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Title: Waves and Oscillations
Description: Waves and Oscillations for undergraduate student
Description: Waves and Oscillations for undergraduate student
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Waves and Oscillations
Periodic & Oscillatory Motion:The motion in which repeats after a regular interval of time is called
periodic motion
...
The periodic motion in which there is existence of a restoring
force and the body moves along the same path to and fro about a
definite point called equilibrium position/mean position, is
called oscillatory motion
...
In all type of oscillatory motion one thing is common i
...
3
...
Example of linear oscillation:1
...
2
...
3
...
4
...
5
...
Example of circular oscillation:1
...
2
...
3
...
4
...
5
...
Oscillatory system:1
...
2
...
3
...
For mechanical oscillation two things are
specially responsible i
...
E
...
4
...
e
...
Simple Harmonic Motion:It is the simplest type of oscillatory motion
...
If F is the restoring force on the oscillator when its displacement
from the equilibrium position is x, then
F –x
Here, the negative sign implies that the direction of
restoring force is opposite to that of displacement of body i
...
F= -kx …………
...
Ma=-kx
M
M
=-kx
kx=0
+ x=0
ω2x=0 ………… (2)
Where ω2=
Here ω=√ is the angular frequency of the oscillation
...
By solving these differential equation
x=
……… (3)
+
Where ,
are two constants which can be determined from the
initial condition of a physical system
...
(5)
x=A sin
Where A=√
2
+D2) &
)
Similarly, the solution of differential equation can be given as
) ………(6)
x=Acos
Here A denotes amplitude of oscillatory system,
) is called
phase and is called epoch/initial phase/phase constant/phase angel
...
Velocity in SHM:=Asin
)
=A cos
v=A cos
)
) ………… (7)
The minimum value of v is 0(as minimum value of Asin
)=0
& maximum value is A
...
Acceleration in SHM:a= -A
2
sin
) …………
...
Also, a=
a
2
...
Time period in SHM:The time required for one complete oscillation is called the time
period (T)
...
T=
……………… (9)
Frequency in SHM:The number of oscillation per time is called frequency or it is the
reciprocal of time period
...
Kinetic energy in SHM:The kinetic energy of oscillator at any instant of time is,
)2
K=
=
K=
v2
A2ω2 cos2
) ……
...
Both kinetic and potential energy attain their maximum value twice in
one complete oscillation
...
E+P
...
Graphical relation between different characteristics in SHM
...
The distance between the point of suspension the centre of gravity is
called the length of length of the pendulum &denoted by
When the pendulum is displaced through a angle θ from the mean
position,a restoring torque come to play which tries to bring the
pendulum back to the mean position
...
Here the restoring force is -mgsinθ
...
If the moment of inertia of the body about “OA” is “I”, the angular
acceleration becomes,
α=τ/I
α=
………………
...
So,
α=-mglθ/I
...
(2)
α=d2θ/dt2
Also
Now we can write
d2θ/dt2+ ( mg /I) θ =0……………
...
(4)
Where,
ω2=
harmonic
...
And eqn(4) is the general equation of simple
T=2π(I/mg )1/2
T=2π( M(k2 +L2)/Mg )1/2
...
Here
+ =L, Called as equivalent length of pendulum
...
we have
another Point on the line called centre of Oscillation is equivalent
Length of pendulum
...
If these two points are interchanged
then “time period” will be constant
...
C CIRCUIT(NON MECHANICAL OSCILLATION ):-
In this region,it is combination “L” &”C” with the DC source through
the key
...
When the switch is off the capacitor gets
discharged
...
So, current at that
situation is given by
I=dq/dt
...
This energy is transferred to magnetic field that
appears around the inductor
...
Even though q
equals to zero,the current is zero at this time
...
So a back emfdevelops which is given by
ε
=-L ……………
...
(3)
...
Here this LC oscillation act as an source of electromagnetic wave
...
Hence oscillation continues indefinitely
...
The oscillators whose amplitude, in successive
oscillations goes on decreasing due to the presence of resistive forces
are called damped oscillators, and oscillation called damping
oscillation
...
Fdam –v
Fdam=-bv
b= damping constant which is a positive quantity defined as
damping force/velocity,
Fnet = Fres+ Fdam
Fnet= -kx –bv
Fnet= -kx– b
M
+kx+ b
+
=0
+ x=0
+2β +ω02x = 0 …………
...
The above equation is second degree linear homogeneous equation
...
As we know,
x(t) =A1
x(t) = A1
x(t) =
+ A2
(
)
√
(A1
√
+ A2
+ A2
(
√
√
)
) … (4)
Depending upon the strength of damping force the quantity (β 2-ω02)
can be positive /negative /zero giving rise to three different cases
...
The constant „r‟ and ‟ θ‟ are determined from initial
potion & velocity of oscillatior
T1=2π/ ω1
T1=2π/√ ω02- β2……(vi) (time period of damped oscillator)
T1 T (where T= time period of undamped oscillator
Implies f1 f
Frequency of damped oscillator is less than that of the
undamped oscillator
...
Mean life time:The time interval in which the oscillation falls to 1/e
of its initial value is called mean life time of the oscillator
...
E = mv2
K
...
E= kx2
= kr2
cos2 (ω1t+ θ)
Total Energy:
T
...
E+P
...
The ratio between amplitude of two successive maxima, is the
decrement of the oscillator
...
=eβt=
Rate of two amplitudes of oscillation whichare separated by one period
Relaxation time( :
It is the time taken by damped oscillation by
decaying of its energy 1/e of its initial energy
...
The motion of simple pendulum in a highly viscous medium is an
example of over damped oscillation
...
Critical damping:
β2 = ω02
The general solution of equation (ii) in this case,
X(t) = (Ct+D) e-βt …………………………………(ix)
Here the displacement approaches to zero asymptotically for given
value of initial position and velocity
a critically damped oscillator
approaches equilibrium position faster than other two cases
...
Curves of three Cases:
Forced Oscillation
The oscillation of a oscillator is said to be forced oscillator or driven
oscillation if the oscillator is subjected to external periodic force
...
Equation (i) is also represented as
̈+
̇+
2
0
x= f0cosωt
Equation (i) represents the general equation of forced oscillation
...
For weak damping (ω02 >β 2) , the general equation
contains,
x(t) = xc(t) + xp(t)
Where xc(t) is called complementary solution and its value is
xc(t)=
A1
√
+A2
√
) ………………(ii)
Now xp(t) is called the particular integral part
...
cos
δ – sin(ωt-δ )
...
(vii)
cos(ωt-δ)
√(
(steady state solution)
)
Now, x(t) = xc(t) + xp(t)
x(t) =
A1
√
+ A2
√
)+
cos(ωt√(
)
δ)
Steady state behavior:
Frequency:-The Oscillator oscillates with the same frequency as that
of the periodic force
...
The
duration between transient beats is determined by the damping
coefficient „β‟
...
(In the above figure fQ= ω0 and fp= ω )
At ω=ω0 , φ₌ , the displacement of the oscillator lags behind the
driving force by
...
If it is very small, then the amplitude of
forced oscillation increases
...
e ω>>ω0 and damping is small
(β is small) or ( β→0)
A=
√
A=
A=
Amplitude is inversely proportional to the mass of the oscillator &
hence the motion is mass controlled motion
...
e
...
The
amplitude of the forced oscillator in the region ω<<ω0and β< ω0
is inversely proportional to the stiffness constant (k) and hence
motion is called the stiffness controlled motion
...
e
...
It is the phenomenon of a body setting
a body into vibrations with its natural frequency by the application of
a periodic force of same frequency
...
(I
...
ω =ω0)
...
Only the denominator of the expression
√
is minimum i
...
[√
=0
=>-4ωω02+4ω3+8β2ω=0
=>-ω02+ω2+2β2=0
=>ω= √
=ω0√
It is the value of angular frequency, where „A‟ will be maximum in
presence of damping force
But when damping is very small,
ω=ω0 (β→0)
The max value of „A „when damping is present
A=f0/√
=f0/√
=f0/√
=f0/√
Amax=f0
√
=f/2mβ√
This is called amplitude Resonance
...
e
...
