Transcript Physics 261 - Purdue University

Lecture 1-1
Coulomb’s Law
• Charges with the same sign repel each other, and charges
with opposite signs attract each other.
• The electrostatic force between two particles is
proportional to the amount of electric charge that each
possesses and is inversely proportional to the distance
between the two squared.
q1q2
F1,2 k 2 rˆ1,2
r1,2
 r1,2
by 1
on 2
q1
q2
r
• Coulomb constant:
k
1
4 0
 8.99  109 N  m 2 / C 2
where 0 is called the permittivity constant.
Lecture 1-2
Electric Field
Define electric field, which is independent of the test charge,
q2 , and depends only on position in space:
F
E
q
Electric Field due to a
Point Charge Q
F
1 Q
E 
rˆ
2
q 4 0 r
Lecture 1-3
Dynamics of a Charged Mass in Electric
Field
For -Q<0 in uniform E downward:
F  ma  ( Q ) E
QE
a  ay j 
j (E  E j)
m
1 2
 y (t )  a y t , x (t )  v x t
2
-Q
2
1  x  QEx 2
y  ay   
2
vy2= at
=
qE/m
t
vx >>0
v
2
mv
x
 x
 QEt 
v (t )  v  v y (t )  v   
 m 
tan   y
2
xv
2
2
x
2
• Oscilloscope
• Ink-Jet Printing
• Oil drop experiment
vx
http://canu.ucalgary.ca/map/content/force/elcrmagn/simulate/electric_single_particle/applet.html
Lecture 1-4
Electric Field from Coulomb’s Law
Bunch of Charges
r
i
+
+
+
-
qi+
-
-
1
P
+
-
Continuous Charge Distribution
P
r
dq
+
k
qi
E
rˆ

2 i
4 0 i ri
Summation over
discrete charges
http://www.falstad.com/vector3de/
1 dq
E
rˆ   d E
2
4 0 r
  dV

dq   dA
  dL

(volume charge)
(surface charge)
(line charge)
Integral over continuous
charge distribution
Lecture 1-5
Gauss’s Law: Quantitative Statement
The net electric flux through any closed surface equals the
net charge enclosed by that surface divided by ε0.
 E ndA  
E

Qenclosed
0
How do we use this equation??
The above equation is TRUE always but it doesn’t
look easy to use.
BUT - It is very useful in finding E when the physical situation
exhibits a lot of SYMMETRY.
Lecture 1-6
Charges and fields of a conductor
• In electrostatic equilibrium, free charges inside a
conductor do not move. Thus, E = 0 everywhere in the
interior of a conductor.
• Since E = 0 inside, there are no net charges anywhere in
the interior. Net charges can only be on the surface(s).
 
0
The electric field must be perpendicular to the surface just
outside a conductor, since, otherwise, there would be
currents flowing along the surface.
Lecture 1-7
Electric Potential Energy of a Charge in
Electric Field
• Coulomb force is conservative
=> Work done by the Coulomb force is path
independent.
• Can associate potential energy
to charge q0 at any point r in space.
dl
U (r )
It’s energy! A scalar measured in J (Joules)

dW  q0 E  d l

dU  dW  q0 E  d l
Lecture 1-8
Electric Potential
• U(r) of a test charge q0 in electric field generated by other source charges is
proportional to q0 .
• So U(r)/q0 is independent of q0, allowing us to introduce electric potential V
independent of q0.
U ( r )
V ( r ) 
q0
• [Electric potential] = [energy]/[charge]
SI units:
J/C = V (volts)
1J
U (r)
V (r) 
q0
taking the same
reference point
Scalar!
Potential energy difference when 1 C of charge is moved between points of
potential difference 1 V
E from V
We can obtain the electric field
computes V from
E from the potential V by inverting the integral that
E:

r


r
V (r )   E  d l   ( Ex dx  E y dy  Ez dz)
V
Ex  
x
Expressed as a vector,
V
Ey  
y
E is the negative gradient of V


