Here students can locate TS Inter 2nd Year Physics Notes 3rd Lesson Wave Optics to prepare for their exam.

## TS Inter 2nd Year Physics Notes 3rd Lesson Wave Optics

→ Huygens principle: Each point on the wave-front is the source of a secondary disturbance and the wavelets emanating from these points spread out in all directions with the speed of the wave. These wave lets emanating from the wavefront are usually referred to as secondary wavelets.

From Huygens principle every wave is a secondary wave to the preceding wave.

→ Wavefront: The locus of points which oscillate in phase is called “wavefront”.

(OR)

A wavefront is defined as a surface of constant phase.

→ Plane wave: Generally wavefront is spherical in nature when radius of sphere is very large. A small portion of spherical wave can be treated as plane wave.

→ Geometrical optics: It is a branch of optics in which we are completely neglects the finiteness of the wavelength is called geometrical optics. A ray of light is defined as the path of energy propagation. In this concept wavelength of light is tending to zero.

→ Snell’s law of refraction: Let nj and n2 are the refractive indices of the two media and i’ and ‘r’ are angle of incidence and angle of refraction then

n_{1} sin i = n_{2} sin r. This relation is called Snell’s law.

From Snell’s law for a given pair of media sin i / sin r = n_{2} / n_{1} is constant also called refractive index of the medium. Where nj is air or vacuum.

→ Critical angle (I_{c}): It is define as the angle of incidence in denser medium i for which angle of refraction in rarer medium r = 90°.

μ = \(\frac{1}{\sin c}\)

i. e, when r = 90° then i = C in denser medium.

→ Doppler’s effect in light: When there is relative motion between source and obser¬ver then there is a change in frequency of light received by the observer.

→ Red shift: If the source moves away from the, observer then frequency measured by observer is less (i.e., wavelength increases) as a result wavelength of received light moves towards red colour. This is known as “red shift”.

→ Blue shift: When source of light is approach¬ing the observer frequency of light received increases, (i.e., wavelength of light decreases.) As a result wavelength of received light will move towards blue colour. This is known as “blue shift”.

→ Superposition principle: According to super¬position principle at a particular point in the medium the resultant displacement produced by a number of waves is the vector sum of the displacements produced by each of the waves.

→ Coherent soures: Two sources are said to be coherent when the phase difference pro¬duced by each of two waves does not change with time.

Note: For a non – coherent waves the phase difference between them changes with time.

→ Interference: Interference is based on the superposition principle. According to which at a particular point in the medium the resultant displacement produced by a number of waves is the vecotor sum of displacements produced by each of the wave.

Note: In light when two coherent waves are superposed we will get dark and bright bands.

→ Constructive interference Or bright band:

When two coherent waves of path difference λ /2 or its integral multiples or a phase difference of or integral multiples of 2π are superposed on one another then the displacements of the two waves are in phase and intensity of light is 4I_{0} where I_{0} is intensity of each wave.

Condition, for constructive interference

is path difference = nλ or Φ = 0, 2π, 4π ………….. etc i.e., even multiples of π.

→ Destructive interference or dark band:

When two coherent waves of path difference \(\frac{\lambda}{2}\) or (n + \(\frac{1}{2}\)) λ or phase difference of \(\frac{\pi}{2}\) or odd multiples of \(\frac{\pi}{2}\) superposed at a given point then their displacements are out of phase and resultant intensity its is zero. This is called dark band.

For dark band to from path difference

= (n + \(\frac{1}{2}\)) λ (OR)

phase difference Φ = π, 3π, 5π …. odd multiples of n.

→ Diffraction: Bending of light rays at sharp edges (say edge of blade) is called”diffraction”.

As a result of diffraction we can see dark and bright bands close to the region of geometrical shadow.

→ Resolving power of Telescope:

Resolving power of telescope Δθ = \(\frac{0.61 \lambda}{\mathrm{a}}\)

Where 2a is aperture or vertical height (dia-meter) of lens used:

Resolving power of telescopes is its ability to show two distant object clearly when angular separation between them is Δθ.

