Recap of last lecture
particles
dsin = m (maximas)
dsin = (m+ ) (minimas)
For small sin ~ tan
or d.ymax /R = m
or ymax = mR/d
I = 0cE2p
= I0 cos2/2
waves
In Youngs experiment, coherent light passing through
two slits (S1 and S2) produces a pattern of dark
Recap of last lecture
For small the maximas are at 2t = mn where
n = 0/n
Maxima for 2t=(m+ ) m=0,1,2
Minima for 2t = m
Phase shift of /2
An air wedge between two glass plates
Just like the thin film, two waves reflect back from the air wedge
in close pro
Recap of last lecture
Bright rings when net phase shift = 2t+ /2 =m or 2t = (m+ )
dark rings when 2t = m m=0,1,2,
Thin film coatings can make perfect
reflectors or perfect absorbers
Diffraction
and
Fraunhofer Diffraction
P will be a dark band if
asin =
O
Recap of last lecture
w1/a
85% of
the power
P will be a dark band if
asin =
Or
sin = m/a m =1, 2,
For small
sin ~ ~ tan
y = xm/a with m =1, 2,
Intensity in a single-slit pattern
Ep = E0sin(/2)/(/2)
sin/2
I = I0
/2
2
= (2/)*[path difference] = (2/
Recap of last lecture
Revisit 2 slit interference: actually diffraction pattern from each
slit interfere to produce the pattern seen which is a little different
from the idealized 2 slit pattern we saw in the last chapter.
I I0 cos
2 sin /2
2
2 /2
f = (
Recap of last lecture
.
When light has a wavelength l much smaller than objects that it
interacts with, we can treat light as composed of straight-line rays.
This regime is called geometric optics.
Reflections from a spherical mirrors
a + b = 2f
tan a = h
Recap of last lecture
Spherical reflecting surfaces can be concave or
convex. (concave is when the mirror is silvered
on the outer surface while convex is the other
kind.)
Under the paraxial approximation (small
angles or light almost parallel to the opti
Recap of last lecture
na/s + nb/s = (nb-na)/R
m = y/y = -nas/nbs
A pair of spherical surfaces can form concave or
convex lenses. Light refracting through these
spherical surfaces tend to converge in convex lenses
and diverge in concave lenses.
For thin le
Recap of last lecture
Human eye: most of the bending (refraction) due to the cornea which has a
small focal length.
The lens is used to get additional variable focusing so that near and far objects
can be on focus at the retina
Normal eye near point = 25
Recap of last lecture
Magnification
by a simple
magnifier
'
M
h/ f
h/ N
N
f
If eye focuses at near p oint,
achieve slightly greater power :
M
N
1
f
N = near point , 25 cm
Microscope magnification = M1*M2 = s25/f1f2
Telescope magnification = -f1/f2
Microsc
Recap of last lecture
Postulate of Classical relativity:
Laws of mechanics (Newtons laws ) are invariant in all inertial reference frames
Crisis in classical relative: In classical relativity the velocity of light would be
different in a frame moving with
Recap of last lecture
Consequences of the postulates of special relativity:
1. Speed of light is a constant
2. Simultaneity of events is frame dependent
3. Time intervals are frame dependent
4. spatial intervals (length) is frame dependent
5. Space-time i
Recap of last lecture
Consequences of the postulates of special relativity:
1. Time in moving frames is dilated t = t
2. Length in moving frames is contracted l = l/
3. = 1/(1-u2/c2)1/2
4. The complete Lorentz transforms are given by:
t = (t - ux/c2)
x= (
Recap of last lecture
Looked at 4 phenomena which are problematic for classical physics
Line spectra and black body radiation spectrum can be explained in
terms of discrete energy levels in atoms where E1 E2 = hf;
Ephoton = hf
The same particle like prope
Recap of last lecture
Evidence for wave like behavior of particles:
Diffraction and two slit interference patterns
These are a consequence of the uncertainty principle
DpDx h
Particles can be described by wave-functions y (x,y,z,t)
Particle wave-functions
Recap of last lecture
Light can also be polarized by reflection. It is 100% polarized in
the direction parallel to the reflecting surface when the angle of
incidence = brewster angle = tan-1 (nb/na)
Scattered light perpendicular to the direction of propag
Recap of last lecture
The wavelength dependence of refraction is called dispersion
na = c/va = 0/a
There are two states of linear polarization of light
Certain materials block EM waves polarized light in a particular
direction
If the angle between the pol
Recap of last lecture
For light traveling from a to b where na > nb, as the angle of
incidence becomes more and more acute, the light ceases to be
transmitted, only reflected. crit = sin-1(nb/na)
Frustrated TIR
TIR
In which of the following
situations is
Recap of last lecture
Chapter 13,
Periodic motion
A restoring force which is directly proportional to the displacement
from equilibrium will cause a periodic motion called SHM.
