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TFY4170 - Fysikk 2 Forelesning 8: Bølgefysikk Electromagnetic waves, Maxwell’s equations, energy density Mansfield & O’Sullivan: 18.1,18.2,18.5. (18.3,18.4) Waves ! ! ! ! ! ! ! Wave phenomena Wave equation in one dimension Energy, power and intensity of waves Plane waves and dispersing waves. Huygens principle. Reflection and refraction (brytning). Interference and diffraction. Young’s double slit, many waves. Diffraction in crystals, Xray, neutron and electron diffraction. Standing waves, resonance. ! ! ! ! ! ! Doppler effect (classic and relativistic). Lasers and coherent waves. The wave equation. Mechanical waves and sound waves. Electromagnetic waves, Maxwell’s equations, polarisation. Wave packets and envelopes, group velocity, dispersion. Fourier analysis, bandwidth. Electromagnetic waves ! ! ! ! ! ! Light is an electromagnetic wave. Electromagnetic waves consist of an oscillating electric field AND magnetic field, which propagate. Electromagnetic waves also follow the wave equation. The wave velocity is c, the speed of light. Light is a transverse wave. The electric and magnetic fields are polarised in the perpendicular directions to the wave propagation. The electromagnetic spectrum. ! ! ! ! Radio-waves few kHz to around 1GHz wavelength from many kilometers to around 1 meter Microwaves more than 1GHz and up to about 100GHz wavelength between 1mm between 1m thermal waves (heat-transport / infra-red) frequency between 1012 Hz and 4x1014Hz wavelength from 800 nm to 1mm Visible light frequency between 4x1014Hz and 8x1014Hz wavelength from 400nm to 800nm The electromagnetic spectrum. Ultra-violet light frequency from 8x1014Hz to 1017Hz wavelength from 1nm to 400nm ! X-ray radiation frequency from 1017 to 1019Hz wavelength from 0.01nm to 1nm ! Gamma-radiation frequency over 1018Hz (overlaps with hard x-rays) wavelength less than 0.1nm ! Electromagnetism - repetition A charge in an electric field experiences a force: A (moving) charge in a magnetic field experiences a force: The total force on the charge is: An electric and/or magnetic field is created by a charge/current in the surrounding area. Begge deler er kjent for de fleste gjennom dagligdagse opplevelser. Elek særstilling, med alt vi omgir oss med av strømkrevende dingser. Magnet i folks bevissthet med ting som kjøleskapsmagneter, men som vi skal mer enn det – og elektrisitet og magnetisme er dessuten nært sammen Electromagnetism Det som kalles Maxwells ligninger er fire ligninger som til sammen dann To find out how an electromagnetic wave propagates, need to find Clerk out tromagnetismen. De er oppkalt etter den skotske we fysikeren James the microscopic description of the electric and magnetic som oppdaget sammenhengene mellom ligningene fields: og brakte det hele teori. Flere av de faktiske ligningene ble oppdaget av, og er oppkalt ett er alle de fire Maxwell-ligningene listet opp: The electromagnetic field is given by Maxwell’s equations: ⇢ r·E= ✏0 Ga r ⇥ B = µ0 J + µ0 ✏ 0 r⇥E= r·B=0 @B @t @E @t Ampère-Maxw Faradays induk Gauss’ lov for mag Disse ligningene er vektorligninger. Det er fordi elektromagnetismen er at den fundamentale bestanddelen i teorien er felter som man tenker se Electric and magnetic fields: Maxwell We introduce the electric displacement field: Therefore: Maxwell’s first equation can be simplified to: magnetisme, og elektromagnetisk teori dreier seg svært konsentrert om disse to gge deler er kjent for de fleste gjennom dagligdagse opplevelser. Elektrisitet er stilling, med alt vi omgir oss med av strømkrevende dingser. Magnetisme er of lks bevissthet med ting som kjøleskapsmagneter, men som vi skal se er mag r enn det – og elektrisitet og magnetisme er dessuten nært sammenknyttet i Electric and magnetic fields: Maxwell H-field or B-field? t som kalles Maxwells ligninger er fire ligninger som til sammen danner grunnla magnetismen. De er oppkalt etter den skotske fysikeren James Clerk Maxwell ( m oppdaget sammenhengene mellom ligningene og brakte det hele frem til e ri. Flere av de faktiske ligningene ble oppdaget av, og er oppkalt etter, andre freeMaxwell-ligningene space, the two fields are opp: very similar: alle deInfire listet M=0 if the medium is not magentized r·E= ⇢ ✏0 @E r ⇥ B = µ0 J + µ0 ✏ 0 @t @B r⇥E= @t r·B=0 Gauss’ lov ⌘ @D r ⇥ H = J Ampère-Maxwells + lov @t Faradays induksjonslov Gauss’ lov for 9magnetisme Maxwell’s equations Maxwell’s equations are: r·D=⇢ r·B=0 @B @t @D r⇥H=J+ @t r⇥E= We will concentrate on electromagnetic waves in materials without free charges and currents, and which are not magnetized. Thus we can further simplify these equations: @B @t r·D=0 r⇥E= r·B=0 @D r⇥H= @t Energy in an electromagnetic system The energy in an electromagnetic system (such as a wave) is the sum of the energy due to the electric field and the magnetic field We will now try to show that the energy density w is given by: We will show this by studying the energy in a electric field (capacitor) and magnetic field (solenoid) separately. Capacitance Capacitance is defined as the ratio between the amount of charge stored, and the potential difference: The work done in moving a charge is: The total energy is therefore: -Q/2 Q/2 Energy in an electric field: The energy stored in a capacitor is: For a parallel-plate capacitor: The energy density in the field is: W w= vol Energy in an electric field: ! Energy density in a constant electric field is: ! This result is independent on the source of the field! We can generalise the result to fields with a position dependence: ! Energy in a magnetic field The energy stored in a solenoid is: The energy density is: W w= vol The self inductance is: Energy in a magnetic field ! The energy in a constant magnetic field is: ! This result is also independent on the source of the field! We can also generalise this equation to include a position dependence: ! Energy in a electromagnetic field We have found the energy density in an electric field: And the energy density in a magnetic field: The total energy is therefore: Repetition – forelesning 8 Maxwell’s equations describe the relationship between electric and magnetic fields. Radio-waves, microwaves, infra-red waves, visible light, X-rays, gamma-rays are all examples of electromagnetic waves, just with different frequencies. The energy density in an electromagnetic wave is: Next time, we will see how we can express electromagnetic waves using the wave equation, and what the consequences of this are!