Β1<β2
fr=ω/2π
=√ ω20-2β2)/2π
Damping is small,
fr=ω0/2π
Here, fr‟ is called resonant frequency
...
Therefore, the driving force always acts in the direction of motion
of oscillator
...
Sharpness of resonance:The amplitude is maximum at resonance frequency which
decreases rapidly as the frequency increases or decreases from the
resonant frequency
...
Different condition:(i) For ω=ω0 β, the amp
...
The width of resonance
curve i
...
the range of frequency over which the amplitude remains
more than Amax/√
...
(ii) For β=0, A→∞ at ω=ω0
...
Resonance occurs lesser value amp is max at
ω=ωr
...
amp
decreases
...
Velocity:X=xp=f0/√
V=-ωf0/√
V=ωf0/√
Vmax=ωf0/√
*here also „v‟is max
...
(Vmax
...
E= kx2
cos2(ωt-δ)
= k
√(
)
= kA2cos2(ωt-δ)
E= mv2
= m
(
cos2(ωt-δ+π/2)
)
= mω2A2cos2(pt-δ+π/2)
Power absorption:
= F Cos (
⁄
√
= A F Cos (
=
=
= m A2
ii)
...
=+b
= 2m
(
(
⁄
)
⁄
= 2m
Thus in the steady state of forced vibration, the average rate of power
supplied by the forcing system is equal to the average of work done
by the forced system against the damping force
...
Q- Factor is defined as,
Q=2
=2
(
=2
=
At
(
)
=
)
(
)
, for weak damping
Q=
=>
Q=
Small, Q Large, sharpness of resonance is more
...
–
1400
Violin string
103
Microwave
105
resonator
Crystalosill
106
Excetetation
108
Amplitude Resonance
Velocity Resonance
1
...
Resonance, the amp
...
The velocity amplitude of the
of oscillator is maximumfor
forced
a particular frequency of the
maximum
applied force
...
2
...
Velocity resonance occurs at
⁄
at
3
...
of the freq
...
The phase of the forced
oscillator with respect to that
3
...
Phase of the forced oscillator
with respect to that of applied
force is …
...
⁄
=>
=
)
)
the
velocity amplitude is zero
...
In this case, the resultant intensity differs from the sum of intensities
of individual waves due to interfering factor
...
e
...
In this case, the resultant intensity is equal to the sum of the
intensities of the individual waves
...
e
...
Let us consider two waves having different amplitude and phase
are propagated in a medium is given as
(1)
(2)
Applying the principle of superposition
(3)
Let
(4)
and
(5)
(6)
Squaring and adding equation (4) and (5)
A
√
(7)
We know,
√ √
(8)
Dividing equation (5) by (4), we get,
Coherent Superposition:
In coherent superposition, the phase difference remains constant
between two beams
...
) vanishes as the cos value varies from -1
√
Here,
Multiple beam superpositions:
When a number of beams having same frequency, wavelength and
different amplitude and phase are undergoing the superposition, such
superposition is known as multiple beam superpositions
...
be the number of beams having same
frequency, wavelength and different in amplitude and phase are
propagating in a medium are given as,
:
:
:
According to principle of superposition,
, , ,
...
Phase of the ith component
...
e
...
i
...
Now,
I coherent N 2 I1
Incoherent Superposition
In incoherent superposition, the phase difference between the beams
changes frequently or randomly due to which the time average of
factor ∑
vanishes as cos value varies from1 to +1
∑
∑
Now
,
I incoherent KA 2
∑
I incoherent NI 1
N
I coherent
I icoherent
Interference:
The phenomenon of modification in distribution of energy due
to superposition of two or more number of waves is known as
interference
...
According to Huygens‟s principle, as each point of a given wavefront
will act as centre of disturbance they will emit secondary wave front
on reaching slit S1 and S2
...
During the propagation, the crest or trough of one wave falls upon the
crest and trough of other wave forming constructive interference,
while the crest of one wave of trough of other wave producing
destructive interference
...
Mathematical treatment:
Let us consider two harmonic waves of same frequency and
wavelength and different amplitude and phase are propagating in a
medium given as
Let
Squaring and adding (2) and (3)
*
√
+
As, I
[
√
√
]
Dividing equation (3) by (2) we get,
Condition for maxima:
The intensity will be maximum when the constructive interference
takes place i
...
, n=0, 1, 2
...
Now, [
difference is even multiple of
√
]
[
√
√
]
If the waves having equal amplitude,
]
Condition for minima
The intensity will be minimum destructive interference takes place
i
...
Where n = 0, 1, 2, 3
...
Now, [
[
√
√
√
]
]
Intensity distribution curve
If we plot a graph between phase difference or path difference along
X-axis and intensity along Y-axis, the nature of the graph will be
symmetrical on either side
...
e
...
As the minima‟s and maxima position changes alternatively so the
disappearance of energy appearing is same as the energy appearing in
other energy which leads to the principle of conservation of energy
...
Condition for Interference
1) The two waves must have same frequency and wavelength
...
3) The amplitude of wave may be equal or nearly equal
...
II
...
The distance between the two slit must be small
...
The background should be perfectly dark
...
The distribution between the slit and the screen should be large
...
The two waves may have equal or nearly equal amplitude (for
sharp superposition)
...
Practical resolution of Coherent
Coherent sources from a single source of light can be realised as
follows
A narrow beam of light can be split into its number of component
waves and multiple reflections
...
[
]
Methods for producing coherent sources/Types of interferences
Coherent sources can be produced by two methods
1) Division of wave front
2) Division of amplitude
Division of Wave front
The process of coherent source or interference by dividing the wave
front of a given source of light is known as division of wave front
...
In this case a
point source is used
...
YDSE
2
...
Fresnel‟s bi-prism
4
...
In this case, extended source of light is used
...
Newton‟s ring method
2
...
Michelson‟s interferometer
Young’sDouble Slit Experiment:
In 1801 Thomas Young demonstrated the phenomenon of interference
in the laboratory with a suitable arrangement
...
The experiential
arrangement consists of two narrow slits, S1 and S2 closely spaced,
illuminated by a monochromatic source of light S
...
In the figure,
d
Slit separation
D
Slit and screen separation
Y
Wavelength of light
distance
x Path
of interfering point from the centre of slit
difference coming from the light S1 and S2
Optical path difference between the rays coming through
S1 and S2
Now the path difference,
In figure,
[
] (Using binomial theorem)
Similarly,
*
+
The alternative dark and bright patches obtained on the interference
screen due to superposition of light waves are known as fringe
...
e
...
e
...
If
and
be the two consecutive bright fringe
...
It is concluded that the separation between the two consecutive bright
fringes is equal to the consecutive dark fringes
...
Discussion:
From the expression for
If young double slit apparatus is shifted from air to any medium
having refractive index (µ), fringe pattern will remain unchanged and
the fringe width decreases (1/µ) as λ decreases
...
When YDSE is performed with white light instead of monochromatic
light we observed,
I
...
Fringe width decreases gradually
III
...
[
So, it decreases 1/µ times
...
The formation of the Newton‟s ring is based on the
principle of interference due to division of amplitude
...
Formation of Newton’s Ring
I
...
II
...
At the point of contact air film is formed whose thickness
gradually goes on increasing towards outside
...
IV
...
V
...
Condition for bright and dark fringe in Reflected light
In Newton‟s ring experiment, the light travels from upper and lower
part of the air film suffers a path difference of λ/2 (phase change of
π)
...
Then the total path travelled by the light is given
as
...
e
...
In transmitted light
(a)Condition for bright ring;
(b)
Condition
for
dark
ring;
(c)Newton‟s rings are less
intense
...
C is the centre
of curvature of curved surface LOL‟
...
From the property of circles,
t = thickness of air film
( t
From the condition for bright Newton‟s ring,
, For the nth ring
...
√
√
√
√
, n = 1, 2, 3……
...
Similarly from the Newton‟s dark ring,
√
√
√
√
√
√
Thus the diameter of Newton‟s dark ring is proportional to square root
of natural numbers
...
