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WARNING:
Exam is Tuesday Nov. 25th in class
Review Sessions:
In class: Friday
Me:
Monday
Office Hours:
Brian F.: Friday
Various: Sunday
Brian P.: Monday
Monday
12-1 pm; 2-3 pm
4-6 pm 35-225
1-2 pm
1-5 pm
3-4 pm
6-8 pm
4-344
TEAL
4-344
6-106
Please email me questions
P28- 1
Class 30: Outline
Hour 1:
Displacement Current
Hour 2:
Electromagnetic waves
P28- 2
Last Time:
Driven RLC Circuits
P28- 3
AC Circuits: Summary
Element
I0
Resistor
V0 R
R
Capacitor
Inductor
 CV0C
V0 L
L
Current
vs.
Voltage
In Phase
Leads
Lags
Resistance
Reactance
Impedance
RR
1
XC 
C
XL  L
Although derived from single element circuits,
these relationships hold generally!
P28- 4
Driven RLC Series Circuit
V0L
V0S

I (t )  I 0 sin(t   )
V0C
VS  V0 S sin   t 
I0
V0R
V0 S  VR 0  (VL 0  VC 0 )  I 0 R  ( X L  X C )  I 0 Z
2
V0 S
I0 
Z
2
Z  R  ( X L  XC )
Impedance
2
2
2
  tan
2
1
 X L  XC 


R


P28- 5
Resonance
V0
I0  
Z
C-like:
<0
I leads
V0
R 2  ( X L  X C )2
;
1
X L   L, X C 
C
  tan
1
 X L  XC 


R


L-like:
>0
I lags
0  1 LC
P28- 6
This Time:
Putting it All Together
P28- 7
Displacement Current
P28- 8
Ampere’s Law: Capacitor
Consider a charging capacitor:
I
Use Ampere’s Law to calculate the
magnetic field just above the top plate
Ampere's law:

B  d s   0 I enc
1) Red Amperian Area, Ienc= I
2) Green Amperian Area, I = 0
What’s Going On?
P28- 9
Displacement Current
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?
Q
E
 Q   0 EA   0 E
0 A
dE
dQ
 0
 Id
dt
dt
This is called (for historic reasons)
the Displacement Current
P28- 10
Maxwell-Ampere’s Law
B

d
s


(
I

I
)
0
encl
d

C
 0 I encl
dE
 0 0
dt
P28- 11
PRS Questions:
Capacitor
P28- 12
In Class Problem:
Displacement Current
P28- 13
Maxwell’s Equations
P28- 14
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
Gauss’s Law
Faraday’s Law
Ampere’s Law
Maxwell’s Addition
• E (B) fields exert forces on (moving) electric charges
Lorentz Force
P28- 15
Maxwell’s Equations
Qin
 E  dA  
S
(Gauss's Law)
0
dB
C E  d s   dt
(Faraday's Law)
 B  dA  0
(Magnetic Gauss's Law)
dE
C B  d s  0 I enc  0 0 dt
(Ampere-Maxwell Law)
F  q (E  v  B)
(Lorentz force Law)
S
P28- 16
Electromagnetic Radiation
P28- 17
A Question of Time…
P28- 18
Electromagnetic Radiation:
Plane Waves
P28- 19
Traveling Waves
Consider f(x) =
x=0
What is g(x,t) = f(x-vt)?
t=0
t=t0
t=2t0
x=0
x=vt0 x=2vt0
f(x-vt) is traveling wave moving to the right!
P28- 20
Traveling Sine Wave
Now consider f(x) = y = y0sin(kx):
Amplitude (y0)
2
Wavelength ( ) 
wavenumber (k )
x
What is g(x,t) = f(x+vt)? Travels to left at velocity v
y = y0sin(k(x+vt)) = y0sin(kx+kvt)
P28- 21
Traveling Sine Wave
y  y0 sin  kx  kvt 
At x=0, just a function of time: y  y0 sin(kvt )  y0 sin(t )
Amplitude (y0)
1
Period (T ) 
frequency (f )
2

angular frequency ( )
P28- 22
Traveling Sine Wave
Wavelength: 
Frequency : f
Wave Number: k 
y  y0 sin(kx  t )
2