E  V
V
Ez  
z
Electric Potential Energy and Electric
Potential
positive
Lecture 1-10
charge
High U
(potential
energy)
High V
High V
(potential)
Low U
negative
charge
Low U
Low V
Low V
High U
Electric field direction
Electric field direction
Lecture 1-11
Two Ways to Calculate Potential
-
• Integrate E from the reference point at (∞) to the point (P) of observation:

V r 

E dl
Q
r
 P
P
 A line integral (which could be tricky to do)
 If E is known and simple and a simple path can be used, it may be reduced to a
simple, ordinary 1D integral.
• Integrate dV (contribution to V(r) from each infinitesimal source charge dq) over all
source charges:
q1
P
q2
Q
P
q3
q4
Lecture 1-12
Capacitance
• Capacitor plates hold charge Q
• The capacitance C of a capacitor is a
measure of how much charge Q it can store
for a given potential difference ΔV between
the plates.
The two conductors hold
charge +Q and –Q,
respectively.
Expect
Q  V
Q
 Let C 
V
Capacitance is an
intrinsic property of the
capacitor.
Q Coulomb

F
C   
V  Volt
(Often we use V to mean ΔV.)
farad
Lecture 1-13
Steps to calculate capacitance C
1.
2.
3.
4.
Put charges Q and -Q on the two plates, respectively.
Calculate the electric field E between the plates due to
the charges Q and -Q, e.g., by using Gauss’s law.
Calculate the potential difference V between
the plates
b
due to the electric field E by Vba    E dl
a
Calculate the capacitance of the capacitor by dividing the
charge by the potential difference, i.e., C = Q/V.
Lecture 1-14
Energy of a charged capacitor
How much energy is stored in a charged capacitor?
Calculate the work required (usually
provided by a battery) to charge a capacitor
to Q
Calculate incremental work dW needed
to move charge dq from negative plate
to the positive plate at voltage V.
dW  V (q)  dq   q / C   dq
Total work is then
Q
1
1 Q2
U   dW   qdq 
C0
2 C
2
1
QV
Q
U  CV 2 

2
2
2C
Lecture 1-15
Dielectrics between Capacitor Plates
free charges
+Q
• Electric field E between plates can be calculated
-Q
from Q – q.









E
(Q  q) / A

neutral
-q
+q
Polarization
Charges ± q
0
, V  Ed
Q
Q
C 
V (Q  q )d /  0 A

0 A
d

1
q
1
Q
Lecture 1-16
Capacitors in Parallel
V is common
q1 q2 q3
V 


C1 C2 C3
Equivalent Capacitor:
C
q
V
where
q  q1  q2  q3
q1  q2  q3
 Ceq 
 C1  C2  C3
V
Lecture 1-17
Capacitors in Series
q is common
 q  CV
1 1  C2V2  C3V3
Equivalent Capacitor:
C
q
V
where
V  V1  V2  V3
1 V1  V2  V3 1
1
1


 

Ceq
q
C1 C2 C3
Lecture 1-18
Electric Current
Current = charges in motion
Magnitude
q dq
I  lim

x 0 t
dt
rate at which net positive charges
move across a cross sectional surface
Units:
[I] = C/s = A
(ampere)
Current is a scalar, signed quantity, whose
sign corresponds to the direction of motion
of net positive charges by convention
I  J dA
A
J = current density
(vector) in A/m²
Lecture 1-19
Ohm’s Law
Current-Potential (I-V) characteristic of a device may or may
not obey Ohm’s Law:
I V
( J  E)
or V = IR with R constant
Resistance
tungsten wire
V V

 R    
I  A
gas in fluorescent tube
diode
(ohms)
Lecture 1-20
Energy in Electric Circuits
• Steady current means a constant amount
of charge ΔQ flows past any given cross
section during time Δt, where I= ΔQ / Δt.
Energy lost by ΔQ is
V
U  Q  (Va  Vb )  I t V
=> heat
So, Power dissipation = rate of decrease of U =
dU
P
 IV  I 2 R  V 2 / R
dt
Lecture 1-21
Resistors in Parallel


i

i
iR




i
R

i
R

i
R 
1
12
23
3

12 3e
q
Devices in parallel has the
1
1
1
1
same potential drop




o
r


R
R
R
1
2R
3R
e
q R
e
qR
1
2R
3
Generally,
1
1

Req i Ri
•••
R 
L
A
Lecture 1-22
Kirchhoff’s Rules
Kirchhoff’s Rule 1: Loop Rule
 When any closed loop is traversed completely in a circuit,
the algebraic sum of the changes in potential is equal to zero.