Apparatus, on focusing the microscope over the ring system and
placing the crosswire of the eye piece on tangent position, the
readings are noted
...
Let
and
be the nth and (n+p)th dark ring, then we have,
Subtracting equation (1) from (2) we get,
This is the required expression from the wavelength of light for
Newton‟s ring method
...
From the graph the wavelength of light can be calculated the slope of
the slope of the graph
...
Now the optical path travelled by the light is to be 2µt, instead
of 2t where µ be the refractive index of the liquid from the condition
for the Newton‟s ring we have,
For nth ring,
Let
and
be the diameter of the (n+p)th and nth dark ring in
presence of liquid then
'
Dn2 p
4n p R
and
'
Dn2
4nR
Now ,
'
'
Dn2 p - Dn2 =
4n p R 4nR
-
= 4 pR
(1)
If the same order ring observed in air then
2
2
Dn p Dn 4 pR
(2)
Dividing equation (2) by (1) ,we have
D
D
2
n p
'2
n p
2
Dn
D
air
'2
n liquid
This is the required expression for refractive index of the liquid
...
Explanation of diffraction:
To explain diffraction, let us consider an obstacle AB is placed
on the path of an monochromatic beam of light coming from a source
„S‟ which produces the geometrical shadow CD on the screen
...
If the dimension or size of the obstacle is comparable with the
wave length of the incident light, then light bends at the edge of the
obstacle and enters in to the geometrical shadow region of the
obstacle
...
This explains the diffraction phenomena
...
a
...
Fraunhoffer Diffraction
Fresnel’s Diffraction
(1)
The type of diffraction
in which the distance of
either source or screen or
both from the obstacle is
finite, such diffraction is
known
as
Fresnel‟s
diffraction
...
(3)
The incident wave
front is either cylindrical or
spherical
...
Fraunhoffer Diffraction
(1)
The type of diffraction
in which the distance of
either source or screen or
both from the obstacle is
infinite, such diffraction is
known
as
Fraunhoffer
diffraction
...
(3)
The incident wave
front is plane
...
The diffraction at the
narrow
...
The rays of the light
which are incident normally on the convex lens „L2‟, they are
converged to a point „P0‟ on the screen forming a central bright
image
...
As the rays get deviated at the slit „AB‟ they suffer a path
difference
...
of equal holes and a be
the amplitude of the light coming from each equal holes
...
phase difference=
1 2
e sin
n
Now the resultant amplitude due to superposition of waves is
given as
n 1 2
nd
e sin a sin e sin
a sin a sin
2 =
2 n
=
R
d
1
1 2
sin
sin
e sin
e sin
n
2
n
sin
2
Let e sin ,then
Since
R
a sin
sin
n
is very small and n is very large so
n
is also very
small
...
Position for/Condition for minima:
The minimum will be obtained when
sin sin m
sin 0 sin(m )
m
e sin m
e sin m
m
where
m 1,2,3,4,
...
e
e
e
e
Position/Condition for secondary maxima:
The maxima‟s occurring in between two consecutive secondary
maxima is known as secondary maxima
...
It can be solved by graphical
method
...
Thus the secondary maxima‟s are obtained at
3
5
7
,
,
...
3!
5!
7!
= A x 1
2
3!
4
...
The
nature of the graph is as follows:
Intensity distribution curve
From the nature of the graph it is clear that
1
...
The maxima are not of equal intensity
3
...
of parallel slits
of equal width separated by an equal opaque space is known as
diffraction grating or plane transmission grating
...
of rulings over a plane
transparent material or glass plate with a fine diamond point
...
N
...
Though the plane transmission grating and a plane glass
piece looks like alike but a plane transmission grating executes
rainbow colour when it exposed to sun light where as a plane
glass piece does not executes so
...
It can be measured by counting the
no
...
Let us consider a diffraction grating having
e = width of the slit
d = width of the opacity
If “N” be the no
...
00016933
15000
cm
Diffraction due to plane transmission grating /Fraunhoffer
diffraction due to N-parallel slit:
Let us consider a plane wave front coming from an infinite
distance is allowed to incident on a convex lens “L” which is
placed at its focal length
...
Again those rays of
light which are diffracted through an angle “θ” are undergoes a
path difference and hence a phase difference producing
diffraction
...
of rulings present in the grating
Now the path difference between the deviated light rays is
S2K = S1S2Sinθ =
(e d )Sin
Therefore, Phase difference =
where
2
x S2K = 2 (e d )Sin = 2 (say)
(e d ) Sin
Now the resultant amplitude due to superposition of “N” no
...
This is called grating equation or condition
for central principal maxima
...
This is the condition for minima due to diffraction at N-parallel
slit
...
The positions for secondary maxima will be obtained as
dI
0
d
d
d
Sin 2 Sin 2 N
0
I 0
2 Sin 2
Sin 2 Sin N N cos N sin sin N cos
2I 0
0
Sin
2
sin 2
N cos N sin sin N cos
sin 2
=0
N cos N sin sin N cos
=0
N cos N sin sin N cos
N tan N tan N
This is a transdectional equation
...
Taking y tan N and y N tan N ,where the two plots are
interests, this intersection points give the position for secondary
maxima
...
2
2
2
Intensity distribution curve:
The graph plotted between phase difference and intensity
of the fringes is known as intensity distribution curve
...
The spectra of different order are situated on either side of
central principal maximum
2
...
The spectra lines are more dispersed as we go to the higher
orders
...
The central maxima is the brightest and the intensity decreases
with the increase of the order of spectra
...
Condition for Missing spectra:
We have,
The condition for central principal maxima due diffraction at Nparallel slit
(e d )Sin m
e sin n
(e d ) Sin m m
e sin
n
n
Special case:
1
...
i
...
2
...
5n 1
n 2
2
i
...
3
...
e Third order spectra or multiple of 3 spectra will found to be
missed or absent on the resulting diffraction pattern
...
Dispersive power:
The variation of angle of diffraction with the wave length
of light is known as dispersive power
...
The parallel beam of monochromatic light coming from source
is allowed to incident on the transmission grating which are now
defracted by different angle of diffraction
...
Using the grating equation ,
(e d )Sin m
(e d ) Sin
m
We can calculate the wave length of the monochromatic light
...
As it was first
observed by Fresnel, these are also known as Fresnel half period
zone
...
Let “P” be a point just ahead of the
plane wave front at a perpendicular distance “b” from the plane
wavefront
...
of concentric circles such that light coming
from each consecutive half period zone will differ by a phase
difference of
...
These half period
zones are known as Fresnel half period zone
...
Properties of Half period Zone:
1
...
Area of half period zone:
The space enclosed between two consecutive half period
zones is called area of Half period zone
...
Radius of half period zone:
We have,
The area of first half period zone is πbλ
i
...
rn nb
...
Factors affecting amplitude of half period zones:
The factors affecting the amplitude are:
a
...
Average distance of half period zone (inversely)
c
...
Rn be the amplitudes of 1st, 2nd, 3rd,……nth
half period zone respectively
...
Rn
R1 R2 R3
...
Rn1
Since
R 1 R2 , R 2 R3
(If n is odd)
(If n is even)
so we have
R1 R3
R R
and R4 3 5 and so on
2
2
R R
R
R
R
R 1 R2 3 1 R4 5
...
n1
2 2
2
2
2
R
R1 Rn
2
2
=
As
if n is odd
if n is odd
R1 Rn 1
2
2
n
if n is even
1
if n is even
and
Rn1 Rn
or
1
2
2
so,
R
R1
2
Thus the net amplitude due to entire half period zone is equal to
half of the amplitude due to first half period zone
...
Construction:
It can be constructed by drawing a series of concentric
circles on a white sheet of paper with radii proportional to
square root of natural number
...
A reduced photograph of this drawing is taken
on a plane glass plate
...
Depending on the initial blackening the zone plate is of
two types
1
...
Negative
center is dark
Working:
When a beam of monochromatic light is allowed to fall on
a zone plate, the light is obstructed from the alternate half period
zone through the alternate transparent zones
...
