Important Formulas: Wave Optics

# Important Wave Optics Formulas for JEE and NEET

``` Page 1

Top Concepts

1. A wave front is the locus of points having the same phase of oscillation.
Rays are the lines perpendicular to the wavefront, which show the
direction of propagation of energy. The time taken for light to travel from
one wavefront to another is the same along any ray.

2. Huygens’ Principle.

According to Huygens’
(a) Each point on the given wave front (called primary wave front)
acts as a fresh source of new disturbance, called secondary
wavelet, which travels in all directions with the velocity of light
in the medium
(b) A surface touching these secondary wavelets, tangentially in the
forward direction at any instant gives the new wavefront at that
instant. This is called secondary wave front,

3. Huygens’ Construction is based on the principle that every point of a
wavefront is a source of secondary wavefront. The envelope of these
wavefronts i.e., the surface tangent to all the secondary wavefront gives
the new wavefront.

4. Refraction and Reflection of Plane Waves Using Huygens’
Principle.
The law of reflection (i = r) and the Snell’s law of refraction
?
? ? ? ?
?
12
21
21
v sini
sinr v

can be derived using the wave theory. (Here v
1
and v
2
are the speed of
light in media 1 and 2 with refractive index
12
and ?? respectively).
The frequency ? remains the same as light travels from one medium to
another. The speed v of a wave is given by
v
T
?
?
where ? is the wavelength of the wave and T (=1/ ?) is the period of
oscillation.

5. Doppler effect is the shift in frequency of light when there is a relative
motion between the source and the observer. The effect can be used to
measure the speed of an approaching or receding object.

Important Terms, Definitions & Formulae

Page 2

Top Concepts

1. A wave front is the locus of points having the same phase of oscillation.
Rays are the lines perpendicular to the wavefront, which show the
direction of propagation of energy. The time taken for light to travel from
one wavefront to another is the same along any ray.

2. Huygens’ Principle.

According to Huygens’
(a) Each point on the given wave front (called primary wave front)
acts as a fresh source of new disturbance, called secondary
wavelet, which travels in all directions with the velocity of light
in the medium
(b) A surface touching these secondary wavelets, tangentially in the
forward direction at any instant gives the new wavefront at that
instant. This is called secondary wave front,

3. Huygens’ Construction is based on the principle that every point of a
wavefront is a source of secondary wavefront. The envelope of these
wavefronts i.e., the surface tangent to all the secondary wavefront gives
the new wavefront.

4. Refraction and Reflection of Plane Waves Using Huygens’
Principle.
The law of reflection (i = r) and the Snell’s law of refraction
?
? ? ? ?
?
12
21
21
v sini
sinr v

can be derived using the wave theory. (Here v
1
and v
2
are the speed of
light in media 1 and 2 with refractive index
12
and ?? respectively).
The frequency ? remains the same as light travels from one medium to
another. The speed v of a wave is given by
v
T
?
?
where ? is the wavelength of the wave and T (=1/ ?) is the period of
oscillation.

5. Doppler effect is the shift in frequency of light when there is a relative
motion between the source and the observer. The effect can be used to
measure the speed of an approaching or receding object.

Important Terms, Definitions & Formulae

for the source moving away from the observer,
0
?? ? and for the source
moving towards the observer,
0
?? ? . The change in frequency is given as
00
v
c
? ? ? ? ? ? ? ? ?
where we are using the approximation v c. ?
So, finally,
0
v
c
?
?
?
??

6. Coherent and Incoherent Addition of Waves. Two sources are
coherent if they have the same frequency and a stable phase difference.
In this case, the total intensity I is not just the sum of individual
intensities I
1
and I
2
due to the two sources but includes an interference
term:

1 2 1 2
I I I 2k.E . E ? ? ?
where E
1
and E
2
are the electric fields at a point due to the sources.
The interference term averaged over many cycles is zero if
(a) the sources have different frequencies; or
(b) the sources have the same frequency but no stable phase
difference.
For such coherent sources,
1 2.
I I I ??
According to the superposition principle when two or more wave
motions traveling through a medium superimpose one another, a new
wave is formed in which resultant displacements due to the individual
waves at that instant.
