maxwell’s equations and electromagnetic...
TRANSCRIPT
![Page 1: Maxwell’s Equations and Electromagnetic Wavespages.erau.edu/~snivelyj/ps250/PS250-Lecture27.pdf · PS 250: Lecture 27 Maxwell’s Equations and Electromagnetic Waves J. B. Snively](https://reader034.vdocuments.mx/reader034/viewer/2022042020/5e77721c8b21ab4c815a60ff/html5/thumbnails/1.jpg)
PS 250: Lecture 27 Maxwell’s Equations and Electromagnetic
WavesJ. B. Snively
November 13th, 2015
![Page 2: Maxwell’s Equations and Electromagnetic Wavespages.erau.edu/~snivelyj/ps250/PS250-Lecture27.pdf · PS 250: Lecture 27 Maxwell’s Equations and Electromagnetic Waves J. B. Snively](https://reader034.vdocuments.mx/reader034/viewer/2022042020/5e77721c8b21ab4c815a60ff/html5/thumbnails/2.jpg)
Today’s Class
Maxwell’s EquationsIntro to WavesSummary
![Page 3: Maxwell’s Equations and Electromagnetic Wavespages.erau.edu/~snivelyj/ps250/PS250-Lecture27.pdf · PS 250: Lecture 27 Maxwell’s Equations and Electromagnetic Waves J. B. Snively](https://reader034.vdocuments.mx/reader034/viewer/2022042020/5e77721c8b21ab4c815a60ff/html5/thumbnails/3.jpg)
Maxwell’s EquationsI
⇥E · d ⇥A =Q
encl
�o
Gauss’s Law:(E Field)
I�B · d �A = 0
Gauss’s Law:(B Field)
I⇥B · d⇥l = µ
o
✓iC
+ �o
d�E
dt
◆
encl
Ampere’s Law:(B Field)
I�E · d�l = �d�B
dt
Faraday’s Law:(E Field)
![Page 4: Maxwell’s Equations and Electromagnetic Wavespages.erau.edu/~snivelyj/ps250/PS250-Lecture27.pdf · PS 250: Lecture 27 Maxwell’s Equations and Electromagnetic Waves J. B. Snively](https://reader034.vdocuments.mx/reader034/viewer/2022042020/5e77721c8b21ab4c815a60ff/html5/thumbnails/4.jpg)
Together with the Lorentz Force (F=qE+qvxB), these equations describe classical electromagnetic interactions.
With some math, can be expressed in “point form” or “differential form”, which allows convenient calculation of the electromagnetic wave equations.
Maxwell’s Equations
![Page 5: Maxwell’s Equations and Electromagnetic Wavespages.erau.edu/~snivelyj/ps250/PS250-Lecture27.pdf · PS 250: Lecture 27 Maxwell’s Equations and Electromagnetic Waves J. B. Snively](https://reader034.vdocuments.mx/reader034/viewer/2022042020/5e77721c8b21ab4c815a60ff/html5/thumbnails/5.jpg)
Maxwell’s EquationsGauss’s Law:
(E Field)
Gauss’s Law:(B Field)
Ampere’s Law:(B Field)
Faraday’s Law:(E Field)
r⇥ ⇤B = µo
⇤J + �
o
⇥ ⇤E
⇥t
!
r⇥ ⇥E = �� ⇥B
�t
� · �B = 0
� · ⇤E =⇥
�o
![Page 6: Maxwell’s Equations and Electromagnetic Wavespages.erau.edu/~snivelyj/ps250/PS250-Lecture27.pdf · PS 250: Lecture 27 Maxwell’s Equations and Electromagnetic Waves J. B. Snively](https://reader034.vdocuments.mx/reader034/viewer/2022042020/5e77721c8b21ab4c815a60ff/html5/thumbnails/6.jpg)
Electromagnetic Radiation (i.e., Radiation of Waves)
Acceleration of Charge
Changing Currentor
Occurs as a result of:
![Page 7: Maxwell’s Equations and Electromagnetic Wavespages.erau.edu/~snivelyj/ps250/PS250-Lecture27.pdf · PS 250: Lecture 27 Maxwell’s Equations and Electromagnetic Waves J. B. Snively](https://reader034.vdocuments.mx/reader034/viewer/2022042020/5e77721c8b21ab4c815a60ff/html5/thumbnails/7.jpg)
Today’s Class
Maxwell’s EquationsIntro to WavesSummary
![Page 8: Maxwell’s Equations and Electromagnetic Wavespages.erau.edu/~snivelyj/ps250/PS250-Lecture27.pdf · PS 250: Lecture 27 Maxwell’s Equations and Electromagnetic Waves J. B. Snively](https://reader034.vdocuments.mx/reader034/viewer/2022042020/5e77721c8b21ab4c815a60ff/html5/thumbnails/8.jpg)
Electromagnetic Waves
Transverse – E and B fields are perpendicular to the direction of propagation. Propagates in direction of ExB.