Hence, the resultant amplitude is sum of the individual
amplitude due to light coming from alternate half period zones
...
Thus a zone plate is equivalent to
that of a convex lens
...
Let „O‟ be a point object
placed at a distance „ OP u ‟ forms a real image „I‟ at a distance „
PI v ‟ from the zone plate
...
of concentric circles such that the light coming from
alternate half period zone will differ by a path difference of
such a way that
OM1 I OPI
2
OM 2 I OPI
2
2
OM 3 I OPI
3
2
…………………
...
n
2
OM n I OPI
OM n I PM n I
n
2
(1)
Now, in right angled ΔOPMn
OM n OP 2 PM 2 n
u2 r
1
2 2
n
1
2
1
r 2n
r 2n 2
u 1 2 = u 1 2
...
v
2v
r 2n
r 2n
v 1 2 v
2v
2v
,
as
rn v
(3)
Using eqn (2) and eqn (3) in eqn (1) we get
u
r 2n
r 2n
n
v
(u v)
2u
2v
2
r 2 n 1 1
n
(u v)
(u v)
2
2 u v
r 2n
2
1 1
u v
=
n
2
1 1
r 2 n n
u v
(4)
u v
= n
r 2n
uv
uv
n
uv
rn
rn cons
...
Expression for primary focal length:
From eqn
...
Let An-1 and An be the area of (n-1) th and nth zone
Then A= A - A = r 2 r 2 = uvn uvn 1 = uv = constat
n
n-1
n
n 1
uv
uv
uv
Thus, the area of zone plate is independent of order of zone i
...
Multiple foci of zone plate:
Now from the expression we have,
1 1
r 2 n n
u v
If the object is situated at infinity (∞), then the first image at
distance ,
rn2
v1 f1 =
n
If we divide the half period zones into half period elements
having equal area, then the 1st half period zone will divided into
three half period zones,2nd half period zone will divided into five
half period elements and so on
The second brightest image will at
1
1 rn2
v3 f 3 = f 1
3
3 n
2
The third brightest image will at v5 f 5 = 1 f1 1 rn and so on…
...
Comparison between the zone plate and the convex lens:
Similarities
1
...
2
...
3
...
Dissimilarities
Convex Lens
a) Image
is
formed
by
refraction
b) It has a single focus
...
d) Image is more intense
e) The optical path is constant
for all the rays of light
...
Huygens’s Principle:
About the propagation of the wave, Huygens suggested a theory
which is based on a principle known as Huygens‟s principle
...
2) The forward tangent envelope to these wave lets gives the
direction of new wave front
...
Let AB be the wave front
at t=0
...
Taking a, b, c, d, e as centre and radii equal to „ct‟ (cvelocity of light &„t‟ time), we can construct a large number
of spheres which represents a centre of disturbance for the
new wave
...
N
...
The backward front is not visible as the intensity of the
backward wave front is very small since for the backward
wave front,
I = k (1+cosθ)
since for backward wave
0
front (θ=180 )
I=k (1-1) =0
Iback=0
POLARISATION
The phenomenon of restricting the vibration of light in a particular
direction perpendicular to the direction of wave motion is called as
polarisation
...
When an ordinary light is incident normally on the two crystal plates
the emergence light shows a variation in intensity as T2 is rotated
...
This shows that the light
emerging from T1 is not symmetrical about the direction of
propagation of light but its vibration are confined only to a single line
in a plane perpendicular to the direction of propagation, such light is
called as polarised light
...
The vibrations are confined
1
...
particle are not confined in a
2
...
occurrence of vibration
2
...
The intensity of light plate
position of the crystal
...
The intensity of light plate
the crystal plate
...
Polarised light:
The resultant light wave in which the vibrations are confined in a
particular direction of propagation of light wave, such light waves are
called Polarised light
...
Circularly polarised light:
When the two plane polarised wave superpose under certain
condition such that the resultant light vector rotate with a constant
magnitude in a plane perpendicular to the direction of propagation
and tip of light vector traces a circle around a fixed point such light is
called circularly polarised light
...
Pictorial representation of polarised light:
Since in unpolarised light all the direction of vibration at right
angles to that of propagation of light
...
In a plane polarised beam of light, the polarisation is along
straight line, the vibration are parallel to the plane and can be
represented by
If the light particles vibrate along the straight line perpendicular
to the plane of paper, then they can be represented by a dot
...
Plane of polarisation:
The plane passing through the direction of propagation and
containing no vibration is called as plane of polarisation
...
Angle between plane of vibration and plane of polarisation is
90˚
...
Since
during the rotation of the crystal, the variation in intensity takes
place, this suggests that light waves are transverse in nature
rather longitudinal
...
Polarisation by Reflection
2
...
Polarisation by Scattering
4
...
Polarisation by reflection:
The production of the polarised light by the method of reflection from
reflecting interface is called polarisation by reflection
...
If the angle of incidence is 0°
or 90° the light is not polarised
...
The angle of incidence for which the reflected component of
light is completely plane polarised, such angle of incidence is called
polarising angle or angle of polarisation or Brewster‟s angle
...
At ip the angle between reflected ray and refracted or transmitted
ray is π/2
...
The incident
ulpolarised light contain both perpendicular and parallel component
of light
...
The parallel
component of light is continues to vibrate and get refracted or
transmitted
...
Conclusion:
Hence, the reflected ray of light contains the vibrations of
electric vector perpendicular to the plane incidence
...
Brewster’s Law:
This law states that when an unpolarised light is incident at polarizing
angle „ip‟ on an interface separating air from a medium of refractive
index “µ” then the reflected light is fully polarized
...
e
...
CBY+ DBY=90˚
90 r 90 r 90
90 i 90 r 90
0
'
0
0
0
0
0
p
i p r 90 0
r ' 90 0 r
From Snell‟s law
sin i p
sin r
=
sin i p
sin(90 0 i p )
=
sin i p
cos i p
= tan i p
Thus the tangent of the angle of polarization is numerically equal to
the refractive index of the medium
...
From Brewster‟s law;
We have
tan i p
sin i p
cos i p
According to Snell‟s law;
sin i p
sin r
From above equations
sin r cos i p sin r sin(90 0 i p ) r 900 i p r i p 90 0
90 0 CBY 90 0 DBY 90 0
CBY+ DBY=90˚
CB BD CBD 90 0
Thus, it is concluded that at polarizing angle or at Brewster‟s angle,
the reflected light and the refracted light are mutually perpendicular to
each other
...
Polarisation by Scattering:
When a beam of ordinary light is passed through a medium
containing particles, whose size is of order of wavelength of the
incident light, then the beam of light get scattered in which the light
particles are found to vibrate in one particular direction
...
Explanation:
To explain the phenomenon of scattering, let us consider a beam of
unpolarised light along z-axis on a scatter at origin
...
When we look along Xaxis we can see the vibrations which are parallel to Y-axis
...
Hence, the light can be scattered perpendicular to incident light is
always plane polarized
...
To explain the polarization by refraction, let us consider an ordinary
light which is incident upon the upper surface of the glass slab at an
polarizing angle i p or Brewster‟s angle B , so that the reflected light is
completely polarized while the rest is refracted and partially
polarized
...
Now,
tan r
sin r
sin r
sin r g
a
0
cos r sin(90 r ) sin i p
tan r g a
Thus according to Brewster‟s law, „ r ‟ is the polarizing angle for the
reflection at the lower surface of the plate
...
Hence, if a beam of unpolarised
light be incident at the polarizing angle on a pile of plates, then some
of the vibrations are perpendicular to the plane of incidence are
reflected at each surface and all those parallel to it are refracted
...
Malus law:
It states that when a beam of completely plane polarized light
incident on the plane of analyser, the intensity of the transmitted light
varies directly proportional to the square of the cosine of the angle
between the planes of the polariser and plane of the analyser
...
The amplitude of the light vector „E‟ is now resolved into two mutual
perpendicular component i
...
E1 E0 cos which is parallel to the plane
of transmission and E2 E0 sin which is perpendicular to the plane of
transmission
...