The average of the total intensity will be
? ? ? ?
1 2 1 2
I I I 2 I I cos ? ? ? ?
where ? is the inherent phase difference between the two
superimposing waves.
The significance is that the intensity due to two sources of light is not
equal to the sum of intensities due to each of them. The resultant
intensity depends on the relative location of the point from the two
sources, since changing it changes the path difference as we go from
one point to another. As a result, the resulting intensity will vary
between maximum and minimum values, determined by the maximum
and minimum values of the cosine function. These will be
? ? ? ?
? ?
? ? ? ?
? ?
2
MAX
1 2 1 2 1 2
2
MIN
1 2 1 2 1 2
I I I 2 I I I I
I I I 2 I I I I
? ? ? ? ?
? ? ? ? ?

7. Young’s experiment, two parallel and very close slits S
1
and S
2

(illuminated by another narrow slit) behave like two coherent sources and
produce on a screen a pattern of dark and bright bands – interference
fringes. For a point P on the screen, the path difference
2
Page 3

Top Concepts

1. A wave front is the locus of points having the same phase of oscillation.
Rays are the lines perpendicular to the wavefront, which show the
direction of propagation of energy. The time taken for light to travel from
one wavefront to another is the same along any ray.

2. Huygens’ Principle.

According to Huygens’
(a) Each point on the given wave front (called primary wave front)
acts as a fresh source of new disturbance, called secondary
wavelet, which travels in all directions with the velocity of light
in the medium
(b) A surface touching these secondary wavelets, tangentially in the
forward direction at any instant gives the new wavefront at that
instant. This is called secondary wave front,

3. Huygens’ Construction is based on the principle that every point of a
wavefront is a source of secondary wavefront. The envelope of these
wavefronts i.e., the surface tangent to all the secondary wavefront gives
the new wavefront.

4. Refraction and Reflection of Plane Waves Using Huygens’
Principle.
The law of reflection (i = r) and the Snell’s law of refraction
?
? ? ? ?
?
12
21
21
v sini
sinr v

can be derived using the wave theory. (Here v
1
and v
2
are the speed of
light in media 1 and 2 with refractive index
12
and ?? respectively).
The frequency ? remains the same as light travels from one medium to
another. The speed v of a wave is given by
v
T
?
?
where ? is the wavelength of the wave and T (=1/ ?) is the period of
oscillation.

5. Doppler effect is the shift in frequency of light when there is a relative
motion between the source and the observer. The effect can be used to
measure the speed of an approaching or receding object.

Important Terms, Definitions & Formulae

for the source moving away from the observer,
0
?? ? and for the source
moving towards the observer,
0
?? ? . The change in frequency is given as
00
v
c
? ? ? ? ? ? ? ? ?
where we are using the approximation v c. ?
So, finally,
0
v
c
?
?
?
??

6. Coherent and Incoherent Addition of Waves. Two sources are
coherent if they have the same frequency and a stable phase difference.
In this case, the total intensity I is not just the sum of individual
intensities I
1
and I
2
due to the two sources but includes an interference
term:

1 2 1 2
I I I 2k.E . E ? ? ?
where E
1
and E
2
are the electric fields at a point due to the sources.
The interference term averaged over many cycles is zero if
(a) the sources have different frequencies; or
(b) the sources have the same frequency but no stable phase
difference.
For such coherent sources,
1 2.
I I I ??
According to the superposition principle when two or more wave
motions traveling through a medium superimpose one another, a new
wave is formed in which resultant displacements due to the individual
waves at that instant.
The average of the total intensity will be
? ? ? ?
1 2 1 2
I I I 2 I I cos ? ? ? ?
where ? is the inherent phase difference between the two
superimposing waves.
The significance is that the intensity due to two sources of light is not
equal to the sum of intensities due to each of them. The resultant
intensity depends on the relative location of the point from the two
sources, since changing it changes the path difference as we go from
one point to another. As a result, the resulting intensity will vary
between maximum and minimum values, determined by the maximum
and minimum values of the cosine function. These will be
? ? ? ?
? ?
? ? ? ?
? ?