V0

 Coulomb force is conservative
i
loop
Kirchhoff’s Rule 2: Junction Rule
 The sum of currents entering any junction in a circuit is equal to
the sum of currents leaving that junction.
I 
I

i
in
j
o
u
t
 Conservation of charge
 In and Out branches
 Assign Ii to each branch
Lecture 1-23
Galvanometer Inside Ammeter and
Voltmeter
Galvanometer: a device that detects small currents and indicates its
magnitude. Its own resistance Rg is small for not disturbing what is
being measured.
galvanometer
Ammeter: an instrument used to
measure currents
shunt resistor
Voltmeter: an instrument used to
measure potential differences
galvanometer
Lecture 1-24
Galvanometer Inside Ammeter and
Voltmeter
Galvanometer: a device that detects small currents and indicates its
magnitude. Its own resistance Rg is small for not disturbing what is
being measured.
galvanometer
Ammeter: an instrument used to
measure currents
shunt resistor
Voltmeter: an instrument used to
measure potential differences
galvanometer
Lecture 1-25
Discharging a Capacitor in RC Circuits
1.
Switch closed at t=0. Initially
C is fully charged with Q0
2.
Loop Rule:
3.
Convert to a differential equation
dQ
I 
dt
4.
Q
 IR  0
C
Q
dQ
R
0
C
dt
Solve it!
Q  Q0 e  t / RC
dQ Q0  t / RC 
I 

e
dt RC
I
Lecture 1-26
Charging a Capacitor in RC Circuits
1.
Switch closed at t=0
C initially uncharged, thus
zero voltage across C.
I0   / R
2.
Loop Rule:
  IR 
Q
0
C
3. Convert to a differential equation
dQ
I 
dt
4.
dQ Q
 R
 0
dt C
Solve it!
Q  C 1  e  t / RC  , I 
dQ   t / RC
 e
dt R
(τ=RC is the time constant again)
Lecture 1-27
Magnetic Field B
• Magnetic force acting on a moving charge q depends on q,
v.Vary q and v in the presence of a given magnetic
field and measure magnetic force F on the charge.
Find:
F varies sinusoidally as
F v
direction of v is changed
F  qv
Fq
vB

(q>0)
direction by Right Hand
Rule. B is a vector field
This defines B.

F

v
,
B
F
q
v
B
s
i
n
F
NN


B
  
T
(
t
e
s
l
a
)


q
v

m
/
sA

m



C
vB
1 T = 104 gauss (earth magnetic field at surface is
about 0.5 gauss)
If q<0
Lecture 1-28
Magnetic Force on a Current
A
• Consider a current-carrying wire in the
presence of a magnetic field B.
• There will be a force on each of the charges
moving in the wire. What will be the total force
dF on a length dl of the wire?
• Suppose current is made up of n charges/volume
each carrying charge q < 0 and moving with
velocity v through a wire of cross-section A.
• Force on each charge =
• Total force =
• Current =
qv  B
dF  n A(dl ) qv  B
I  n Av q
For a straight length of wire L carrying a current I,
the force on it is:
dF  Idl  B
F  IL  B
Lecture 1-29
Both B and E present
F

q
v
B
u
p
m
F

q
E
d
o
w
n
e
E
v 
B
when balanced
velocity selector
No deflection when
E=3 kV/m, B=1.4 G

4
v

3
0
0
0
/
1
.
4

1
0
0
7

2
.1
4
3

1
0
(
m
/)
s
http://canu.ucalgary.ca/map/content/force/elcrmagn/simulate/exb_thomson/applet.html
Lecture 1-30
Magnetic Torque on a Current Loop
Definition
of torque:
  rF
abut a chosen
point
• If B field is parallel to plane of
loop, the net torque on loop is 0.
B




• If B is not zero, there is net torque.
n
magnetic moment direction
n
so that n is twisted
to align with B
Lecture 1-31
Potential Energy of Dipole
• Work must be done to change the
orientation of a dipole (current loop)
in the presence of a magnetic field.
•
Define a potential energy U (with zero at
position of max torque) corresponding to this
work.
B
x
F

F
.

