1 P28- Class 28: Outline Hour 1: Displacement Current Maxwell’s Equations Hour 2: Electromagnetic waves
1P28-
Class 28: Outline
Hour 1: Displacement Current Maxwell’s Equations
Hour 2: Electromagnetic waves
2P28-
Finally: Bringing it All Together
3P28-
Displacement Current
4P28-
Ampere’s Law: Capacitor
Consider a charging capacitor: I Use Ampere’s Law to calculate the
magnetic field just above the top plate
1) Red Amperian Area, Ienc= I 2) Green Amperian Area, I = 0
What’s Going On?
0Ampere's law: enc d Iµ⋅ =∫ B s
5P28-
Displacement Current
0 E
d ddQ I
dt dtε Φ = ≡
We don’t have current between the capacitor plates but we do have a changing E field. Can we “make” a current out of that?
0 0 0
E
QE Q EA A
ε εε
= ⇒ = = Φ
This is called (for historic reasons) the Displacement Current
6P28-
Maxwell-Ampere’s Law
0
0 0 0
( )encl d C
E encl
d I I
dI dt
µ
µ µ ε
⋅ = +
Φ = +
∫ B s
7P28-
PRS Questions: Capacitor
8P28-
Maxwell’s Equations
9P28-
Electromagnetism Review
• E fields are created by: (1) electric charges (2) time changing B fields
• B fields are created by (1) moving electric charges
(NOT magnetic charges) (2) time changing E fields
• E (B) fields exert forces on (moving) electric charges
Gauss’s Law Faraday’s Law
Ampere’s Law
Maxwell’s Addition
Lorentz Force
10P28-
Maxwell’s Equations
0
0 0 0
(Gauss's Law)
(Faraday's Law)
0 (Magnetic Gauss's Law)
(Ampere-Maxwell Law)
( (Lorentz force Law)
in
S
B
C
S
E enc
C
Qd
dd dt
d
dd I dt
q
ε
µ µ ε
⋅ =
Φ ⋅ = −
⋅ =
Φ ⋅ = +
= + ×
∫∫
∫
∫∫
∫
E A
E s
B A
B s
F E v B)
11P28-
Electromagnetic Radiation
12P28-
A Question of Time…
http://ocw.mit.edu/ans7870/8/ 8.02T/f04/visualizations/light/ 05-CreatingRadiation/05-pith_f220_320.html
13P28-
14P28-
Electromagnetic Radiation: Plane Waves
http://ocw.mit.edu/ans7870/8/8.02T/f04/visualizations/light/07-EBlight/07-EB_Light_320.html
15P28-
Traveling Waves
Consider f(x) =
x=0
What is g(x,t) = f(x-vt)?
x=0
t=0
x=vt0
t=t0
x=2vt0
t=2t0
f(x-vt) is traveling wave moving to the right!
16P28-
Traveling Sine Wave Now consider f(x) = y = y0sin(kx):
x
Amplitude (y0) 2Wavelength ( )
wavenumber ( )k πλ =
What is g(x,t) = f(x+vt)? Travels to left at velocity v y = y0sin(k(x+vt)) = y0sin(kx+kvt)
17P28-
Traveling Sine Wave
Amplitude (y0)
1Period ( ) frequency ( )
2 angular frequency ( )
T f π
ω
=
=
( )0 siny y kx kvt= +
0 0sin( ) sin( )y y kvt y tω= ≡At x=0, just a function of time:
18P28-
Traveling Sine Wave
0 sin( )y y kx tω= −Wavelength: Frequency :
2Wave Number:
Angular Frequency: 2 1 2Period:
Speed of Propagation:
Direction of Propagation:
f
k
f
T f
v fk x
λ
π λ ω π
π ω
ω λ
=
=
= =
= =
+
i i
i
i
i
i
i
19P28-
Electromagnetic Waves
Remember: f cλ =
Hz
20P28-
Electromagnetic Radiation: Plane Waves
Watch 2 Ways: 1) Sine wave
traveling to right (+x)
2) Collection of out of phase oscillators (watch one position)
Don’t confuse vectors with heights – they are magnitudes of E (gold) and B (blue)
http://ocw.mit.edu/ans7870/8/8.02T/f04/visualizations/light/07-EBlight/07-EB_Light_320.html
21P28-
PRS Question: Wave
22P28-
Group Work: Do Problem 1
23P28-
Properties of EM Waves
8
0 0
1 3 10 m v c sµ ε
= = = ×
0
0
EE cB B
= =
Travel (through vacuum) with speed of light
At every point in the wave and any instant of time, E and B are in phase with one another, with
E and B fields perpendicular to one another, and to the direction of propagation (they are transverse):
Direction of propagation = Direction of ×E B
Direction of Propagation E E= E sin( k p̂ ⋅r −ωt); B B = B sin( k p̂ ⋅r −ωt)ˆ
0 ( ) ˆ0 ( )
P28-24
( ) ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆˆ
ˆ ˆˆ ˆ ˆ ˆ
ˆ ˆˆ ˆ ˆˆ
z x y z x y
⋅
− − − − − −
E B p p r i j k
j k i k i j j i k
k j i i k j
ˆ ˆ ̂× =E B p
25P28-
PRS Question: Direction of Propagation
26P28-
In Class Problem: Plane EM Waves
27P28-
Energy & the Poynting Vector
28P28-
Energy in EM Waves 2 2
0 0
1 1 ,2 2E Bu E u Bε
µ = =Energy densities:
Consider cylinder: 2 2
0 0
1( ) 2E B
BdU u u Adz E Acdtε µ
⎛ ⎞ = + = +⎜ ⎟
⎝ ⎠
What is rate of energy flow per unit area?
0 02
c EB cEB c
ε µ
⎛ ⎞ = +⎜ ⎟
⎝ ⎠
( )2 0 0
0
1 2 EB cε µµ
= + 0
EB
µ =
1 dUS A dt
= 2
2 0
02 c BEε
µ ⎛ ⎞
= +⎜ ⎟ ⎝ ⎠
29P28-
Poynting Vector and Intensity
0
: Poynting vector µ × = E BS
units: Joules per square meter per sec
Direction of energy flow = direction of wave propagation
Intensity I: 2 2 0 0 0 0
0 0 02 2 2 E B E cB I S
cµ µ µ ≡< > = = =
30P28-
Energy Flow: Resistor
0µ × = E BS
On surface of resistor is INWARD
31P28-
PRS Questions: Poynting Vector
32P28-
Energy Flow: Inductor On surface of inductor with increasing current is INWARD
0µ × = E BS
33P28-
Energy Flow: Inductor On surface of inductor with decreasing current is OUTWARD
0µ × = E BS