2
MAX
1 2 1 2 1 2
2
MIN
1 2 1 2 1 2
I I I 2 I I I I
I I I 2 I I I I
? ? ? ? ?
? ? ? ? ?

7. Young’s experiment, two parallel and very close slits S
1
and S
2

(illuminated by another narrow slit) behave like two coherent sources and
produce on a screen a pattern of dark and bright bands – interference
fringes. For a point P on the screen, the path difference
2
1
22
1
yd
S P S P
D
??
where d is the separation between two slits, D
1
is the distance
between the slits and the screen and y
1
is the distance of the point of
P from the central fringe.
For constructive interference (bright band), the path difference must
be an integer multiple of ?, i.e.,

11
1
1
y d D
n or y n
Dd
?
? ? ?
The separation ?y
1
between adjacent bright (or dark) fringes is.

?
? ?
1
1
D
y
d

using which ? can be measured.

8. Diffraction refers to light spreading out from narrow holes and slits, and
bending around corners and obstacles. The single-slit diffraction pattern
shows the central maximum ( at ? = 0), zero intensity at angular
separation ? = ? (n + ½) ?… (n ? 0).
Different parts of the wavefront at the slit act as secondary sources:
diffraction pattern is the result of interference of waves from these
sources.
The intensity plot looks as follows, with there being a bright central
maximum, followed by smaller intensity secondary maxima, with there
being points of zero intensity in between, whenever dsin n ,n 0 ?? ??

9. Emission, absorption and scattering are three processes by which
In emission, an accelerated charge radiates and loses energy.
In absorption, the charge gains energy at the expense of the
electromagnetic wave.
In scattering, the charge accelerated by incident electromagnetic wave

10. Polarization specifies the manner in which electric field E oscillates
in the plane transverse to the direction of propagation of light. If E
oscillates back and forth in a straight line, the wave is said to be
2
Page 4

Top Concepts

1. A wave front is the locus of points having the same phase of oscillation.
Rays are the lines perpendicular to the wavefront, which show the
direction of propagation of energy. The time taken for light to travel from
one wavefront to another is the same along any ray.

2. Huygens’ Principle.

According to Huygens’
(a) Each point on the given wave front (called primary wave front)
acts as a fresh source of new disturbance, called secondary
wavelet, which travels in all directions with the velocity of light
in the medium
(b) A surface touching these secondary wavelets, tangentially in the
forward direction at any instant gives the new wavefront at that
instant. This is called secondary wave front,

3. Huygens’ Construction is based on the principle that every point of a
wavefront is a source of secondary wavefront. The envelope of these
wavefronts i.e., the surface tangent to all the secondary wavefront gives
the new wavefront.

4. Refraction and Reflection of Plane Waves Using Huygens’
Principle.
The law of reflection (i = r) and the Snell’s law of refraction
?
? ? ? ?
?
12
21
21
v sini
sinr v

can be derived using the wave theory. (Here v
1
and v
2
are the speed of
light in media 1 and 2 with refractive index
12
and ?? respectively).
The frequency ? remains the same as light travels from one medium to
another. The speed v of a wave is given by
v
T
?
?
where ? is the wavelength of the wave and T (=1/ ?) is the period of
oscillation.

5. Doppler effect is the shift in frequency of light when there is a relative
motion between the source and the observer. The effect can be used to
measure the speed of an approaching or receding object.

Important Terms, Definitions & Formulae

for the source moving away from the observer,
0
?? ? and for the source
moving towards the observer,
0
?? ? . The change in frequency is given as
00
v
c
? ? ? ? ? ? ? ? ?
where we are using the approximation v c. ?
So, finally,
0
v
c
?
?
?
??

6. Coherent and Incoherent Addition of Waves. Two sources are
coherent if they have the same frequency and a stable phase difference.
In this case, the total intensity I is not just the sum of individual
intensities I
1
and I
2
due to the two sources but includes an interference
term:

1 2 1 2
I I I 2k.E . E ? ? ?
where E
1
and E
2
are the electric fields at a point due to the sources.
The interference term averaged over many cycles is zero if
(a) the sources have different frequencies; or
(b) the sources have the same frequency but no stable phase
difference.
