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Physics 272 November 25 Fall 2014 http://www.phys.hawaii.edu/~philipvd/pvd_14_fall_272_uhm.html Prof. Philip von Doetinchem [email protected] Phys272 - Fall 14 - von Doetinchem - 105 Electromagnetic waves ● What is light? → electromagnetic wave → electromagnetism is needed (not the complete story → QFT) ● Time varying magnetic field creates electric field Time varying electric field creates magnetic field → sustain each other and create an electromagnetic wave that propagates through space Phys272 - Fall 14 - von Doetinchem - 106 Electromagnetic waves Source: http://en.wikipedia.org/wiki/Light ● Electromagnetic waves carry energy and momentum ● Electric and magnetic fields in sinusoidal waves have defined frequency ● ● Infrared, visible light, UV, X-ray, -ray, etc. all follow the same principle, but at different wavelength Electromagnetic waves do not require medium (like mechanical waves) Phys272 - Fall 14 - von Doetinchem - 107 Electricity, magnetism, and light ● ● ● ● Maxwell's equation relate magnetic and electric field Moving charges produce both electric and magnetic fields An electromagnetic wave is formed when a charge accelerates We never spoke about how fast a magnetic field can be measured at a certain distance after a charge starts moving: – Electromagnetic waves do not travel with infinite speed Phys272 - Fall 14 - von Doetinchem - 108 Generating electromagnetic radiation ● ● Look at oscillating L-C circuit Transformation of the LC circuit into a conducting rod: → this is still an oscillating LC circuit Phys272 - Fall 14 - von Doetinchem - 109 Generating electromagnetic radiation ● What do the electric and magnetic fields look like: ● In initial configuration fields are contained in capacitor and inductor ● In the rod configuration the electric and magnetic field overlap ● Current is moving up and down the rod → charge in “capacitor” and current in “inductor” are changing with time → electric and magnetic field change with time → electric and magnetic field propagate with finite velocity → electromagnetic wave Phys272 - Fall 14 - von Doetinchem - 110 Hertzian dipole Source: http://de.wikipedia.org/wiki/Hertzscher_Dipol Change of electric field over time Phys272 - Fall 14 - von Doetinchem - 111 Plane electromagnetic waves ● ● ● Electromagnetic waves are transverse waves with an electric and a magnetic component → similar to waves on a string Make the following assumption: – Electric and magnetic field configuration with wave-like behavior – Electric field has only a y component – Magnetic field has only a z component – Both move in x direction with velocity c We have to test if this assumption is consistent with Maxwell's equation Phys272 - Fall 14 - von Doetinchem - 112 Plane electromagnetic waves ● Why perpendicular? ● Faraday's law: ● Integration path perpendicular to electric field: no contribution Propagation velocity ● Magnetic flux change: ● Magnetic field component in z direction is crucial to comply to Faraday's law Phys272 - Fall 14 - von Doetinchem - 113 Plane electromagnetic waves ● ● In a very similar way: use Ampere's law (no conduction current): any components of E and B not perpendicular to each other and not perpendicular to the direction do not play a role to fulfill Maxwell's equation → Electric and magnetic fields must be perpendicular ● To fulfill Maxwell's equations → wave must also fulfill: Propagation velocity Phys272 - Fall 14 - von Doetinchem - 114 Key properties of electromagnetic waves ● ● Electromagnetic waves fulfill the general wave equation for both electric and magnetic fields (not discussed here): Wave is transverse: – ● ● Electric field and magnetic field are perpendicular to the direction of the wave Magnitudes of E and B are related by the propagation velocity: E=cB In vacuum (or if medium is not changing) the electromagnetic wave is traveling at a constant velocity ● Electromagnetic waves do not need a medium ● Simple plane waves can be generalized – E and B fields can be superposed – For each superposition the simple principles of the plane electromagnetic waves apply – Also wave packets or sinusoidal waves fulfill Maxwell's equations Phys272 - Fall 14 - von Doetinchem - 115 Sinusoidal electromagnetic waves ● ● ● ● At any instant the fields are uniform over any plane perpendicular to the direction of propagation with speed c The entire pattern travels in the direction of propagation Electric and magnetic field are still transverse to the propagation direction at any instant In a small region of space at great distance from source → electromagnetic waves can be treated as plane waves Phys272 - Fall 14 - von Doetinchem - 116 Fields of a sinusoidal waves ● ● ● ● Electric and magnetic fields oscillate in phase The fields are at the same time at maximum, zero, and minimum Vector product of electric field and magnetic field always points in the propagation direction Electromagnetic wave can be described by wave function depending on location and time k is the