Download TFY4170 - Fysikk 2

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
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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.
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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
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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.
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!
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!