Thus,
IE1
2
2
I kE0 cos 2 I 0 cos 2
I cos 2
, where
2
I 0 kE0
Which is Mauls law
Double refraction:
The phenomenon of splitting of ordinary light into two
refracted ray namely ordinary and extra ordinary ray on passing
through a double refracting crystal is known as double refraction
Explanation:
To explain the double refraction, let us consider an ordinary
light incident upon section of a doubly refracting crystal
When the light passing through the crystal along the optic axis
then at the optic axis the ray splits up into two rays called as
ordinary and extraordinary ray which get emerge parallel from
the opposite face of the crystal through which are relatively
displaced by a distance proportional to the thickness of the
crystal
...
Difference between the Ordinary (O-ray)and Extra ordinary
ray(E-ray)
Ordinary ray
Extraordinary ray
1
...
For ordinary ray
plane of vibration lies
perpendicular
to
the
direction of
propagation
3
...
4
...
5
...
6
...
1
...
For extraordinary ray the
plane of vibration parallel to
the direction of propagation
3
...
4
...
5
...
6
...
But it
travel with equal speed along
optics axis
Double refracting crystal:
The crystal which splits a ray of light incident on it into two
refracted rays such crystal are called double refracting crystal
...
Uniaxial
2
...
Uniaxial: The double refracting crystal which have one optic
axis along which the two refracted rays travel with same
velocity are known as uniaxial crystal
Ex: Calcite crystal, tourmaline crystal, quartz
Biaxial: The double refracting crystal which have two optic axis
are called as biaxial crystal
Ex: Topaz, Agromite
Optic axis: It is a direction inside a double refracting crystal
along which both the refracted behave like in all respect
...
The ordinary ray from the first crystal passes undeviated
through the 2nd
crystal and emerges as O1 ray
...
Hence the image O1 and E1 are seen
separately
...
As the
rotation is continued , O1 and O2 remained fixed while E1 and
E2 rotate around O1 and O2 respectively and images are found
to be equal intensities
...
r
...
When the 2nd crystal is rotated at an angle 135˚ w
...
t the 1st , four
images once again appear with equally intense
...
r
...
This is how we are able to produce the plane polarised light by
the method of double refraction
...
Principle:
It is based on the principle that it eliminates the ordinary ray by total
internal so that the extraordinary ray became plane polarised emerges
out from it
...
Construction:
A calcite crystal about the three times as long as the wide is taken
...
The crystal is cut apart
along a plane which is perpendicular to both the principal section
...
They are then
cemented together by Canada balsam whose refractive index is 1
...
Action:
When a ray of unpolarised light is incident on the nicol prism it
splits up into two refracted ray as O & E ray
...
55 is less than the refractive index of
calcite for the ordinary ray (O- ray), so the O- ray on reaching the
Canada balsam get totally reflected and is absorbed by the tube
containing the crystal while E-ray on reaching the Canada balsam is
get transmitted
...
Uses:
The nicol prism can be used both as apolariser and also an analyser
...
As this, ray falls on a second nicol which is parallel
to that of 1st, its vibration will be in the principal section of 2nd and
will be completely transmitted and the intensity of emergent light is
maximum, thus the nicol prism behaves as a polariser
...
Hence the ray will behave as a ray inside the 2nd and will lost by total
reflection at the balsam surface
...
Limitations:
1
...
2
...
Quarter wave plate: A double refracting crystal plate having a
thickness such as to produce a path difference of
4
or a phase
difference of between the ordinary and extra ordinary wave is called
2
as quarter wave plate or
4
plate
...
Working:
When a beam of monochromatic light incident on the plate it
will be broken up into O-ray and E-ray which will perpendicular to
the direction of wave propagation and vibrating in the direction of
incidence respectively
...
The crystal in which the O-ray
travels with a less velocity than E-ray called positive crystal
...
O
The crystal in which the O-ray travels with a greater velocity
than E-ray called positive crystal
...
It is used for producing circularly and elliptically polarised
light
...
In addition with nicol prism it is used for analysing all kind of
polarised light
...
Construction: It can be constructed by cutting a plane from double
refracting crystal such that its face parallel to the optic axis
...
Let us consider a doubly refracting crystal
Let t= thickness of crystal plate
O be the refractive index of the crystal for O-ray
E be the refractive index of the crystal for O-ray
O t = optical path for O ray
E t = optical path for E ray
then the path difference between the waves is ( O E ) t
If the plate acts as quarter plate, then ( O E ) t = λ/2
t
2( O E )
This is for positive crystal
...
For positive crystal VO VE and E O
Ex: calcite, tormulaline etc
...
For a-ve crystal and VO VE and E O
Ex: quartz, ice
t
2( E O )
Uses: 1
...
It is used to produce the plane polarised light
...
It produces a path difference
of λ /4 between O and E wave
2
...
3
...
4
...
1
...
2
...
3
...
4
...
Production and Analysis Polarised Light
1
...
Inside the prism the beam is broken upto two
components „O‟ and „E‟ ray
...
The „E‟ component emerges out which is plane polarised with
vibration parallel to the end faces of the Nicol
...
Production of circularly polarised light:
The circularly polarised light can be produced by allowing
plane-polarised light
obtained from the Nicol to fall normally on a quarter wave plate
such that the
direction of vibration in the incident plane polarised light makes
an angle of 45⁰ with
the optic axis of the crystal
...
45⁰= a of the axis of x
) a cos wt
2
and
y a sin wt
Eliminating t from both the equation, we have
x 2 y 2 a 2 which represents a circle
...
3
...
In this case the incident wave is divided inside the plate into E and O
components of unequal amplitude A cos 300 and A sin 30 0 respectively
which emerge from the plate with a phase difference of
...
Hence the emerging
light coming from
plate is elliptically polarised
Analysis of different polarised light:
The whole analysis of different type of polarised light can be
represented in algorithm form with figure as follows:
Case: 1
Case: 2
Case:
3
POLARISATION IN SUMMARY
VECTOR CALCULUS
The electric field ( E ) , magnetic induction ( B) , magnetic intensity ( H ) ,
electric displacement ( D) , electrical current density ( J ) , magnetic
vector potential ( A) etc
...
These are vector fields
...
are also function of position and
time
...
Time Derivative of a Vector Field
If
A(t ) time
dependent vector field, then the Cartesian coordinates
ˆ
ˆ
A(t ) iAx (t ) ˆ y (t ) kAz (t )
jA
dA ˆ Ax (t ) ˆ Ay (t ) ˆ Az (t )
i
j
k
dt
t
t
t
Notes:
d
dB dA
( A B) A
( ) B
dt
dt
dt
Gradient of a Scalar Field
The change of a scalar field with position is described in terms
of gradient operator
...
The gradient of a scalar is a vector
...
(iAx ˆ y kAz ) = ( Ax y k Az )
ˆ
...
Notes:
...
A
...
(V A) (V )
...
A)
where V is a scalar field
If the divergence of a vector field vanishes everywhere, it is
called a solenoidal field
...
Curl of a Vector Field
The curl of a vector field is given by
Curl A= A
i
j
k
Ax
Ay
x
x
ˆ
i(
Az
x
A A
Az Ay
ˆ Ay Ax )
) ˆ( x z ) k (
j
y
z
z
x
x
y
Curl of a vector field is a vector
If V is a scalar field,
A and B are
two vector fields, then
( A B) A B
(V A) (V ) A V ( A)
If curl of a vector field vanishes, then it is called an irrotational
field
...
(i ˆ k ) 2 2 2
x
y
z
x
y
z
x y z
2
2
2
2 2 2
x y z
2
(ii)
This is called Laplacian Operator
Curl of gradient of ascalar
ˆ
V i
V ˆ V ˆ V
j
k
x
y
z
Where V is a scalar field
ˆ
(V ) i [(
V
V
V
V
V
ˆ V
)( ) ( )( )] ˆ[( )( ) ( )( )] k[( )( ) ( )( )]
j
y z
z y
z x
x z
x y
y x
ˆ
i
(V )
ˆ
j
k
x
V
x
x
V
x
x
V
x
2
2
2
2
2
2
ˆ
ˆ V V ) ˆ( V V ) k ( V V ) 0
(V ) i (
j
yz zy
zx xz
xy yx
Thus Curl of gradient of a scalar field is zero
...
e
...