For such coherent sources,
1 2.
I I I ??
According to the superposition principle when two or more wave
motions traveling through a medium superimpose one another, a new
wave is formed in which resultant displacements due to the individual
waves at that instant.
The average of the total intensity will be
? ? ? ?
1 2 1 2
I I I 2 I I cos ? ? ? ?
where ? is the inherent phase difference between the two
superimposing waves.
The significance is that the intensity due to two sources of light is not
equal to the sum of intensities due to each of them. The resultant
intensity depends on the relative location of the point from the two
sources, since changing it changes the path difference as we go from
one point to another. As a result, the resulting intensity will vary
between maximum and minimum values, determined by the maximum
and minimum values of the cosine function. These will be
? ? ? ?
? ?
? ? ? ?
? ?
2
MAX
1 2 1 2 1 2
2
MIN
1 2 1 2 1 2
I I I 2 I I I I
I I I 2 I I I I
? ? ? ? ?
? ? ? ? ?

7. Young’s experiment, two parallel and very close slits S
1
and S
2

(illuminated by another narrow slit) behave like two coherent sources and
produce on a screen a pattern of dark and bright bands – interference
fringes. For a point P on the screen, the path difference
2
1
22
1
yd
S P S P
D
??
where d is the separation between two slits, D
1
is the distance
between the slits and the screen and y
1
is the distance of the point of
P from the central fringe.
For constructive interference (bright band), the path difference must
be an integer multiple of ?, i.e.,

11
1
1
y d D
n or y n
Dd
?
? ? ?
The separation ?y
1
between adjacent bright (or dark) fringes is.

?
? ?
1
1
D
y
d

using which ? can be measured.

8. Diffraction refers to light spreading out from narrow holes and slits, and
bending around corners and obstacles. The single-slit diffraction pattern
shows the central maximum ( at ? = 0), zero intensity at angular
separation ? = ? (n + ½) ?… (n ? 0).
Different parts of the wavefront at the slit act as secondary sources:
diffraction pattern is the result of interference of waves from these
sources.
The intensity plot looks as follows, with there being a bright central
maximum, followed by smaller intensity secondary maxima, with there
being points of zero intensity in between, whenever dsin n ,n 0 ?? ??

9. Emission, absorption and scattering are three processes by which
In emission, an accelerated charge radiates and loses energy.
In absorption, the charge gains energy at the expense of the
electromagnetic wave.
In scattering, the charge accelerated by incident electromagnetic wave

10. Polarization specifies the manner in which electric field E oscillates
in the plane transverse to the direction of propagation of light. If E
oscillates back and forth in a straight line, the wave is said to be
2
linearly polarized. If the direction of E changes irregularly the wave
is unpolarized.
When light passes through a single polaroid P
1
light intensity is
reduced to half, independent of the orientation of P
1
. When a second
Polaroid P
2
is also included, at one specific orientation wrt P1, the net
transmitted intensity is reduced to zero but is transmitted fully when
P
1
is turned
o
90 from that orientation. This happens because the
transmitted polarization by a polaroid is the component of E parallel to
its axis.
Unpolarized sunlight scattered by the atmosphere or reflected from a
medium gets (partially) polarized.
Linearly Polarized light passing through some substances like sugar
solution undergoes a rotation of its direction of polarization,
proportional to the length of the medium traversed and the
concentration to the substance. This effect is known as optical activity.

11. Brewster’s Law:  When an incident light is incident at the polarizing
angle, the reflected & the refracted rays are perpendicular to each other.
The polarizing angle, also called as Brewster’s angle, is given by
tan ?
p
= ?
this expression is also called Brewster’s law.
12. Polarization by scattering: Light is scattered when it meets a particle of
similar size to its own wavelength. For e.g. scattering of sunlight by dust
particles.
Rayleigh showed that  the scattering of light is proportional to the fourth
power of the frequency of the light or varies as 4
1
?
where ? is the
wavelength of light incident on the air molecules of size d where d << ?.
Hence blue light is scattered more than red. This explains the blue colour
of the sky.

