Revision: Optics >> Wave Optics Physics Science (English Medium) Class 12 CBSE

Definitions [20]

Define a wavefront.

Wavefront is defined as the locus of all the points in space that reach a particular distance by a propagating wave at the same instant.

A wave front is defined as a surface of constant phase.

What is a Polaroid?

A Polaroid is a material which polarises light. The phenomenon of selective absorption is made use of in the construction of polariods. There are different types of polaroids.

A Polaroid consists of micro crystals of herapathite (an iodosulphate of quinine). Each crystal is a doubly refracting medium, which absorbs the ordinary ray and transmits only the extra ordinary ray. The modern polaroid consists of a large number of ultra microscopic crystals of herapathite embedded with their optic axes, parallel, in a matrix of nitro - cellulose.

Recently, new types of polariod are prepared in which thin film of polyvinyl alcohol is used. These are colourless crystals which transmit more light, and give better polarisation.

Answer briefly.

What is Doppler effect?

The apparent change in the frequency of sound heard by a listener, due to relative motion between the source of sound and the listener is called Doppler effect in sound.

When the source and the observer are in relative motion with respect to each other and to the medium in which sound propagates, the frequency of the sound wave observed is different from the frequency of the source. This phenomenon is called Doppler Effect.

Definition: Critical Angle

The critical angle is the angle of incidence in the denser medium for which the angle of refraction in the rarer medium is 90°.

Definition: Plane of Vibration

The plane containing the electric field vectors of plane polarized light is called the plane of vibration.

Definition: Plane of Polarization

The plane perpendicular to the plane of the vibration and the electric field vector is called plane of polarization.

Definition: Fresnel Diffraction

Diffraction observed when the source or screen (or both) are at finite distances from the obstacle, and the incident wavefront is spherical or cylindrical.

Definition: Interference

Interference is the phenomenon in which the intensity of light (or any wave) at a point becomes non-uniform due to the superposition of two or more coherent waves, resulting in regions of constructive and destructive interference.

Definition: Resolving Power

The ability of an optical instrument to distinguish two closely spaced objects as separate and distinct is called its resolving power.

Definition: Limit of Resolution

The minimum visual angle between two objects that can be just resolved by an instrument is called the limit of resolution.

Definition: Ray of Light

The path along which light travels is called a ray of light.
A ray is defined as the path of energy propagation in the limit of wavelength tending to zero.

Definition: Ray Optics or Geometrical Optics

The study of optical phenomena under the assumption that it travels in a straight line as a ray is called ray optics or geometrical optics, as geometry is used in this study.
The branch of optics in which one completely neglects the finiteness of the wavelength is called geometrical optics.

Definition: Fraunhofer Diffraction

Diffraction observed when the source and screen are at large distances from the diffracting element, so that the incident wavefront is plane.

Definition: Resolving Power of Telescope

The resolving power of a telescope is then defined as the reciprocal of the least angular separation between the objects that are just resolved.

Defintiion: Diffraction of Light

Diffraction of light is the phenomenon in which light spreads into the geometrical shadow region when it passes around the edges of an obstacle or through a narrow aperture whose size is comparable to its wavelength.

Definition: Wave Optics

The branch of optics which uses the wave nature of light to explain the optical phenomena is called wave optics.

Definition: Unpolarized Light

Light in which the electric field vectors vibrate in all possible directions perpendicular to the direction of propagation.

Definition: Plane Polarized Light

Light in which the electric field vectors vibrate only in one particular direction perpendicular to the direction of propagation.

Definition: Polarization by Scattering

Polarisation by scattering is the phenomenon in which unpolarized light becomes partially or completely plane polarised when it is scattered by small particles such as air molecules or dust particles.

Definition: Polarizer

A material that allows only those light waves to pass whose electric field is along a particular direction (polarizing axis).

Formulae [3]

Formula: Critical Angle

\[
\sin C = \frac{n_2}{n_1}
\]

Where:

\[
n_1
\] = refractive index of denser medium
\[
n_2
\]= refractive index of rarer medium

For air:

\[
\sin C = \frac{1}{\mu}
\]

Formula: Fraunhofer Diffraction at a Single Slit

\[a\sin\theta=\pm\left(n+\frac{1}{2}\right)\lambda\]

Formula: Width of the Central Bright Fringe

\[W_{c}=2y_{1d}=2W=2\left(\frac{\lambda D}{a}\right)\]

Theorems and Laws [8]

With the help of a diagram, show how a plane wave is reflected from a surface. Hence, verify the law of reflection.

According to the laws of reflection:

At the point of incidence, the incident rays, reflected rays, and normal to the reflecting surface all lie in the same plane.
On opposing sides of the normal are the incident and reflected rays.
The angle of incidence and the angle of reflection are the same. i.e., ∠i = ∠r.