Divergence of Curl of a Vector Field
A
A A
ˆ A A
ˆ A
A i ( z y ) ˆ( x z ) k ( y x )
j
y
z
z
x
x y
ˆ
...
[(i ( z y ) ˆ( x z ) k ( y x )]
j
x
y
z
y
z
z
x
x y
...
A
xy xz yz yx zx zy
...
e
...
Conversely, if the divergence of a vector field is zero, then
the vector field can be expressed as the curl of a vector
...
A) 2 A
(v)
...
( A) A
...
dl
a
dl
elemental length along the given path between a and b
...
b
b
a
a
ˆ
ˆ
I L (iAx ˆ y kAz )
...
The line integral of a conservative field A along a closed path
vanishes
i
...
A
...
Surface integral of a Vector
The surface integral of a vector field
A,
I s A
...
ˆ
ˆ
Writting ds nds , where n is unit vector normal to the surface at a
given point
...
ds A
...
n , normal to the component of the vector at the area
element
...
Surface area of a vector field is a scalar
...
ds
S
Volume integral of a Vector
The volume integral of a vector field
A over
a given volume V is
IV AdV
V
Where dV is the elemental volume (a scalar)
Volume integral of a vector field is a vector
...
e
...
...
ds
S
V
The curl of a vector field is the limiting value of its line integral
along a closed path per unit area bounded by the path, as the area
tends to zero,
A lim
S 0
A
...
Gauss Divergence Theorem
The volume integral of divergence of a vector A over a given
volume V is equal to the surface integral of the vector over a closed
area enclosing the volume
...
AdV A
...
Stokes Theorem
The surface integral of the curl of a vector field A over a given
surface area S is equal to the line integral of the vector along the
boundary C of the area
( A)
...
dl
S
C
For a closed surface C=0
...
Green’s Theorem
If there are two scalar functions of space f and g, then Green‟s
theorem is used to change the volume integral into surface integral
...
Electric Polarization ( P )
Electric polarization P is defined as the net dipole moment (
induced in a specimen per unit volume
...
So p E ,
proportionality cons tan t , known as polarizability
p)
If N is the number of molecules per unit volume then polarization
is given by
P N E
Electric Displacement Vector D
The electric displacement vector D is given by
D P 0 E -------------------- (1)
where is the P polarization vector
Unit of D 1 ampere sec
2
m
In linear and isotropic dielectric,
D E 0 r E ---------------(2)
Comparing equations (1) and (2), we get
0 r E P 0 E
P 0 ( r 1) E
Electric Flux (φE)
The number of lines of force passing through a given area is known as
electric flux
...
dS
S
Unit of flux-1 N m
2
Coul
Gauss’ Law in Electrostatic:
Statement: The total electric flux (φE) over a closed surface is equal to
1
times the net charge enclosed by the surface
...
dS
S
qnet
0
Here S is known as Gaussian surface
...
dS
S
qnet
-
Permittivity of the medium
...
dS qnet
S
Notes:
The charges enclosed by the surface may be point charges or
continuous charge distribution
...
Electric flux is independent of shape & size of Gaussian
surface
...
Limitation of Gauss‟ Law
(a)Since flux is a scalar quantity Gauss‟ law enables us to find
the magnitude of electric field only
...
Gauss’ Law in Differential form
Gauss‟ law is given by
E
...
dS
...
E dV
0 V
V
(
...
E
...
Magnetic Intensity (H) and Magnetic Induction ( B)
The magnetic intensity
( H ) is
related to the magnetic field induction
( B ) by
(H )
( B)
0
Unit: in SI system ( H ) is in amp/m and
Magnetic Flux (m )
( B ) in
tesla
...
dS B dS cos
S
S
where angle between magnetic field B and normal to the surface
Unit of flux:
1 weber in SI
1 maxwell in cgs(emu)
So 1T= 1 weber/m2
1 gauss= 1maxwell/cm2
Gauss’ Law in magnetism
Since isolated magnetic pole does not exist, by analogy with Gauss‟
law of electrostatics, Gauss‟ law of magnetism is given by
B
...
dS
...
B 0
This is the differential form of Gauss‟ law of magnetism
...
dl 0 I
C
I net current enclosed by the loop
Where
C closed
path enclosing the current (called ampere loop)
...
dl I
C
Ampere’s Law in Differential form
Ampere‟s law is
B
...
dl ( B)
...
ds --------------------------------------(iii)
S
Using (ii) and (iii) in equation (i) we have
( B)
...
ds (o J )
...
Faraday’s Law of electromagnetic induction
Statement :-The emf induced in a conducting loop is equal to the
negative of rate of change of magnetic flux through the surface
enclosed by the loop
...
E
...
ds
S
So from the above
E
...
ds
t
S
This is Faraday‟s law of electromagnetic induction in terms of
B
Differential form of Faraday’s Law
Now using Stokes‟ theorem
E
...
ds
C
But
S
E
...
ds
t
t S
From above two equations
B
( E )
...
ds
...
ds 0
t
S
Or
B
E
0
t
This is differential form of Faraday‟s law electromagnetic
induction
...
ds
S
Using Gauss divergence theorem
I
S
J
...
J dV ----------------
(i)
V
Where S is boundary of volume V
...
J dV
V
V
(
...
J
dV
t
)dV 0
t
0
t
This is equation of continuity
...
A parallel plate capacitor connected to a cell is considered
...
q
instantaneous charge on capacitor plates
...
ds 0 tE
t S
Where E is electric flux
...
dl I I
o
C
d
This law is sometimes referred as Ampere- Maxwell law
...
free charge carriers
...
(ii) Does not obey ohm‟s law
...
The ratio of their peak values
J max E0
J d max 0 E0 0
It means this ratio depends upon frequency of alternating field
...
So normal
conductors behave as dielectric at extremely high frequencies
...
D
--------------(1)
...
Equation (2) is the differential form of Gauss‟ law of
magnetism
...
Equation (4) is the generalized form of Ampere‟s circuital law
...
They are also unaffected by the presence of free
charges or currents
...
Equations (1) and (4) depend upon the presence of free charges
and currents and also the medium
...
Maxwell’s Equations in terms of E and B
...
B 0
-----------------(2)
B
E
t
------------------(3)
E
B
J
t
-------------------(4)
In absence of charges
...
B 0
-----------------(2)
B
E
0
t
E
B 0 0
0
t
------------------(3)
-------------------(4)
Maxwell’s Equations in Integral Form
E
...
dS 0
--------------(2)
S
E
...
dS
--------------(3)
S
E
B
...
dS
C
S
--------------(4)
Physical Significance of Maxwell’s Equation
(i)
Maxwell equations incorporate all the laws of
electromagnetism
...
(iii) Maxwell equations are consistent with the special theory of
relativity
...
(v) Maxwell equations provided a unified description of the
electric and magnetic phenomena which were treated
independently
...
E 0
...
E ) E 0 0 2
t
2
Since (
...
Now taking curl of equation (4)
B 0 0
( E )
t
Using equation (3)
B
2 B
B 0 0 (
) 0 0 2
t
t
t
2 B
(
...
B) 0,
2 B
2 B 0 0 2
t
Taking
0 0
1
, where c velocity of light
c2
1 2 B
B 2 2
c t
We have
2
This is the wave equation for B
...
We know that
Then
...
ofcurl of a vector is zero)
The vector A is called magnetic vector potential
...
Scalar Potential
The scalar potential in a scalar field is defined as when the curl of a
field is zero the vector can be expressed as the negative gradient of a
potential called scalar potential ( )
...
So we can write
E
A
where is a scalar functioncalled the scalar potential
...
E 0 (1)
B 0 0
Writing
E
t
(2)
E
A
; A vector potential
t
In free space and absence of
charge
We have
A
...
0
t
or 2
...
A
1 2
0
c 2 2t
We have 2
1 2
0
c 2 2t
This is the wave equation in terms of scalar potential
...