TOP Formulae
1. Snell’s law of refraction:

1
12
2
c speed of light in f isrt medium
c speed of light in second medium
? ? ?

2. Relation between phase difference & path difference:
2
. x
?
? ? ? ?
?

where ? ? is the phase difference & ?x is the path difference
3. Young’s double slit interference experiment:
Fringe width:
D
w
d
?
?

where D is the distance between the slits & the screen
d is the distance between the two slits
2
Page 5

Top Concepts

1. A wave front is the locus of points having the same phase of oscillation.
Rays are the lines perpendicular to the wavefront, which show the
direction of propagation of energy. The time taken for light to travel from
one wavefront to another is the same along any ray.

2. Huygens’ Principle.

According to Huygens’
(a) Each point on the given wave front (called primary wave front)
acts as a fresh source of new disturbance, called secondary
wavelet, which travels in all directions with the velocity of light
in the medium
(b) A surface touching these secondary wavelets, tangentially in the
forward direction at any instant gives the new wavefront at that
instant. This is called secondary wave front,

3. Huygens’ Construction is based on the principle that every point of a
wavefront is a source of secondary wavefront. The envelope of these
wavefronts i.e., the surface tangent to all the secondary wavefront gives
the new wavefront.

4. Refraction and Reflection of Plane Waves Using Huygens’
Principle.
The law of reflection (i = r) and the Snell’s law of refraction
?
? ? ? ?
?
12
21
21
v sini
sinr v

can be derived using the wave theory. (Here v
1
and v
2
are the speed of
light in media 1 and 2 with refractive index
12
and ?? respectively).
The frequency ? remains the same as light travels from one medium to
another. The speed v of a wave is given by
v
T
?
?
where ? is the wavelength of the wave and T (=1/ ?) is the period of
oscillation.

5. Doppler effect is the shift in frequency of light when there is a relative
motion between the source and the observer. The effect can be used to
measure the speed of an approaching or receding object.

Important Terms, Definitions & Formulae

for the source moving away from the observer,
0
?? ? and for the source
moving towards the observer,
0
?? ? . The change in frequency is given as
00
v
c
? ? ? ? ? ? ? ? ?
where we are using the approximation v c. ?
So, finally,
0
v
c
?
?
?
??

6. Coherent and Incoherent Addition of Waves. Two sources are
coherent if they have the same frequency and a stable phase difference.
In this case, the total intensity I is not just the sum of individual
intensities I
1
and I
2
due to the two sources but includes an interference
term:

1 2 1 2
I I I 2k.E . E ? ? ?
where E
1
and E
2
are the electric fields at a point due to the sources.
The interference term averaged over many cycles is zero if
(a) the sources have different frequencies; or
(b) the sources have the same frequency but no stable phase
difference.
For such coherent sources,
1 2.
I I I ??
According to the superposition principle when two or more wave
motions traveling through a medium superimpose one another, a new
wave is formed in which resultant displacements due to the individual
waves at that instant.
The average of the total intensity will be
? ? ? ?
1 2 1 2
I I I 2 I I cos ? ? ? ?
where ? is the inherent phase difference between the two
superimposing waves.
The significance is that the intensity due to two sources of light is not
equal to the sum of intensities due to each of them. The resultant
intensity depends on the relative location of the point from the two
sources, since changing it changes the path difference as we go from
one point to another. As a result, the resulting intensity will vary
between maximum and minimum values, determined by the maximum
and minimum values of the cosine function. These will be
? ? ? ?
? ?
? ? ? ?
? ?
2
MAX
1 2 1 2 1 2
2
MIN
1 2 1 2 1 2
I I I 2 I I I I
I I I 2 I I I I
? ? ? ? ?
? ? ? ? ?

7. Young’s experiment, two parallel and very close slits S
1
and S
2

(illuminated by another narrow slit) behave like two coherent sources and
produce on a screen a pattern of dark and bright bands – interference
fringes. For a point P on the screen, the path difference
2
1
22
1
yd
S P S P
D
??
where d is the separation between two slits, D
1
is the distance
between the slits and the screen and y
1
is the distance of the point of
P from the central fringe.
For constructive interference (bright band), the path difference must
be an integer multiple of ?, i.e.,

11
1
1
y d D
n or y n
Dd
?