θ
θ
U   τdθ
U   μB sin θdθ

90
90
Therefore,
U  μB cos θ 90
θ

U   μB cos θ

U    B
Lecture 1-32
Sources of Magnetic Fields
• Moving point charge:
μ0 q v  r
dB 
4π r 2
also
N


0  4  107  2  T  m / A
A 
Permeability constant
• Bits of current:
I
The magnetic field “circulates” around the
wire.
μ0 I d l  r
dB 
4π r 2
Biot-Savart Law
http://falstad.com/vector3dm/
Lecture 1-33
Gauss’s Law for Magnetism
sources
Gauss’s Law
E 

S
E da 
qS
0
Gauss’s Law for Magnetism
No sources
B 

S
B da  0
Lecture 1-34
Ampere’s Law in Magnetostatics
Biot-Savart’s Law can be used to derive another relation: Ampere’s Law
The path integral of the dot product of magnetic field and unit vector along a closed loop,
Amperian loop, is proportional to the net current encircled by the loop,

C
Bt dl 
0 (i1  i2 )

C
B d l  0 I C
•
Choosing a direction of
integration.
•
A current is positive if it
flows along the RHR normal
direction of the Amperian
loop, as defined by the
direction of integration.
Lecture 1-35
Potential Energy of Dipole
• Work must be done to change the
orientation of a dipole (current loop)
in the presence of a magnetic field.
•
Define a potential energy U (with zero at
position of max torque) corresponding to this
work.
B
x
F

F

θ
U   τdθ

U = +μ B cos θ
90
Therefore,
U   μB cos θ
U  μB cos θ 90
θ


.
Lecture 1-36
Faraday’s Law of Induction
The magnitude of the induced EMF in conducting loop is equal to the rate at which
the magnetic flux through the surface spanned by the loop changes with time.
dΦB
ε
dt
wher  B 
e

S
B ndA
N
Minus sign indicates the sense of EMF: Lenz’s Law
• Decide on which way n goes
Fixes sign of ΔϕB
• RHR determines the
positive direction for EMF
N
Lecture 1-37
Ways to Change Magnetic Flux
 B  BA cos
• Changing the magnitude of the field within a conducting loop (or coil).
• Changing the area of the loop (or coil) that lies within the magnetic field.
• Changing the relative orientation of the field and the loop.
motor
generator
Lecture 1-38
Motional EMF of Sliding Conductor
Induced EMF:
 Lenz’s Law gives direction
counter-clockwise
Faraday’s Law
 
 FM decelerates the bar
dv
B 2l 2 v
m 
dt
R
dv
B 2l 2

dt
v
mR
v(t )  v  0 e
 B 2l 2 

t
mR


dB
dx
  Bl
  Blv
dt
dt
 This EMF induces current I

Blv
I  
R
R
 Magnetic force FM acts on this I
B 2l 2 v
FM  I lB 
R
Lecture 1-39
Self-Inductance
• As current i through coil increases,
magnetic flux through itself increases.
This in turn induces counter EMF
in the coil itself
• When current i is decreasing, EMF is
induced again in the coil itself in such a way
as to slow the decrease.
Self-induction
B
NB
L
L
(if flux linked)
i
i
 L  T  m2 / A  Wb / A  H (henry)
Faraday’s Law:
dB
dI
 
 L
dt
dt
Lecture 1-40
1.
2.
Energy Stored By Inductor
Switch on at t=0
As the current tries to begin flowing,
self-inductance induces back EMF,
thus opposing the increase of I.
+
Loop Rule:
dI
  IR  L
0
dt
-
3. Multiply through by I
dI
 I  I R  LI
dt
2
Rate at which battery is
supplying energy
Rate at which energy is
stored in inductor L
dU m
dI
 LI
dt
dt
Rate at which energy is
dissipated by the resistor
Um 
1 2
LI
2
Lecture 1-41
1.
RL Circuits – Starting Current
Switch to e at t=0
As the current tries to begin flowing,
self-inductance induces back EMF,
thus opposing the increase of I.
+
I0  0
2.
Loop Rule:
-
dI
  IR  L
0
dt
3. Solve this differential equation
I