Example a spring driven glider:
Restoring force = Fx = -kx or ax = d2x/dt2 = -
Recap of last lecture
Chapter 13,
Periodic motion
SHM can be described as uniform circular motion projected on a plane
This gives x = A cos and
ax = -2x
Hence for the spring driven glider : = (k/m)
and
1k
m
f=
T = 2
frequency and period are independent
2
Recap of last lecture
Chapter 13,
Periodic motion
A simple pendulums period is given by
The period of a physical pendulum is
T = 2
L
g
Damped oscillations
The decrease in the amplitude due to dissipative forces
is called damping and these oscillations are
Recap of last lecture
Chapter 13,
Periodic motion
For damped spring with a damping force of Fx = -bvx
d2 x
k
b dx
= x
dt 2
m
m dt
With:
solution
x = Ae
Forced (driven)
oscillator
(b / 2 m )t
A=
cos( t + )
we have
&=
k
b2
m 4m 2
Fmax
2
2
(k md ) 2 + b2 d
Recap of last lecture
Waves are transverse or longitudinal or a combination
If the oscillations producing the wave is a SHM
velocity v = f where is the wavelength
Wave function or amplitude of wave at
Any arbitrary position x and time t is:
y(x,t) = Acos(
Recap of last lecture
The wave equation for a periodic wave is:
2y 1 2y
=22
2
x
v t
This relates the curvature of the wave to the transverse acceleration.
The velocity of a wave in string is given by
Where F is the tension and
is its linear mass density
Recap of last lecture
Instantaneous power =
Max Power = F 2A2
F 2A2 sin 2 (kx t)
Average power = Max power
Intensity = Power per unit area
W/m2
I1/I2 = r22/r12
Waves reflect at boundaries
(changes in medium)
If Particles at the boundary are:
Fixed (feel a
Recap of last lecture
Sound waves:
y(x,t) = Acos(kx-t) for wave traveling in the +x direction
(same as any other wave).
But can also be described in terms of pressure fluctuation.
P = BkAsin(kx-t)
Pmax = BkA
Pressure varies sinusoidally but has a /2 (90)
Recap of last lecture
v=
v fluid =
restoring force
inertia resisting change
B
vsolid =
Y
v gas =
RT
M
Intensity = <Power>av/area = BkA2
= B 2A2
= P2max/B
= (10dB)log(I/I0)
where I0 = 1x10-12W/m2
(I0 threshold for human hearing at 1000Hz)
Applications suc
Recap of last lecture
Standing waves in longitudinal waves (sound waves)
open columns
f1 = v/2L, f2 = v/L, f3 = 3v/2L
http:/The_Gas_Tube
closed columns
f1 = v/4L, f3 = 3v/4L, f5 = 5v/4L
Wave interference
Destructive interference when the distance between
Recap of last lecture
Just like other waves sound waves interfere
When the frequency of two sound waves is the same:
Destructive interference when the distance between the speaker is /2, 3/2, 5/2
Constructive interference when the distance between the spe