Explanation:

Reflection of light

XY: Plane reflecting surface

AB: Plane wavefront

RB₁: Reflecting wavefront

A₁M, B₁N: Normal to the plane

∠AA₁M = ∠BB₁N = ∠i = Angle of incidence

∠TA₁M = ∠QB₁N = ∠r = Angle of reflection

A plane wavefront AB is advancing obliquely towards the plane reflecting surface XY. The AA₁ and BB₁ are incident rays.

When ‘A’ reaches XY at A₁, then the ray at ‘B’ reaches point ‘P’, and it has to cover the distance PB₁ to reach the reflecting surface XY.

Let ‘t’ be the time required to cover the distance PB₁. During this time interval, secondary wavelets are emitted from A₁ and will spread over a hemisphere of radius A₁R, in the same medium. The distance covered by secondary wavelets to reach from A₁ to R in time t is the same as the distance covered by primary waves to reach from P to B₁. Thus, A₁R = PB₁ = ct.

All other rays between AA₁ and BB₁ will reach XY after A₁ and before B₁. Hence, they will also emit secondary wavelets of decreasing radii.

The surface touching all such hemispheres is RB₁ which is the reflected wavefront, bounded by reflected rays A₁R and B₁Q.

Draw A₁M ⊥ XY and B₁N ⊥ XY.

Thus, the angle of incidence is ∠AA₁M = ∠BB₁N = i, and the angle of reflection is ∠MA₁R = ∠NB₁Q = r.

∠RA₁B₁ = 90 − r

∠PB₁A₁ = 90 − i

In ΔA₁RB₁ and ΔA₁PB₁

∠A₁RB₁ = ∠A₁PB₁

A₁R = PB₁....(Reflected waves travel an equal distance in the same medium in equal time.)

A₁B₁ = A₁B₁....(Common side)

∴ ΔA₁RB₁ ≅ ΔA₁PB₁

∴ ∠RA₁B₁ = ∠PB₁A₁

∴ 90 − r = 90 − i

∴ i = r

Also from the figure, it is clear that incident rays, reflected rays, and normal lie in the same plane.

This explains the laws of reflection of light from a plane reflecting surface on the basis of Huygen’s wave theory.

Frequency, wavelength, and speed of light do not change after reflection. If reflection takes place from a denser medium, then the phase changes by π radians.

AB = Incident wavefront

CD = Reflected wavefront

XY = Reflecting surface

If c be the speed of light and t be the time taken by light to go from B to C or A to D or E to G through F, then

t = `(EF)/C + (FG)/C`

= `(AF sin i)/C + (FC sin r)/C`

= `(AC sin r + AF(sin i - sin r))/C`

For rays of light from different parts of the incident wavefront, the values of AF are different. But light from different points of the incident wavefront should take the same time to reach the corresponding points on the reflected wavefront.

So, ‘t’ should not depend upon AF.

This is possible only if sin i – sin r = 0.

i.e., sin i = sin r

⇒ i = r

Hence proved.

State law of Malus.

It states that when a completely plane polarised light beam is incident on an analyzer, the intensity of the emergent light varies as the square of the cosine of the angle between the plane of transmission of the analyzer and the polarizer.

I = I₀cos²θ

Law: Rayleigh’s Criterion

According to Rayleigh’s criterion, two objects are just resolved when the central maximum of one diffraction pattern coincides with the first minimum of the other.

Law: Malus’ Law

$I 2 = I 1 cos 2 θ$

It gives the intensity of plane polarized light after passing through a second polarizer, where θ is the angle between the axes of the two polarizers.

Law: Brewster’s Law

Statement

When unpolarized light is incident on a transparent surface at a particular angle (called Brewster’s angle), the reflected light is completely plane polarised.
At this angle, the reflected and refracted rays are perpendicular to each other.

\frac {n_2}{n_1}

where
= Brewster’s angle
, = refractive indices of the two media

Proof

At Brewster’s angle,

From Snell’s law:

Since ,

\frac {n_2}{n_1}

Conclusion

At Brewster’s angle, the reflected light is completely plane polarized and the reflected and refracted rays are mutually perpendicular.

Law: Huygens' Principle

"Each point on a wavefront acts as a secondary source of light emitting secondary light waves called wavelets in all directions which travel with the speed of light in the medium. The new wavefront can be obtained by taking the envelope of these secondary wavelets travelling in the forward direction and is thus, the envelope of the secondary wavelets in forward direction. The wavelets travelling in the backward direction are in effective".

Law: Laws of Reflection

First Law of Reflection:
The angle of incidence is equal to the angle of reflection.