A A 0 0 0 0 2
t
t
2
A
2
...
So 2 A 0 0
2 A
0
t 2
This is the wave equation in terms of vector potential
...
A 0 (Coulomb gauge condition)
2
...
r t )
(1)
i ( k
...
E0 , B0 amplitudes of E and B respectively
...
E 0 in
equation (1) we have
...
r t ) 0
e
...
r t )
...
r t ) 0
as
...
A V
...
e 0
e
...
r t ) 0
or e
...
r t ) 0
Since E0 0, ei ( k
...
k 0 (3)
This shows the transverse nature of electric field
...
B 0
We have
...
r t ) 0
b
...
r t )
...
r t ) 0
Since
ˆ
b
constant,
ˆ
...
B0ei ( k
...
ikB0ei ( k
...
r t ) 0,
b
...
Mutual orthogonality of E, B and k
Now from Maxwell‟s 3rd equation we have
[eE0ei ( k
...
r t ) ]
t
-----------(5)
( AV ) V ( A) (V ) A ,
we have
[eE0ei ( k
...
r t ) ( e) [( E0ei ( k
...
r t ) ) E0ikei ( k
...
r t ) ] E0ikei ( k
...
r t ) (k e)
Now
[bB0ei ( k
...
r t ) } bB0ei ( k
...
5
E0iei ( k
...
r t ) (i ) bB0iei ( k
...
Thus electric field, magnetic field and propagation vector are
mutually orthogonal
...
Phase relation between E and B
In an electromagnetic wave electric and magnetic field are in phase
...
Electromagnetic Energy Density
The electric energy per unit volume is
uE
1 1 2
E
...
H H 2
2
2
(2)
The electromagnetic energy density is givenby
1
uEM ( E 2 H 2 )
2
In vacuum
1
uEM ( 0 E 2 0 H 2 )
2
Poynting Vector
The rate of energy transport per unit area in electromagnetic wave is
described by a vector known as Poynting vector ( S ) which is given as
E B
S E H
Poynting vector measures the flow of electromagnetic energy per unit
time per unit area normal to the direction of wave propagation
...
m2
Poynting Theorem
We have the Maxwell equations
B
E
t
(i )
D
H
J (ii )
t
Taking dot product
H with
(i) and
E with
(ii) and subtracting
B D
H
...
H H
...
E
...
( E H )
B ( H ) H 2
H
...
(
)
t
t
t 2
Similarly
D ( E ) E 2
E
...
(
)
t
t
t 2
Then from (iii)
E2 H 2
...
J
t 2
2
...
J
t
as E H S
and
uEM
E2
2
H 2
2
This is sometimes called differential form of Poynting theorem
...
S dV
V
V
uEM
dV E
...
d A
A
V
uEM
dV E
...
LHS of the equation rate of flow of electromagnetic energy
through the closed area enclosing the
given volume
1st term of RHS
volume
rate of change of electromagnetic energy in
1st term of RHS work done by the electromagnetic field on the
source of current
...
In absence of any source, J=o
then
...
Poynting Vector & Intensity of electromagnetic wave
Since E and H are mutually perpendicular
S EH
EB
Here E and H are instantaneous values
...
But some new phenomenon observed during the last decade of
19th century which is not explained by classical physics
...
The quantum idea was 1st introduced by Max Planck in 1900 to
explain the observed energy distribution in the spectrum of
black body radiation which is later used successfully by Einstein
to explain Photoelectric Effect
...
The concept of dual nature of radiation was extended to Louis
De Broglie who suggested that particles should have wave
nature under certain circumstances
...
The concept of Uncertainty Principle was introduced by
Heisenberg which explains that all the physical properties of a
system cannot even in principle, be determined simultaneously
with unlimited accuracy
...
Every system is characterized by a wave function ψ which
describes the state of the system completely and developed by
Max Born
...
The relativistic quantum mechanics was formulated by P
...
M
...
In this way, this leads to the development of quantum field
theory which successfully describes the interaction of radiation
with matter and describes most of the phenomena in Atomic
physics, nuclear physics, Particle physics, Solid state physics
and Astrophysics
...
All the laws of quantum physics reduces to the laws of classical
physics under certain circumstances of quantum physics are a
super set then classical physics is a subset
...
e
...
BLACK BODY RADIATION
A black body is one which absorbs all them radiations
incident on it
...
The black body emits radiation when it is heated at a fixed
temperature and it contains all frequencies ranging from
zero to infinity
...
The energy distribution curve for black body radiation
shows the following characteristics such as
At a given temperature the energy density has
maximum value corresponding to a value of
frequency or wavelength
...
The energy density decreases to zero for both higher
and lower values of frequency or wavelength
...
Many formulations are formulated to explain the above
experimental observations like Stefan-Boltzmann law,
Wein‟s displacement law and Planck‟s radiation formula
...
PLANCK’S RADIATION FORMULA
According to Planck the black body was assumed to be cavity
which consists of a large no
...
e ν
⁄
,
]
Therefore,
(ν)dν=
which is called Rayleigh-Jeans law
...
e
,
,
⁄
⁄
Therefore,
⁄
,
which
is
called Wein‟s radiation formula
...
Experimental Arrangement
The experimental arrangement consists of the following parts
...
It is directly proportional to intensity of incident light
...
Stopping potential depends upon the frequency but independent
of intensity
...
Saturation current is independent of frequency
...
i
...
ii
...
E to the electron ( m
)
...
E of the emitted electron, we have
(4)
Using eqn (4) in eqn(1), we get
implies that
(5)
Calculation of threshold frequency
We have
We have
⁄
(6)
Substituting eqn(2) in eqn(6) we get
...
e
Calculation of Planck Constant (h)
If we multiply the slope of plot of stopping potential
with „e‟
we get „h‟
i
...
Is wave nature of radiation successfully explains the Compton
effect? Justify your answer
...
No
Compton effect
The phenomena in which a beam of high frequency radiation like xray &γ-ray is incident on a metallic block and undergoes scattering is
called Compton effect
...
It depends on the angle of scattering (angle between the
scattered & incident x-ray)
...
*Wave nature of radiation is unable to explain Compton shift as the
Compton shift depends on angle of scattering and wavelength of
scattered x-ray is different from that of the incident x-rays
...
Though the electron is closely bound with the nucleus, but a
small fraction of energy is used to free the electron
...
E and recoils at an angle ϕ to the incident photon
direction after collision and the photon with decrease energy hν' will
emerges at an angle θ to the initial direction after collision
...
62 1034 JS
9
...
426 1012 m
c 0
...
2, c
, 2c (Maximum
shift)
Pair Production:
The phenomenon in which some γ-rays are converted into electronpositron pair on passing near an atomic nucleus is called Pair
production
...
Pair production is not possible if the γ-rays are treated as EM
waves for which the pair production is not possible in vacuum
...
The minimum frequency 0 of γ-rays for which
h 0 2mec 2 and
the
pair production takes place is called threshold frequency
...
(γ-ray)
Compton effect takes place for intermediate frequency value
...
Matter waves and De-Broglie Hypothesis
The waves associated with all material particles are called Matter
waves
...
The wave nature of electron was demonstrated by division and
Germer
...
If Δx= uncertainty in x-component of the position of a particle
Δpx= uncertainty in x-component of its linear momentum
x px
then,
2
Similarly for y and z-component
y p y
2
, z pz
2
Again uncertainty in energy and time is given by
t E
2
Application of the uncertainty principle;
i
...
P and x
i
...
px
2
x
...
x
p
(2)
2
Using eqn(2) in eqn(1),we get
E
2
1
m 2 x 2
2mx 2
(3)
2
Since the energy E of the oscillator is minimum in the ground state, so
E
0
x x x0
2
E
0
3 m 2 x0
x x x0 mx0
2
x0
Where
x0 corresponds
(4)
m
to the ground state
...
ii
...
p
p
2
p
6
...
m
5
...
14 10 m
s
2x
The minimum energy of the electron in the nucleus is
2
E m0 c 4 p 2c 2
9
...