When Δθ is less then resolving power of that telescope is high.

From above equation to increase resolving power of telescope its aperture or diameter of lens used must be high.

→ Resolving power of microscope:

The resolving power of the microscope is given by the reciprocal of the minimum separation of two points seen as distinct.

Minimum distance d_{min} = \(\frac{1.22 \lambda}{2 n \sin \beta}\), The term n sin β is called numerical aperture.

Resolving power of microscope = \(\frac{2 \mathrm{n} \sin \beta}{1.22 \lambda}\)

Note: The resolving power of microscope increases if refractive index n is high. In oil immersion objectives the lenses are placed in a transparent oil with refractive index close to that of objective lens to increase magnification.

→ Fresnel distance: The term z = a^{2}/ λ is called fresnel distance.

In explaining the spreading of beam due to diffraction we will use the equation z = a^{2}/ λ,

Where after travelling a distance z\(\frac{\lambda}{\mathrm{a}}\) size of beam is comparable to size of slit or hole ‘a’.

Note: When we are travelling from aperture to screen a distance z then width of diffrated beam z\(\frac{\lambda}{\mathrm{a}}\) is equals to aperture a’.

Beyond this distance ‘z’ divergence of the beam of width ‘a’ becomes significant. When distances are smaller than z spreading due to diffraction is small when compared with size of beam.

→ Polarisation: If is a process in which vib-rations of electric vectors of light are made ot oscillate, in a single direction.

→ Polaroids: A polaroid consists of a long chain of molecules aligned in a particular direction.

→ Malus’ Law: Let two polariods say P_{1} and P_{2} are arranged with some angle ‘θ’ between their axes. Then intensity of light coming

I = I_{0} cos^{2}θ

where I_{0} is intensity of polarised light after passing through 1st polaroid P_{1}. This is known as Malus Law.

→ Uses of Polaroids: Polaroids are used

- to control intensity of light.
- They are used in photography,
- polaroids are used in sunglasses and in window panes.

→ Unpolarised light: In an unpolarised light electric vectors can vibrate in 360° direc¬tion perpendicular to direction of propaga¬tion. All these electric vectors can be grouped into two groups.

- Dot components they are vibrating perpendicular to the plane of the paper.
- Arrow components ‘↔’ i.e., their plane of vibration is along the plane of the paper.

Thus an unploarised light can be shown as a group of dot components and arrow components.

→ Polorisation by scattering: The sky appears blue due to scattering of light. The light coming from clear blue portion of sky is made to pass through a polariser when it is rotated the intensity of light coming from polariser is found to be changing. Which shows that the scattered light consists of polarised light.

→ Polarisation by reflection:

- When unpolarised light falls on the boundary layer sepa-rating two transparent media the reflected light is found to be partially polarised. The amount of polarisation depends on angle of incidence i.
- It is found that when reflected ray and refracted ray are perpendicular the reflected ray is found to be totally plane polarised. The angle of incidence at this stage is known as Brewster angle.

→ Brewster’s angle: When reflected ray and refracted ray are mutually perpendicular then reflected ray is plane polarised. This particular angle of incidence i_{B} for which the reflected ray is plane polarised is called Brewster angle.

Explanation:

At Brewster angle i_{B} + r = \(\frac{\sin \mathrm{i}_{\mathrm{B}}}{\sin \mathrm{r}}=\frac{\sin \mathrm{i}_{\mathrm{B}}}{\sin \left(\pi / 2-\mathrm{i}_{\mathrm{B}}\right)}\)

n (OR) μ = \(\frac{\sin \mathrm{i}_{\mathrm{B}}}{\sin \mathrm{r}}=\frac{\sin \mathrm{i}_{\mathrm{B}}}{\sin \left(\pi / 2-\mathrm{i}_{\mathrm{B}}\right)}\)

= \(\frac{\sin \mathrm{i}_{\mathrm{B}}}{\cos \mathrm{i}_{\mathrm{B}}}\) = tani_{B} (OR) μ = tan i_{B}

∴ The tangent of Brewster’s angle tan (i_{g}) is equals to refractive index, i.e., μ = tan i_{B}.

