Download Asin

Waves can be represented by simple harmonic motion
Standing wave
y = Asin(kx − ωt) + Asin(kx + ωt)
y (t )  A sin( kx  t )
The amplitude of a wave is a measure of the maximum disturbance in the medium
during one wave cycle. (the maximum distance from the highest point of the crest to
the equilibrium).
The wavelength (denoted as λ) is the distance between two sequential crests (or
troughs). This generally has the unit of meters.
A wavenumber
k  2
2
Period T 


Phase velocity:
v

k


T
 f
Electromagnetic waves
f 
c

Light as a Wave (1)

c = 300,000 km/s =
3*108 m/s
• Light waves are characterized by a
wavelength  and a frequency f.
• f and  are related through
f = c/
The Electromagnetic Spectrum
Wavelength
Frequency
Need satellites
to observe
High
flying air
planes or
satellites
Dual, wave-particle nature of light
E  hf  h
c

h  6.6  1034 Joule  sec
1 eV = 1.6x10-19 J
c = 3x108 m/s
1 Angstrom = 10-10 m
Speed of light in matter:
cm = c/n, where
n is refractive index
Note: n is a function of 
Light as a Wave (2)
• Wavelengths of light are measured in units
of nanometers (nm) or Ångström (Å):
1 nm = 10-9 m
1 Å = 10-10 m = 0.1 nm
Visible light has wavelengths between
4000 Å and 7000 Å (= 400 – 700 nm).
Light as Particles
• Light can also appear as particles, called
photons (explains, e.g., photoelectric effect).
• A photon has a specific energy E,
proportional to the frequency f:
E = h*f
h = 6.626x10-34 J*s is the Planck constant.
The energy of a photon does not
depend on the intensity of the light!!!
Maxwell’s Equations
  Q
 E  dS 
 1
E  
 
 B  dS  0

 B  0
 
d  
 E  dr   dt  B  dS

 
 E   B
t
0
 
d  
 B  dr 0i  0 0 dt  E  dS
0


 
  B  0 j  0 0 E
t
No charges, no real currents
 
 E  dS  0
 
 B  dS  0
 
d  
 E  dr   dt  B  dS
 
d  
 B  dr 0 0 dt  E  dS

 


E  0ix  E y ( x, t )iy  0iz




B  0ix  0iy  Bz ( x, t )iz
Wave equation
2Ey
x
2
  0 0
2Ey
t 2
 2 Bz
 2 Bz
  0 0 2
2
x
t
2Ey
x
2
  0 0
2Ey
t 2
E y  A sin( kx  t )
k
2
2

T

k is a wave number,  is a wave length, T is the period
Velocity of propagation
v

k

1
0 0
c
Coulomb’s Law

1 q1q2
FE 
40 x 2
1
40
 9 109 Newton  meter 2 / coulomb 2  9 109 N  m 2 / C 2
 0 is the permittivity of free space
Charge
Charge
q1
q2
Conservation of electric charge
Charge is conserved: in any isolated
system, the total charge cannot
change.
If it does change, then the system is
not isolated: charge either went
somewhere or came in from
somewhere
r̂12
r̂21
Charge
Charge
q1
q2
rˆ12  rˆ21

F2
Let’s denote the force that q1 exerts on q2 as
and force exerted
by q2 on q1 as F1 ; r is the distance between charges.


1 q1q2
1 q1q2
F1 
rˆ12  
rˆ21   F2
2
2
40 r
40 r
(Newton’s third law works!)
Like charges repel; opposites attract
8
Exercise: If two electrons are placed 10 meters apart, what is
the magnitude of the Coulomb force between them? Compare this to the
gravitational force between them.
r
Solution: The magnitude of electric force
19 2
q2
(
1
.
6

10
)
9
12
FE 

9

10

N

2
.
3

10
N
2
8 2
40 x
(10 )
1
The magnitude of gravitational force
31 2
m1m2
(
9

10
)
11
55
FG  G 2  6.67 10
N

5
.
4

10
N
8 2
x
(10 )
FE
q2
2.3 43


10
FG 40Gm1m2 5.4
(no matter what the separation is)
Gauss’s Law
  Q
 E  dS 
0
A conducting sphere, conducting shell, insulating sphere, shell …..
Two parallel conducting plates