Second Law of Reflection:
The incident ray, reflected ray, and the normal at the point of incidence lie in the same plane.

Law: Snell’s Law of Refraction

n1 sin i = n2 sin r

\[\frac{\sin i}{\sin r}=\frac{v_1}{v_2}\]

Key Points

Key Points: Total Internal Reflection

Total internal reflection occurs when light travels from a denser medium to a rarer medium.
It takes place only when the angle of incidence is greater than the critical angle.
At angles greater than the critical angle, no refraction occurs.
The entire light ray is reflected back into the denser medium.
The intensity of the reflected ray becomes maximum and the phenomenon is used in optical fibres.

Key Points: Refraction of Light at a Plane Boundary Between Two Media

Refraction at a plane boundary can be explained using Huygens’ principle and secondary wavelets.
When light enters a denser medium, its speed decreases and the wavefronts become closer.
The refracted image is not laterally inverted, but it appears bent (broken) at the boundary for oblique incidence.
The wavelength of light changes when it enters a different medium; it decreases in a denser medium.
The frequency remains unchanged while passing from one medium to another.

Key Points: Light Sources and Wavefronts

Primary sources emit their own light (e.g., the Sun, stars, a bulb); secondary sources reflect or scatter light (e.g., the Moon, planets).
A wavefront is the locus of all points having the same phase at a given instant of time.
The direction of propagation of light is perpendicular to the wavefront (along the rays).
A point source produces spherical wavefronts; far from the source, they appear as plane wavefronts.
A line source produces cylindrical wavefronts; the wave speed equals the speed at which the wavefront moves.

Key Points: Resolving Power

Resolution depends on diffraction effects caused by the optical instrument's aperture.
According to Rayleigh’s criterion, two objects are just resolved when the central maximum of one diffraction pattern coincides with the first minimum of the other.
For a single slit (linear objects), the limit of resolution:
$\frac {λ}{a}$
In a microscope, resolving power increases with numerical aperture (N.A. = n sin α) and decreases with wavelength:
$\frac {N.A.}{λ}$
For self-luminous point objects (microscope):
$\frac {0.61λ}{N.A.}$
For a telescope, angular resolution is:
$\frac {1.22λ}{D}$ where $D$ is the aperture diameter.
Resolving power improves when:
Wavelength is smaller
Aperture diameter is larger
The numerical aperture is higher

Key Points: Light as a Wave

Light is a transverse electromagnetic wave consisting of oscillating electric and magnetic fields perpendicular to each other and to the direction of propagation.
Light does not require a medium and travels in a vacuum at the speed of light
Refractive index $\frac {c}{v}$
Visible light has wavelengths from 400–700 nm; different wavelengths produce different colours and cause dispersion (spectrum formation).

Key Points: Reflection of Light at a Plane Surface

Reflection at a plane surface can be explained using Huygens’ principle and secondary wavelets.
The reflected wavefront is formed as the envelope of secondary wavelets produced at the reflecting surface.
The distance travelled by incident and reflected waves in the same time is equal .
The size of the image formed by a plane mirror is equal to the size of the object.
The image formed in a plane mirror shows lateral reversal (right and left are interchanged).

Key Points: Interference

Coherent sources emit waves of the same frequency with a constant phase difference.
In Young’s double slit experiment, two coherent sources are obtained from a single source.
Condition for constructive interference:
Path difference
Condition for destructive interference:
Path difference $\[\frac {1}{2}\])λ$
Position of bright fringe:
$\frac {nλD}{d}$
Fringe width:
$\frac {λD}{d}$ (Bright and dark fringes are equally spaced.)
For a clear and steady interference pattern:
Sources must be coherent, monochromatic, of nearly equal amplitude, and slits must be narrow with .

key Points: Nature of Light

Corpuscular theory (Newton): Light consists of particles called corpuscles that travel in straight lines; it explains reflection but fails to account for the correct speed in denser media.
Wave theory (Huygens): Light behaves as a wave and accounts for reflection, refraction, interference, diffraction, and polarisation.
Wave theory correctly states that the speed of light is lower in denser media, so light bends towards the normal.
Geometrical (ray) optics studies light as straight-line rays; wave optics explains light using its wave nature.
Dual nature of light: Light exhibits both particle nature (photons) and wave nature under different conditions.

Important Questions [180]

Electrostatic Potential and Capacitance Dual Nature of Radiation and Matter

Revision: Optics >> Wave Optics Physics Science (English Medium) Class 12 CBSE

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Definitions [20]

Formulae [3]

Theorems and Laws [8]

Statement

Proof

Conclusion

Key Points

Important Questions [180]

Concepts [28]

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