3 1021 kg
...
6 1012 J 10MeV
As energy of the electrons emitted in β-decay process is much less
than this estimated value, so the electrons cannot be a part of the
nucleus
...
Ground State energy of the H-Atom
The energy of the H-atom is given as
E
p2
e2
2m 4 0 r
(1)
Let r Uncertainty in position of the electron in the orbit of radius r
in the ground state
...
Then using principle,
r
...
p
r
...
P
1
...
px
1
...
22h
2sin
x
...
2) Electron Diffraction:
Let a be the width of the slit through which the electron beam is
diffracted along y-direction
...
py a 2 p sin 2 p a sin
(2)
2 p 2 p
...
p y
h
2h
p
2
which satisfies the uncertainty principle
...
If the initial conditions of a system are
known, its subsequent configurations can be determined by using the
relevant laws of physics applicable to the system
...
But this deterministic description is
inconsistent with observation
...
This
probabilistic description is the basic characteristic of quantum physics
and is achieved by the wave function
...
It describes all information‟s like amplitude, frequency,
wavelength etc
...
It is a mathematical entity by which the observable physical
properties of a system can be determined
...
i
...
r, t x, y, z; t
It is a complex function having both real and imaginary part
...
The wave function and its first derivative
x
are continuous at
all places including boundaries
...
e
...
The quantity
2
represents the probability density
...
Superposition principle
This principle states that “Any well behaved state of a system can be
expressed as a linear superposition of different possible allowed states
in which the system can exists
...
be the wave functions representing the allowed states,
then the state of the system can be expressed as
1 2 3
...
is
i
...
2
As the probability density is proportional to square of the wave
function, so the wavefunction is called “probability amplitude”
...
the wavefunction has dimension
Dimension of 1-D wave function is L
1
2
...
Observables
The physical properties associated with the wave function provides
the complete description of the system state or configuration are
called observables
...
Operators
The tools used for obtaining new function from a given function are
called operators
...
Physical Quantity
Operator
Energy-E
i
Momentum- p
i
Potential Energy(V)
V
Kinetic
2
Energy( p
2m
)
t
2 2
2m
Eigen States:
The number of definite allowed states for the system are called
eigen states
...
For any operator
ˆ
A having
eigen values
i corresponding
states i
equation is
to the eigen
the eigen value
ˆ
A i i i
Expectation Values:
The expectation values of a variable is the weighted average of the
eigen values with their
relative
probabilities
...
are
p1 , p2 , p3 ,
...
pn qn
p1 p2
...
1
Q p1q1 p2 q2
...
For any function to be normalized is given as
r, t
2
dV 1
The expectation value of energy,
ˆ
E H dV i
dV
t
i
dV
t
Schrodinger‟s Equation:The partial differential equation of a wave function involving the
derivatives of space and time coordinates is called Schrodinger equation
...
If the particle is in a potential V(x), then
E
i
p2
V
2m
2 2
V
t
2m x 2
(4)
Similarly along Y and Z-axis is given as
i
2 2
V
t
2m y 2
i
2 2
V
t
2m z 2
Time-dependent Schrodinger equation in 3-D:
i
2 2
2
2
2 2 V
t
2m x 2 y z
i
2 2
V
t
2m
Time-independent Schrodinger equation:
If the energy of the system does not change with time then
E remains
constant
Now from eqn (1),
i
i
t
E
i E
2 2
V
2m
[fromeqn (5)]
(5)
2
2m
2
E V 0
This is time-independent Schrodinger equation in 3-D
...
The potential step can be given as
V ( x) 0, x 0
V0 , x 0
Let us consider the particle incident on the potential step from left to
right
...
If the energy of the particle is greater than the
height of the potential step the particle will go to the region-2
...
Thuseqn (8) becomes
2 ( x) Ceik x
2
(9)
Using boundary condition,
1 x x 0 2 x x 0
1
x
x 0
2
x
We have
x 0
A+B=C and
B
ik1 ( A B) ik2e
k1 k2
A
k1 k2
(10)
Thus it is observed that
1
...
2
...
3
...
4
...
Case-2: (E
d 2 2 2m
2 V0 E 2 0
dx 2
Where
2m
2
d 2 2
2 2 0
2
dx
V0 E
Thus the solution of the equation is given as
2 ( x) Ce x De x
And
C
2k1
A
k1 k2
(1)
Reflection Coefficient
It is defined as the ratio of reflected flux to incident flux of the particle
...
e R
V1 ref
2
V1 inc
2
, V1
k1
m
ref ref
B
k k
2 1 2 2
inc inc
A
k1 k2
2
2
R
E V
E E V0
2
E
2
0
Transmission Coefficient:It is defined as the ratio between transmitted flux to incident flux
...
e
V2 tran
V1 inc
2
2
, V2
k2
m
2
k C
k
2 tran tran 2 2
k1 inc inc
k1 A
T
4k1k2
k1 k2
2
4 E E V0
E E V0
2
As x , 0 De x ,for which D=0
So
2 x Ce x
(2)
Which indicates that the probability of finding of particle in region-2 is
not zero which is classically forbidden as there is some particles on
region-2 according to quantum mechanics
...
The potential of such region is given by, V(x) =0, x<0 and x>a
=V0, 0≤x≤a
Let us consider a particle is travelling from left to right
...
Case-1(E
particles in the region-1, then
d 2 1 2mE
2 1 0
dx 2
d 2 1
k 2 1 0
2
dx
where k
(1)
2mE
2
In region-2, Schrodinger wave equation is given as
d 2 2 2m
2 V0 E 2 0
dx 2
d 2 2
2 2 0
dx 2
Where
2m
2
(2)
V0 E
In region-3,
d 2 3 2m
2 3 0
dx 2
d 2 3
k 2 3 0
2
dx
(3)
The general solution of Schrodinger equation in the three region is given
by
1 x Aeikx Beikx
(4)
2 x Ce x De x
(5)
3 x Feikx Geikx
(6)
Where Aeikx , Beikx → the incident and reflected waves in region-1
Feikx →
Transmitted wave in region-2
Geikx →wave
incident from right in region-3
=0
3 Feikx
(7)
The wave function 1 and 2 and their derivatives continuous at x=0
...
But
in the region-2 the wave function is non-zero
...
Thus there is non-zero probability of finding the particles in region-3
even if the incident particle energy is less than the barrier height
...
Ex: Emission of α-particle, nuclear fission, tunnel diode, Josephson
junction, scanning tunneling microscope
...
The transmission co-efficient is given as
T 16
Case-2(E>V0)
E
E a
1 e
V0 V0
According to quantum mechanics if E>V0, all particles should be
transmitted to the region-3 without any reflection but it is not possible
for all the values of incident energy
...
n
2
Particle in a one dimensional box:
The physical situation in which the potential between the boundary wall
is zero and is infinite at the rigid walls is called one dimensional box or
one dimensional infinite potential well
...
Thus eqn(2) becomes,
0 A sin kx B cos kxx0 0 B
B0
Thus the wave function inside the well is given as
x A sin kx
0 x a
Energy eigen Values:From eqn(3),
0 A sin kxx0 A sin ka
at x=a
(3)
ka n
an
n=1,2,3…
...
Since
k2
k 2a2
En
2mE
2
2mEa 2
2
2
n 2 2
2
2ma
2
n2
Thus the energy of the particle in the infinite well is quantized
...
The energy of the higher allowed levels are multiple of E1 and
proportional to square of natural numbers
...
Eigen Functions
The eigen functions of the allowed states can be obtained as
2
dx 1
a
0 x a
A sin 2 kxdx 1
2
0
1 cos 2kx
dx 1
2
0
a
A
2
a
2
a
sin 2kx
2 x 0
a
A
2k 0
A2
2
a
A
2
a
n x
2
n
sin
x
a
a
Thus the eigen function for each quantum state are obtained by
1 x
2
sin x
a
a
2 x
2
2
sin
x
a
a
3 x
2
3
sin
x etc
Title: Waves and Oscillations
Description: Waves and Oscillations for undergraduate student
Description: Waves and Oscillations for undergraduate student