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11. 8. 20031 VI–2 Electromagnetic Waves

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11. 8. 20032 Main Topics Properties of Electromagnetic Waves: Generation of electromagnetic waves Relations of and. The speed of Light c. Energy Transport. Radiation Pressure P.

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11. 8. 20033 Generation of Electromagnetic Waves Since changes of electric field produce magnetic field and vice versa these fields once generated can continue to exist and spread into the space. This can be illustrated using a simple dipole antenna and an AC generator. Thisillustrated Planar waves will exist only far from the antenna where the dipole field disappears. Planar

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11. 8. 20034 Relations of and I All properties of electromagnetic waves can be calculated as a general solution of Maxwell’s equations. This needs understanding fairly well some mathematical tools or it is not illustrative. We shall show the main properties for a special case of planar waves and state what can be generalized.

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11. 8. 20035 Relations of and II Let us have a polarized planar wave:polarized in free space with no charges nor currents which moves in the positive x direction the electric field has only y component the magnetic field has only z component. We shall prove relations between time and space derivatives of E and B which are the result of special Maxwell’s equations.

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11. 8. 20036 Maxwell’s Equations

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11. 8. 20037 Relations of and III Let’s first use the Faraday’s law: The line integral of the electric intensity counterclockwise around a small rectangle hdx in the xy plane must be equal to minus the change of magnetic flux through this rectangle:rectanglexy plane

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11. 8. 20038 Relations of and IV Now, let’s similarly use the Ampere’s law: The line integral of magnetic induction counterclockwise around a small rectangle hdx in the xz plane must be equal to the change of electric flux through this rectangle:rectanglexz plane

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11. 8. 20039 Relations of and V Note the symmetry in these equations! Where B decreases in time E grows in x and where E decreases in time B grows in x. This is the reason why E and B must be in-phase.

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11. 8. 200310 General Harmonic Waves I Waves can exist in elastic environment and are generally characterized by the transport of energy (or information) in space but not mass. Deflection of a planar harmonic wave propagating in the +x axis direction by the speed c is either in the direction of propagation or perpendicular: In the point x the deflection is the same as was in the origin before the wave has reached point x. That is x/c =

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11. 8. 200311 General Harmonic Waves II Deflection is periodic both in time and in space: We have used the definitions of the angular frequency, the wavelength and the wave number

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11. 8. 200312 Relations of and VI Now, let us suppose polarized planar harmonic transversal waves: E = E y =E 0 sin( t - kx) B = B z =B 0 sin( t - kx) E and B are in phase Vectors,, form right (hand) turning system

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11. 8. 200313 Relations of and VII From: Since E and B are in-phase, generally: E = c B The magnitude of the magnetic field is c-times smaller!

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11. 8. 200314 Relations of and VIII From: Together if gives the relation of the speed of electromagnetic waves, the permitivity and the permeability of the free space

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11. 8. 200315 The Speed of Light The speed can be found generally from: A t-derivative of the first equation compared to the x-derivative of the second gives the general wave equation for B.wave equation for B Changing the derivatives we get the general wave equation for E. wave equation for E

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11. 8. 200316 General Properties of EMW The solution of ME without charges and currents satisfies general wave equations. Through empty space waves travel with the speed of light c = 3 10 8 m/s. Vectors,, form right turning system The magnitude of the magnetic field is c-times smaller than that of the electric field. Electromagnetic waves obey the principle of superposition.

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11. 8. 200317 Energy Transport of EMW I The energy density of EMW at any instant is a sum of energies of both electric and magnetic fields: From B = E/c and c = ( 0 0 ) -1/2 we get:

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11. 8. 200318 Energy Transport of EMW II We see that the energy density associated with the magnetic field is equal to that associated with the electric field, so each contributes half of the total energy in spite of the peak value difference! ( 0 / 0 ) 1/2 = 0 c is the impedance of the free space = 377 .

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11. 8. 200319 Energy Transport of EMW III The energy transported by the wave per unit time per unit area is given by a Poynting vector, which has the direction of propagating of the wave. The units W/m 2.area The energy which passes in 1 second through some area A is the energy density times the volume: U = uAct

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11. 8. 200320 Energy Transport of EMW IV For general direction of the EMW a vector definition of the Poynting vector is valid: Of course, is parallel to. This is the energy transported at any instant. We are usually interested in intensity, which is the mean (in time) value of S.

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11. 8. 200321 Energy Transport of EMW V For a harmonic wave we can use a result we found when dealing with AC circuits: So we can express the intensity using the peak or rms values of the field variables:

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11. 8. 200322 Radiation Pressure I If EMW carry energy, it can be expected that they also carry linear momentum. If EMW strikes some surface, it can be fully or partly absorbed or fully reflected. In either case a force will be exerted on the surface according to the second Newtons law: The force per unit area is the radiation pressure.

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11. 8. 200323 Radiation Pressure II It can be shown that p = U/c, where is a parameter between 1 for total absorption to 2 for total reflection. So from: F = dp/dt = /c dU/dt = A/c. we can readily get the pressure: P = F/A = /c. This can be significant on the atomic scale or for ‘sailing’ in the Universe.

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11. 8. 200324 The Spectrum of EMW Effects of very different behavior are in fact the same EMW with ‘just’ different frequency.different Radio waves > 0.1 m Microwaves 10 -1 > > 10 -3 m Infrared 10 -3 > > 7 10 -7 m Visible 7 10 -7 > > 4 10 -7 m Ultraviolet 4 10 -7 > > 6 10 -10 m X - rays 10 -8 > > 10 -12 m Gamma rays 10 -10 > > 10 -14 m

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11. 8. 200325 Radio an TV In transmitter a wave of some carrier frequency is either AM or FM modulated, amplified and broadcasted.transmitterAMFM Receiver must use an antenna sensitive either to electric or magnetic component of the wave. Receiverelectricmagnetic Its important part is a tuning stage where the proper frequency is selected.tuning

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11. 8. 200326 Homework No homework assignment today!

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11. 8. 200327 Things to read and learn This lecture covers: Chapter 32 – 4, 5, 6, 7, 8, 9 Advance reading Chapter 33 – 1, 2, 3, 4 Try to understand the physical background and ideas. Physics is not just inserting numbers into formulas!

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A Rectangle in xy Plane ^ We are trying to find an increment dE in the +x axis direction and assume hdx is fixed.

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A Rectangle in xz Plane ^ We are trying to find an increment dB in the +x axis direction and assume hdx is fixed.

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Generalized Ampere’s Law ^ I encl sum of all enclosed currents taking into account their directions and 0 d e /dt is the displacement current due to change-in-time of the electric flux.

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General Wave Equations I ^ After comparing these equations we get the wave equation for B :

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General Wave Equations II ^ After comparing these equations we get the wave equation for E :

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