R
1  et /( L / R )  , VL  L
τ=L/R is the inductive
time constant
dI
  e  t /( L / R )
dt
T  m2 / A T  m2 / A

s
 L / R 

V/A
Lecture 1-42
Alternating Current (AC)
= Electric current that changes direction periodically
ac generator is a device which creates an ac emf/current.
A sinusoidally oscillating EMF is induced in a loop of wire that rotates in a uniform
magnetic field.
B  NBA cos  NBA cos t   
dB
 
  NBA  sin t   
dt
where
2
  2 f 
T
ac motor =
ac generator
run in reverse
Lecture 1-43
Resistive Load
Start by considering simple circuits with one element (R,
C, or L) in addition to the driving emf. Pick a resistor R
first.
+
I(t)
Kirchhoff’s Loop Rule:
 t   I t  R  0 ,  t    peak sin t
vR  t   I  t  R   peak sin t
V peak   peak
I t  
Ipeak
Vpeak
R
sin t
vR(t) and I(t) in phase
--
Lecture 1-44
Capacitive Load
Loop Rule:
+
--
q(t )
 (t ) 
0
C
   peak sin  t
q(t )
  peak sin t
C
dq (t )
d  (t )
I (t ) 
C
dt
dt
 C peak cos  t
 v(t ) 
I(t) leads v(t) by 90o (1/4 cycle)


Power: p (t )  I (t )v(t )  I peak cos t VC , peak sin t

VC , peak I peak
2
sin 2t

Pav  0
Lecture 1-45
Inductive Load
Kirchhoff’s Loop Rule:
+
--
dI (t )
 (t )  L
0
dt
 (t )   peak sin t
dI (t )
vL (t )  L
  peak sin t
dt
 peak
I (t )  
cos t
L
vL(t) leads I(t) by 90o (1/4 cycle)


Power: p (t )  I (t )vL (t )   I peak cos t VL , peak sin t

VL , peak I peak
2
sin 2t

Pav  0
Lecture 1-46
(Ideal) LC Circuit
• From Kirchhoff’s Loop Rule
Q
dI
L 0
C
dt
• From Energy Conservation
2
Q 2 1 2  Q peak
E
 LI  
2C 2
 2C
dQ
I
dt
dE
0
dt
same

  const.

Q dQ
dI
 LI
0
C dt
dt
d 2Q  1 

Q  0
2
dt
 LC 
Q  Q peak cos(0t   )
Q
dI
L 0
C
dt
harmonic oscillator with angular
1
frequency
0 
LC
Natural Frequency
Lecture 1-47
Impedance in Driven Series RLC Circuit
 peak  I peak R   X L  X C 
2
2
Z
1 

Z  R   L  
C 

2
2
impedance, Z
1
L 
VL  VC
R

C
tan  

, cos  
VR
R
Z
ϕ
Lecture 1-48
Resonance
For given ε peak , R, L, and C, the current amplitude Ipeak will be at the maximum when the
impedance Z is at the minimum.
 peak  I peak R 2   X L  X C 
2
Z
X L  XC
res L 
1
resC
ε and I in phase
i.e., load purely resistive
This is called resonance.
Resonance angular frequency:
, Z  R, and I peak 
 peak
1
res 
LC
R
Lecture 1-49
Transformer
• AC voltage can be stepped up or down by using a
transformer.
• AC current in the primary coil creates a timevarying magnetic flux through the secondary coil
via the iron core. This induces EMF in the
secondary circuit.
Ideal transformer (no losses and magnetic
flux per turn is the same on primary and
secondary).
(With no load)
d  B V1 V2
 turn 
 
dt
N1 N 2
N1  N2  V1  V2
N1  N2  V1  V2
step-up
step-down
With resistive load R in secondary, current I2 flows in secondary by the induced
EMF. This then induces opposing EMF back in the primary. The latter EMF must
somehow be exactly cancelled because V1 is a defined voltage source. This occurs by
another current I1 which is induced on the primary side due to I2.
Lecture 1-50
Maxwell’s Equations