? ? ?
The separation ?y
1
between adjacent bright (or dark) fringes is.

?
? ?
1
1
D
y
d

using which ? can be measured.

8. Diffraction refers to light spreading out from narrow holes and slits, and
bending around corners and obstacles. The single-slit diffraction pattern
shows the central maximum ( at ? = 0), zero intensity at angular
separation ? = ? (n + ½) ?… (n ? 0).
Different parts of the wavefront at the slit act as secondary sources:
diffraction pattern is the result of interference of waves from these
sources.
The intensity plot looks as follows, with there being a bright central
maximum, followed by smaller intensity secondary maxima, with there
being points of zero intensity in between, whenever dsin n ,n 0 ?? ??

9. Emission, absorption and scattering are three processes by which
In emission, an accelerated charge radiates and loses energy.
In absorption, the charge gains energy at the expense of the
electromagnetic wave.
In scattering, the charge accelerated by incident electromagnetic wave

10. Polarization specifies the manner in which electric field E oscillates
in the plane transverse to the direction of propagation of light. If E
oscillates back and forth in a straight line, the wave is said to be
2
linearly polarized. If the direction of E changes irregularly the wave
is unpolarized.
When light passes through a single polaroid P
1
light intensity is
reduced to half, independent of the orientation of P
1
. When a second
Polaroid P
2
is also included, at one specific orientation wrt P1, the net
transmitted intensity is reduced to zero but is transmitted fully when
P
1
is turned
o
90 from that orientation. This happens because the
transmitted polarization by a polaroid is the component of E parallel to
its axis.
Unpolarized sunlight scattered by the atmosphere or reflected from a
medium gets (partially) polarized.
Linearly Polarized light passing through some substances like sugar
solution undergoes a rotation of its direction of polarization,
proportional to the length of the medium traversed and the
concentration to the substance. This effect is known as optical activity.

11. Brewster’s Law:  When an incident light is incident at the polarizing
angle, the reflected & the refracted rays are perpendicular to each other.
The polarizing angle, also called as Brewster’s angle, is given by
tan ?
p
= ?
this expression is also called Brewster’s law.
12. Polarization by scattering: Light is scattered when it meets a particle of
similar size to its own wavelength. For e.g. scattering of sunlight by dust
particles.
Rayleigh showed that  the scattering of light is proportional to the fourth
power of the frequency of the light or varies as 4
1
?
where ? is the
wavelength of light incident on the air molecules of size d where d << ?.
Hence blue light is scattered more than red. This explains the blue colour
of the sky.

TOP Formulae
1. Snell’s law of refraction:

1
12
2
c speed of light in f isrt medium
c speed of light in second medium
? ? ?

2. Relation between phase difference & path difference:
2
. x
?
? ? ? ?
?

where ? ? is the phase difference & ?x is the path difference
3. Young’s double slit interference experiment:
Fringe width:
D
w
d
?
?

where D is the distance between the slits & the screen
d is the distance between the two slits
2
Constructive interference:
Phase difference :
??
= 2 ?n where n is an integer
Path difference:
x ?
=n ?, where n is an integer

Destructive interference:
Phase difference :
1
n2
2
??
? ? ? ? ?
??
??
where n is an integer
Path difference:
1
xn
2
??
? ? ? ?
??
??
, where n is an integer

4. Diffraction due to single slit:
Angular spread of the central maxima=
2
d
?

Width of the central maxima:
2D
d
?

where D is the distance of the slit from the screen
d is the slit width

Condition for the minima on the either side of the central maxima:
d sin ? = n ? , where n = 1,2,3,….

5. Intensity of the light due to polarization:
I = I
o
cos
2
?
where I is the intensity of light after polarization
I
o
is the original intensity
? is the angle between the axis of the analyzer & the polarizer

Brewster’s Law:
? = tan ?
p
where ?
p
is the polarizing angle, that is, the angle of
incidence at which the angle of refraction in the second medium is
right angle

2
```

## Physics for JEE Main & Advanced

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289 videos|635 docs|179 tests

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,

,

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;