NEET Physics Wave Optics Notes

Doppler Effect in Light

The fractional change in frequency of light for an observer on earth is given by,

\(\frac{\Delta v}{v}=-\frac{V_{\text {radix }}}{C}\)

Where \(V_{\text {radinal }}\) is the component of the source velocity along the line joining the observer to the
source relative to the observer.

\(V_{\mathrm{radial}}\) is taken positive when the source moves away from the observer.

The above equation is valid if \(\mathrm{V}_{\text {radial }}<<\mathrm{C}\).

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Interference of Light

The phenomenon of redistribution of light energy in a medium as a result of the superposition of light waves from two coherent sources is called interference.

Theory of interference

If two waves \(\mathrm{y}_1=\mathrm{a}_1 \cos \omega \mathrm{t} \text { and } \mathrm{y}_2=\mathrm{a}_2 \cos (\omega \mathrm{t}-\phi)\), interfere, then,

The resultant amplitude is given by,

\(R=\sqrt{a_1^2+a_2^2+2 a_1 a_2 \cos \phi}\)

Case 1:

Constructive interference,

When , \(\cos \phi=1\)

\(\begin{aligned}
& R_{\max }=\sqrt{a_1^2+a_2^2+2 a_1 a_2} \\
& =\sqrt{\left(a_1+a_2\right)^2} \\
& \mathrm{R}_{\max }=\mathrm{a}_1+\mathrm{a}_2 \\
&
\end{aligned}\)

If a1 = a2 = a, then, Rmax = 2a.

Note:

Condition for constructive interference,

\(\cos \phi=1 \Rightarrow \phi=2 \mathrm{n} \pi, \mathrm{n}=0,1,2,3, \ldots \ldots\)

In terms of path difference,

\(\Delta=\mathrm{n} \lambda, \mathrm{n}=0,1,2,3, \ldots\)

Case 2:

Destructive interference,

When \(\cos \phi=-1\)

\(\begin{aligned}
R_{\min } & =\sqrt{a_1^2+a_2^2-2 a_1 a_2} \\
& =\sqrt{\left(a_1-a_2\right)^2} \\
R_{\min } & =a_1-a_2
\end{aligned}\)

If a1 = a2 then, Rmin = 0

Note:

Condition for destructive interference,

\(\begin{aligned}
&\cos \phi=-1\\
&\phi=(2 n-1) \pi, \mathrm{n}=1,2,3, \ldots \ldots
\end{aligned}\)

In terms of path difference,

\(\Delta=(2 \mathrm{n}-1) \frac{\lambda}{2}, \mathrm{n}=1,2,3, \ldots .\)

Young’s Double Slit Experiment

The position of bright fringes is given by,

\(\mathrm{x}_{\mathrm{n}}=\frac{\mathrm{n} \lambda \mathrm{D}}{\mathrm{d}}\) n=0,1,2…..

The position of dark fringes is given by,

\(\mathrm{x}=(2 \mathrm{n}-1) \frac{\lambda \mathrm{D}}{\mathrm{d}}\)

Expression of fringe width, \(\beta=\frac{\lambda D}{d}\)

Note:

w.k.t., \(\Delta=\frac{\lambda}{2 \pi} \phi\)
Intensity ∝ (amplitude)²
If two waves \(\), superpose each other, then the resultant intensity is given by,

\(\mathrm{y}_1=\mathrm{a} \cos (\omega \mathrm{t}) \text { and } \mathrm{y}_2=\mathrm{a} \cos (\omega \mathrm{t}+\phi)\)

In the case of coherent addition, the maximum intensity is given by,

\(I=4 I_0 \cos ^2\left(\frac{\phi}{2}\right)\)

I_max = 4I0

In case of incoherent addition, I_max = 2I0

Diffraction of Light

The phenomenon of bending of light around the corners of obstacles or apertures is called diffraction of light.

In Fresnel’s diffraction, we use spherical or cylindrical wavefront.

In Fraunhoffer’s diffraction, we use plane wavefront.

Path difference in single slit diffraction is given by,

\(\text { a } \sin \theta=\lambda\)

Where ‘a’ is the slit width,

Condition for minima, \(\text { a } \sin \theta=\mathrm{n} \lambda, \mathrm{n}=1,2,3, \ldots \ldots\)

Condition for secondary maxima,

\(\mathrm{a} \sin \theta=(2 \mathrm{n}+1) \frac{\lambda}{2} \quad \mathrm{n}=1,2,3, \ldots \ldots\)

Fresnel Distance

This is the distance up to which ray optics is valid.

\(Z_F=\frac{a^2}{\lambda}\)

Limit of Resolution of a Telescope

It is the smallest angle subtended at the center of the objective of the telescope by two distinct objects whose images are just resolved.

\(\Delta \theta=\frac{1.22 \lambda}{\mathrm{a}}\)

The resolving power of a telescope is the reciprocal of the limit of resolution.

i.e., Resolving power of telescope, \(\frac{1}{\Delta \theta}=\frac{D}{1.22 \lambda}\)

where, ‘D’ is the diameter of the objective.

Limit of resolution of a microscope

It is the minimum distance between two point objects whose images appear just resolved.

\(\mathrm{d}_{\min }=\frac{1.22 \lambda}{2 \mathrm{n} \sin \beta}\)

The product is called the numerical aperture and is the semi-vertical angle.

Resolving Power of a Microscope

It is the reciprocal of the minimum distance between two-point objects whose images appear just resolved.

Resolving power of a microscope, \(\frac{1}{\mathrm{~d}_{\min }}=\frac{2 \mathrm{n} \sin \beta}{1.22 \lambda}\)

Malus’ Law

When a beam of completely plane polarized light is passed through the analyzer, the intensity ‘I’ of the transmitted light varies directly as the square of the cosine of the angle θ between the transmission direction of the polarizer and analyzer.

\(i.e., I \propto \cos ^2 \theta
\mathrm{I}=\mathrm{I}_0 \cos ^2 \theta
\)

Where I₀ is the maximum intensity of transmitted light.

Brewster’s law states that tangent of an angle of polarization is equal to the refractive index of the medium.

\(\text { i.e., } \mathrm{n}=\tan \theta_{\mathrm{p}}\)

Where θ_p is the polarizing angle.

Note:

For a particular angle of incidence, the reflected ray is completely polarized. This angle is called the Brewster angle or polarizing angle (θ_p)
At polarizing angle, \(\theta_{\mathrm{p}}+\mathrm{r}=90^{\circ}\)

NEET Physics Wave Optics Notes