S
E dA
Qinside
0
dB
 C E d l   dt

S
B dA0
dE
C B d l  0 I  0 0 dt
Basis for electromagnetic waves!
The equations are often written in slightly different (and more convenient)
forms when dielectric and/or magnetic materials are present.
Lecture 1-51
Electromagnetic Wave Propagation in Free
Space
So, again we have a traveling electromagnetic wave
Em 
 c
Bm k
c
Em
1

Bm 0 0c
0  4  107 (T  m / A)
1
0 0
speed of light in
vacuum
 0  8.85  1012 (C 2 / N  m2 )
B
1 E
 2
x
c t
E
B

x
t
Ampere’s Law
Faraday’s Law
 2 B 1  2 B Wave Equation
 2 2
2
x
c t
c  3.00 108 (m / s)
Speed of light in vacuum is
currently defined rather than
measured (thus defining meter and
also the vacuum permittivity).
Lecture 1-52
Maxwell’s Rainbow
Light is an
Electromagnetic Wave
f   c
Lecture 1-53
©2008 by W.H. Freeman and Company
Lecture 1-54
Snell’s Law of Refraction
n1 sin θ1 = n 2 sin θ2
Lecture 1-55
Summary: Laws of Reflection and
Refraction
Law of Reflection
• A reflected ray
lies in the plane of incidence
• The angle of reflection is equal to the angle
of incidence
Medium 1
1  1
Law of refraction
• A refracted ray
lies in the plane of incidence
• The angle of refraction is related to the angle
of incidence by
n2 sin 2  n1 sin 1
Snell’s Law
c / ni c 1 
i 
  
f
f ni ni
Where λ is the
wavelength in
vacuum
Medium 2
 n2  n1
Lecture 1-56
Total Internal Reflection
All light can be reflected, none refracting, when light travels from a
medium of higher to lower indices of refraction.
medium 2
e.g., glass (n=1.5) to air (n=1.0)
sin  2 n1

1
sin 1 n2
2  1
But θ2 cannot be greater
than 90O !
medium 1
In general, if sin θ1 > (n2 / n1), we have NO refracted ray;
we have TOTAL INTERNAL REFLECTION.
c  sin
1
 n2 / n1 
Critical angle above which this occurs.
Lecture 1-57
Polarization of Electromagnetic Waves
Polarization is a measure of the degree to which the electric field (or the
magnetic field) of an electromagnetic wave oscillates preferentially along
a particular direction.
Linear combination
of many linearly
partially
polarized rays of
polarized
random orientations
unpolarized
linearly
polarized
Looking at
E head-on
components
equal y- and zamplitudes
unequal y- and zamplitudes
Lecture 1-58
©2008 by W.H. Freeman and Company
Lecture 1-59
Focal Point of a Spherical Mirror
When parallel rays incident upon a
spherical mirror, the reflected rays or the
extensions of the reflected rays all converge
toward a common point, the focal point of
the mirror. Distance f is the focal length.
concave mirror:
Real focal point: the point to which the
reflected rays themselves pass through.
This is relevant for concave mirrors.
f
convex mirror
Virtual focal point: the point to which
the extensions of the reflected rays pass
through. This is relevant for convex
mirrors.
Rays can be traversed in reverse. Thus,
rays which (would) pass through F and
strike the mirror will emerge parallel to
the central axis.
f
Lecture 1-60
Mirror Equation and Magnification
(f = r/2)
1
1
1