Angular Frequency:   2 f
1 2
Period: T  
f


Speed of Propagation: v 
f
k
Direction of Propagation:  x
P28- 23
Electromagnetic Waves
Hz
Remember:
f c
P28- 24
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)
P28- 25
PRS Question:
Wave
P28- 26
Group Work:
Java Problem 1
P28- 27
Properties of EM Waves
Travel (through vacuum) with
speed of light
1
m
vc
 3 10
s
0 0
8
At every point in the wave and any instant of time,
E and B are in phase with one another, with
E E0

c
B B0
E and B fields perpendicular to one another, and to
the direction of propagation (they are transverse):
Direction of propagation = Direction of E  B
P28- 28
Direction of Propagation
ˆ E sin(k  pˆ  r   t ); B  B
ˆ B sin(k  pˆ  r   t )
EE
0
0
ˆ B
ˆ  pˆ
E
ˆ
E
ˆi
Bˆ
ˆj
pˆ
kˆ
 pˆ  r 
ˆj
kˆ
kˆ
ˆi
ˆi
ˆj
x
ˆj
kˆ
ˆi
ˆj
kˆ
ˆi
z
x
ˆi
kˆ
ˆj
y
z
y
P28- 29
PRS Question:
Direction of Propagation
P28- 30
Energy & the Poynting Vector
P28- 31
Energy in EM Waves
Energy densities:
Consider cylinder:
1
1
2
2
uE   0 E , uB 
B
2
20
2

1
B
2
dU  (uE  uB ) Adz    0 E   Acdt
2
0 
What is rate of energy flow per unit area?
2
 c
1 dU c 
B
EB 
2
S
  0 E 
    0cEB 

A dt
2
0  2 
c 0 
EB
EB
2

 0 0c  1 
0
20


P28- 32
Poynting Vector and Intensity
Direction of energy flow = direction of wave propagation
S
E B
0
: Poynting vector
units: Joules per square meter per sec
Intensity I:
2
0
2
0
E0 B0
E
cB
I  S 


20 20c 20
P28- 33
Energy Flow: Resistor
S
EB
0
On surface of resistor is INWARD
P28- 34
PRS Questions:
Poynting Vector
P28- 35
Energy Flow: Inductor
S
EB
0
On surface of inductor with increasing
current is INWARD
P28- 36
Energy Flow: Inductor
S
EB
0
On surface of inductor with decreasing
current is OUTWARD
P28- 37
In Class Problem:
Poynting Vector
P28- 38
Momentum & Radiation Pressure
EM waves transport energy: S 
They also transport momentum:
And exert a pressure:
EB
0
U
p
c
F 1 dp 1 dU S
P 


A A dt cA dt
c
This is only for hitting an absorbing surface. For
hitting a perfectly reflecting surface the values are
doubled:
2U
2S
Momentum transfer: p 
; Radiation pressure: P 
c
c
P28- 39
Standing Waves
P28- 40
Standing Waves
What happens if two waves headed in opposite
directions are allowed to interfere?
E1  E0 sin(kx  t )
E2  E0 sin(kx  t )
Superposition: E  E1  E2  2E0 sin(kx) cos(t )
P28- 41
Standing Waves: Who Cares?
Most commonly seen in resonating systems:
Musical Instruments, Microwave Ovens
E  2E0 sin(kx) cos(t )
P28- 42
Standing Waves: Bridge
P28- 43
Microwave Ovens: Hot Spots
Can you measure the speed of light with marshmallows?
P28- 44
Microwave Ovens: Hot Spots
P28- 45
Microwave Ovens: Hot Spots
OR
P28- 46
Microwave Ovens: Hot Spots
P28- 47
Microwave Ovens: Hot Spots
P28- 48
Group Work: Standing Waves
Play with Problem 2 in Java Example!
E1  E0 sin(kx  t )
E2  E0 sin(kx  t )
Superposition: E  E1  E2  2E0 sin(kx) cos(t )
P28- 49