-
+
+
+
l
-
+
a
+
-
+
d
 
 E  dS 
 EdS  Ea
cap
a
Ea 
0

E
0
(the total field at any point
between the plates)
Capacitors
Consider two large metal plates which are parallel to each other
and separated by a distance small compared with their width.
y
Area A
L








The field between plates is






 
 
V

E
0

 [V (top)  V (bottom)]   E y dy 
dy   L
0
0
0
0
L
L


 A
QL
 [V (top)  V (bottom)]   L  
L
0
0 A
0 A
QL
V 
A 0
The capacitance is:
A 0
Q
Q
C


QL
V
L
A 0
A 0
Q
Q
C


V QL
L
A 0
Capacitors in series:
1
1
1
1



 ...
Ctot C1 C2 C3
Capacitors in parallel: Ctot  C1  C2  C3  ...
1
1 2
2
W  CV 
Q
2
2C
[C ]  farad
Current, Ohm’s Law, Etc.
dQ
i
dt
V
Ohm ' s Law : R  ; R  Const (independent of V )
i
l
R
A


j  E


E  j
Current Density
 
i   j  dS
S
Consider current flowing in a homogeneous wire with cross sectional
area A.
 
i   j  dS   jdS  j  dS  jA
A
A
i
j
A
A
For steady state situation
 
j

d
S

0

 
 E dr  0
Circuits will be included!
Joule’s Law
2
V
P  Vi  i R 
R
2
The force

F
on a charge q moving with a velocity

  
F  q( E  v  B)
If

E 0

v
the magnitude of the force
F  qvB sin 
[ B]  Newtons /(Coulomb  meter / sec)
1T (tesla)  1w / m  1Newton / C  m / s  10 gauss
2
4
4
BEarth  1Gauss  10 T
 
v || B F  0
 
v  B F  qvB


F  ma

ir
Fr  mar
F  0
2
v
 qvB  mr 2  m
r
mv
r
qB
The angular velocity
v
v
qB
 

r mv
m
qB
Using Crossed

E
and

B Fields
Velocity selector
qvB  qE  0
E  vB
E
v
B
independent of the mass of the particle!
Ampere’s Law
 
B

d
r


i
0

The field produced by an infinite wire
0 i
B
2 a
Biot-Savart Law
Infinitesimally small element of a current carrying wire produces an
infinitesimally small magnetic field

dS


 i ( ds  r )
dB 
3
r
i

r
  0 i (ds  r )
dB 
4
r3
0
is called permeability of free space
0  4 10 7 webers /( amp  meter)  4 10 7 N /( amp) 2
(Also called Ampere’s principle)
Force exerted on a current carrying wire

 
dF  ids  B
For a straight, finite wire of length
  
F  il  B
l
and uniform magnetic field
Faraday’s Law of Induction
The induced EMF in a closed loop equals the negative of the
time rate of change of magnetic flux through the loop
d B
EMF  
dt
 
d B
d  
 E  dr   dt   dt  B  dS
There can be EMF produced in a
number of ways:
•
•
•
•
A time varying magnetic field
An area whose size is varying


A time varying angle between B and dS
Any combination of the above
From Faraday’s law: a time varying
flux through a circuit will induce an
EMF in the circuit. If the circuit
consists only of a loop of wire with
one resistor, with resistance R, a
current
R
EMF
i
R
Which way?
Lenz’s Law: if a current is induced by some change, the direction of
the current is such that it opposes the change.
 
d B
 E  dr   dt
A Simple Generator
Faraday’s Law is used to obtain differential
equations for some simple circuits.
 
 
 E  dr   B  dS
Self-inductance L
 
 B   B  dS   Li
Displacement current
d  
iD   0  E  dS
dt
 
d  
 B  dr 0i  0 0 dt  E  dS
Thank you for a great semester!
I’ll miss this class!