In a vacuum (Free Space), the wave propagates at the speed of light “c”.
Magnitudes of E and B are related by
![Page 9: Maxwell’s Equations and Electromagnetic Wavespages.erau.edu/~snivelyj/ps250/PS250-Lecture27.pdf · PS 250: Lecture 27 Maxwell’s Equations and Electromagnetic Waves J. B. Snively](https://reader034.vdocuments.mx/reader034/viewer/2022042020/5e77721c8b21ab4c815a60ff/html5/thumbnails/9.jpg)
Electromagnetic Waves Wavelength and Frequency
Propagate at the speed of light:
c =1
p�o
µo
�
k= c
k=wavenumber, where wavelength: � =2⇥
k
and frequency (Hertz) is given by:
We can thus write: � =c
f
f =⇥
2�
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Electromagnetic Waves The Spectrum
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Electromagnetic Plane Waves
![Page 12: Maxwell’s Equations and Electromagnetic Wavespages.erau.edu/~snivelyj/ps250/PS250-Lecture27.pdf · PS 250: Lecture 27 Maxwell’s Equations and Electromagnetic Waves J. B. Snively](https://reader034.vdocuments.mx/reader034/viewer/2022042020/5e77721c8b21ab4c815a60ff/html5/thumbnails/12.jpg)
Electromagnetic Waves Sinusoidal Solutions...
E = Emax
cos(kx� �t)
B = Bmax
cos(kx� �t)
Emax
= cBmax
Field amplitudes determined by speed of light:
c =1
p�o
µo
For a plane wave traveling in the x-direction:
![Page 13: Maxwell’s Equations and Electromagnetic Wavespages.erau.edu/~snivelyj/ps250/PS250-Lecture27.pdf · PS 250: Lecture 27 Maxwell’s Equations and Electromagnetic Waves J. B. Snively](https://reader034.vdocuments.mx/reader034/viewer/2022042020/5e77721c8b21ab4c815a60ff/html5/thumbnails/13.jpg)
Electromagnetic Waves Sinusoidal Solutions...
⇥E =
ˆjEmax
cos(kx� �t)⇥B =
ˆkBmax
cos(kx� �t)
⇥B = �ˆkBmax
cos(kx+ �t)
⇥E =
ˆjEmax
cos(kx+ �t)
![Page 14: Maxwell’s Equations and Electromagnetic Wavespages.erau.edu/~snivelyj/ps250/PS250-Lecture27.pdf · PS 250: Lecture 27 Maxwell’s Equations and Electromagnetic Waves J. B. Snively](https://reader034.vdocuments.mx/reader034/viewer/2022042020/5e77721c8b21ab4c815a60ff/html5/thumbnails/14.jpg)
Electromagnetic Waves Materials other than “Free Space”
Define wave speed v, where v≤c:
v =1
p�µ
� = K�o
µ = Km
µo
Recall that: ,
v =1
p�µ
=1p
KKm
1p�o
µo
=cp
KKm
![Page 15: Maxwell’s Equations and Electromagnetic Wavespages.erau.edu/~snivelyj/ps250/PS250-Lecture27.pdf · PS 250: Lecture 27 Maxwell’s Equations and Electromagnetic Waves J. B. Snively](https://reader034.vdocuments.mx/reader034/viewer/2022042020/5e77721c8b21ab4c815a60ff/html5/thumbnails/15.jpg)
Electromagnetic Waves “Index of Refraction”
Can define index of refraction, relating the wave speed v in a material with speed of light c:
n =c
v=
pKKm
Keep in mind, however, that dielectric “constant” K varies with frequency.
(Also note that Km = 1 for many materials.)
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Summary / Next Class:
Mastering Physics for Monday.
Homework for next-next Wednesday