s
s'
f
y' s'
m

y
s
m
s'
s
• s is positive if the object is in front of the mirror (real object)
• s is negative if it is in back of the mirror (virtual object)
• s’ is positive if the image is in front of the mirror (real image)
• s’ is negative if it is in back of the mirror (virtual image)
• m is positive if image and object have the same orientation (upright)
• m is negative if they have opposite orientation (inverted)
• f and r are positive if center of curvature in front of mirror (concave)
• f and r are negative if it is in back of the mirror (convex)
Lecture 1-61
Locating Images
only using the
parallel, focal,
and/or radial
rays.
Real images form on the side of a mirror where the objects are,
and virtual images form on the opposite side.
Lecture 1-62
Thin Lenses nomenclature
1 1
1
  n  1   
• A lens is a piece of transparent material with two
f
refracting surfaces whose central axes coincide. A
 r1 r2 
lens is thin if its thickness is small compared to all
other lengths (s, s’, radii of curvature).
f>0
Convergent lens
• Net convex – thicker in the middle
• Parallel rays converge to real focus.
r1>0
r2<0
•f>0
f<0
Divergent lens
• Net concave – thinner in the middle
• Parallel rays diverge from virtual focus.
•f<0
r1<0
r2>0
Lecture 1-63
Signs in the Lens Equation for Thin Lenses
1 1 1
 
p q f
q
m
p
• p is positive for real object
• p is negative for virtual object
• q is positive for real image
yi
f q
q


yo
f
p
• q is negative for virtual image
• m is positive if image is upright
• m is negative if image is inverted
• f is positive if converging lens
• f is negative if diverging lens
Lecture 1-64
Geometric Optics vs Wave Optics
• Geometric optics is a limit of the general optics where wave effects such as interference
and diffraction are negligible.
 Geometric optics applies when objects and
apertures involved are much larger than the
wavelength of light.
 In geometric optics, the propagation of light can
be analyzed using rays alone.
• Wave optics (sometimes also called physical optics) - wave effects
play important roles.
 Wave optics applies when objects and
apertures are comparable to or smaller than
the wavelength of light.
 In wave optics, we must use the concepts
relevant to waves such as phases, coherence,
and interference.
Lecture 1-65
Thin-Film Interference-Cont’d
(Assume near-normal
incidence.)
Path length difference:
l  2t
destructive
m


(m  1/ 2) constructive
where

0
n
• ray-one got a phase change of 180o due to reflection from air to glass.
• the phase difference due to path length is:
•then total phase difference:  = ’+180.
'
l

 2 
l

'
n
 2
Lecture 1-66
Interference Fringes
For D >> d, the difference in path lengths
between the two waves is
L  d sin 
•
A bright fringe is produced if the path
lengths differ by an integer number of
wavelengths,
d sin   m , m  0, 1,
•
A dark fringe is produced if the path
lengths differ by an odd multiple of
half a wavelength,
d sin   (m  1/ 2), m  0, 1,
Lecture 1-67
Intensity of Interference Fringes-Cont’d
For Young’s double-slit experiment, the phase difference is
  2 
d sin 


maxima
 22 (mm, for
1/2), for mimina
4 I 0 cos
2

2
Lecture 1-68
Dark and Bright Fringes of Single-Slit
Diffraction
Lecture 1-69
Intensity Distribution 1
2
  
 sin  2  
2 a sin 
I ( )  I max 
 where  

  
 2 
maxima:
 0
or
sin
central maximum because
sin x
 1 as x  0
x

1
 1 or a sin   ( m  )
2
2
minima:
sin

2
 0 or a sin   m
m  1, 2, 3,... m  0
Lecture 1-70
Intensity Distribution from Realistic DoubleSlit Diffraction
I  4 I 0 cos 
2
double-slit intensity
 sin  
replace by I m 

  
2
 sin  
2
I ( )  I m (cos  ) 





d
sin 

single-slit intensity envelope
2
a

sin 

d sin  m2
How many maxima will fit
d
m

between central max and
2
a
first envelope min?:
Lecture 1-71
Diffraction Gratings
• Devices that have a great number of slits or rulings to produce an
interference pattern with narrow fringes.
• One of the most useful optical tools. Used to analyze wavelengths.
D
Types of gratings:
• transmission gratings
• reflection gratings
up to thousands
per mm of rulings
Maxima are produced when every pair of
adjacent wavelets interfere constructively, i.e.,
D
d
d sin   m , m  0, 1,
mth order maximum