wave number: 2/ Phys272 - Fall 14 - von Doetinchem - 117 Fields of a sinusoidal waves ● ● The electric field and magnetic field amplitudes are related by: The sign in front of t denotes the direction of the wave: – Negative sign: positive x direction – Positive sign: negative y direction Phys272 - Fall 14 - von Doetinchem - 118 Why are magnetic and electric field in phase? ● ● Assume arbitrary phase angle: Relation between electric and magnetic field using Faraday's law: Phys272 - Fall 14 - von Doetinchem - 119 Why are magnetic and electric field in phase? ● Relation between electric and magnetic field using variation of B is Faraday's law: small for small x calculate magnetic flux through area Phys272 - Fall 14 - von Doetinchem - 120 Why are magnetic and electric field in phase? ● ● Use our approach for the wave function assuming a phase angle: To make these equations equal at all times we have to require Phys272 - Fall 14 - von Doetinchem - 121 Electromagnetic waves in matter ● The wave speed changes in matter (we assume non-conducting dielectric) ● We have to use the dielectric constant and the permittivity of the material: ● Electromagnetic waves can never travel faster than the speed of light in vacuum ● ● Definition of index of refraction n of a material (→ optics) Careful: – dielectric “constant” depends on frequency – for high frequencies materials cannot be polarized as fast as for constant electric fields → reduces dielectric “constant” Phys272 - Fall 14 - von Doetinchem - 122 Energy and momentum in electromagnetic waves ● Electromagnetic waves transport energy: – ● ● Microwave ovens, radio transmitters, lasers for eye surgery, … Combined energy density of electric and magnetic components: In vacuum electric field and magnetic field carry half of the total energy density Phys272 - Fall 14 - von Doetinchem - 127 Electromagnetic energy flow and the Poynting vector ● ● ● Energy is required to establish electric and magnetic fields → electromagnetic waves transport energy from one region to the other Definition: – energy transferred per unit time per unit cross-sectional area – or power per unit area In a certain time the wave moves in space with the propagation velocity → energy in a particular region: Phys272 - Fall 14 - von Doetinchem - 128 Electromagnetic energy flow and the Poynting vector ● Poynting vector is a vector quantity: it points in the direction of the propagation of the wave and depends on time ● SI unit: 1 W/m2 John Henry Poynting (1852-1914) ● Total energy flow per unit time: Phys272 - Fall 14 - von Doetinchem - 129 Electromagnetic energy flow and the Poynting vector ● ● Frequency of electromagnetic waves is fast → average Poynting vector value (intensity) For our sinusoidal wave: zero on average ● ● The intensity for a sinusoidal wave is exactly half of the maximum value: This is what we sense when looking at, e.g., light from the sun → variations are too fast to be noticeable for us Phys272 - Fall 14 - von Doetinchem - 130 Electromagnetic momentum flow and radiation pressure ● ● ● ● Electromagnetic waves also carry momentum (→ deeper explanation requires quantum physics) Electromagnetic wave's momentum is transferred to a surface Transferred momentum per time equals the force on the surface Sun and stars create radiation pressure that effects the surrounding material → star forming regions Massive Star Forming Region DR21 in Infrared Credit: A. Marston (ESTEC/ESA) et al., JPL, Caltech, NASA Phys272 - Fall 14 - von Doetinchem - 131 Standing electromagnetic waves ● Electromagnetic waves can be reflected on surfaces – ● ● ● Dielectrics or conductors can serve as reflectors Superposition principle of electric and magnetic fields also applies to electromagnetic waves Superposition of incident and reflected wave forms a standing wave Electric force is conservative → it is not possible to do work on a test charge like that: Phys272 - Fall 14 - von Doetinchem - 134 Additional Material Phys272 - Fall 14 - von Doetinchem - 135 Energy in a sinusoidal wave ● ● A radio station on the earth's surface emits a sinusoidal wave with average total power of 50kW Assume that transmitter radiates equally in all directions ● Area of hemisphere: ● Intensity: Phys272 - Fall 14 - von Doetinchem - 136 Energy in a sinusoidal wave ● ● A radio station on the earth's surface emits a sinusoidal wave with average total power of 50kW Assume that transmitter radiates equally in all directions ● Electric field amplitude: ● Magnetic field amplitude: magnetic field associated with electromagnetic wave is extremely small compared to what we saw before → it is easier to have device sensitive to the electric field Phys272 - Fall 14 - von Doetinchem - 137 Solar sail ● ● ● Solar sails try to make use out of the radiation pressure as a propulsion system Usage of sails big as football fields should catch the radiation pressure by the sun Concept is proven, but not yet part of real spacecrafts Source: http://en.wikipedia.org/wiki/Solar_sail Phys272 - Fall 14 - von Doetinchem - 138