Note: Refractive index can be shown by the symbol μ or n.

→ From the super position principle the resultant displacement is y = y_{1} + y_{2}.

For constructive interference (bright band) y = y_{1} + y_{2}) ; Intensity I = (y_{1} + y_{2})^{2}

For destructive interference (dark band) y = y_{1} ~ y_{2}; Intensity I = (y_{1} – y_{2})^{2}

→ In interference, the resultant intensity

I = 4I_{0} cos^{2}\(\frac{\phi}{2}\) (Where I_{0} is maximum intensity)

Resultant phase θ = \(\frac{\phi}{2}\) (Where Φ is initial phase difference)

→ Condition for formation of bright band is

(a) Path difference x = mλ, where m = 0,1, 2 ………….. etc.

(b) Phase difference Φ = 0, 2π ………… even multiples of π.

→ Condition for formation of dark band

(a) Path difference x = \(\frac{λ}{2}\) and odd multiples

(b) Phase difference Φ = π, 3π, 5π, ………….. odd multiples of n.

→ Relation between path difference (x) and

phase difference (Φ) is λ = \(\frac{\lambda}{2 \pi}\). Φ

→ Fringe with β = \(\frac{\lambda L}{\mathrm{~d}}\); Angular fringe width \(\frac{\beta}{L}=\frac{\lambda}{d}\)

→ Distance of mth bright band from central bright band is x_{2} = \(\frac{\mathrm{m} \lambda \mathrm{L}}{\mathrm{d}}\)

→ Distance of m dark band from central dark band x_{2} = \(\left(m+\frac{1}{2}\right) \lambda \frac{L}{d}\)

→ For two waves with intensities I and 12 with phase 4 resultant intensity

I = I_{1} + I_{2} + 2\(\sqrt{I_1 I_2}\)cos Φ

→ When a glass plate of thickness (t) is introduced in the path of one light wave then interference fringes will shift. Thickness glass plate t = \(\frac{m \lambda}{(\mu-1)}\)

m = number of fringes shifted;

λ = wavelength of light used.

→ When unpolarized light of intensity I, passes through a polarizer Intensity of emergent light I = \(\frac{\mathrm{I}_0}{2}\)

→ When polarized light falls on a polarizer with an angle θ to the axis then Intensity of refracted light I = I_{0}cos^{2}θ

→ If polarized light falls on 1st polarIzer with an angle θ, and angle between the axes of given two polarizers is θ then intensity of light coming out of 2nd polarlzer I = I_{0}cos^{2}θ_{1}cos^{2}θ_{2}

→ For polarizatIon through reflection wIth Brewster angle i_{B} then μ or n = tan i_{B}

→ In diffraction radIus of central bright region

r_{0} = \(\frac{1.22 \lambda \mathrm{f}}{2 \mathrm{a}}=\frac{0.61 \lambda \mathrm{f}}{\mathrm{a}}\)

→ ResolvIng power of telescope Δθ = \(\frac{0.61 \lambda}{\mathrm{a}}\)

Where Δθ Is the minimum angular separation between two distant objects.

→ Resolving power of microscope

d_{min} = \(=\frac{1.22 \mathrm{f} \lambda}{\mathrm{D}}=\frac{1.22 \lambda}{2 \tan \beta}=\frac{1.22 \lambda}{2 \sin \beta}=\frac{1.22 \lambda}{2 \mathrm{n} \sin \beta}=\frac{1.22 \lambda}{2 \mathrm{~N} \cdot \mathrm{A}}\)

Where d_{min} is the minimum separation required between two points in object.

N.A is numerical aperture (n sin β)

β is the angle subtended by the object at object lens.

→ Fresnel distance z = \(\frac{\mathrm{a}^2}{\lambda}\) Where ‘a’ is size of hole or slit.