. While the reflected light is polarized, the refracted component is partially polarized. component perpendicular to page When the light hits the surface at θ p, refracted and reflected beams are perpendicular to each other. Brewster s law: tan θ p = n b n a