Class 12 Physics Wave Optics Formulas

Wavefront: It is a surface as a set of points with a constant (same) phase.

Amplitude and Wavefront: 

• Spherical Wavefront: A ∝ 1/ r
• Cylindrical Wavefront: A ∝ 1/ √r

Basic Concepts

• Coherent sources: They have the same frequency with a constant phase difference.
• Resultant Intensity when two light waves interfere:

$$I_R=I_1+I_2+2\sqrt{I_1{I}_2}\cos \mathrm{\Delta }\varphi$$

• If amplitudes are $A_1$and $A_2$​

$$A = \sqrt{A_1^2 + A_2^2 + 2A_1A_2\cos \Delta \phi}$$

• Relation between Intensity and Amplitude:

$$I\propto A^2$$

Constructive Interference

• Path Difference between two waves:                                             $\Delta x = n\lambda$
• Amplitude ( $\Delta \phi = 0$):

$$A=A_1+A_2$$

• Intensity ( $\Delta \phi = 0$):

$$I_R=(\sqrt{I_1}+\sqrt{I_2}{)}^2$$

Destructive Interference

• Path Difference

$$\begin{array}{cccc}& \mathrm{\Delta }x=(n+\frac{1}{2})\lambda & & \text{<br>}\end{array}$$

• Amplitude ( Δϕ= π):

$$A=A_1-A_2$$

• Intensity ( Δϕ=π):

$$I_R=(\sqrt{I_1}-\sqrt{I_2}{)}^2$$

• Fringe Width:

• Angular Fringe Width

$$\mathrm{\Delta }\theta =\frac{\lambda }{d}$$

• Position of the nth bright fringes:

$$\begin{array}{cccc}& x_n=\frac{n\lambda D}{d}& & \text{<br>}\end{array}$$

• Position of Dark Fringes:

$$\begin{array}{cccc}& x_n=\frac{(n+\frac{1}{2})\lambda D}{d}& & \text{<br>}\end{array}$$

• Path Difference

$$\mathrm{\Delta }\mathrm{x\; \approx \; dsin\; \theta \; =\; n\lambda }=\frac{yD}{d}$$

Resultant Intensity for two same amplitude waves:

$$I=I_0{\cos }^2\left(\frac{\pi dy}{\lambda D}\right)$$

• Angular Width of Central Maxima

$$\theta =\frac{\lambda }{a}$$

•  Width of Central Maxima

$$\text{β}=2y_1=\frac{2\lambda D}{a}$$

• Intensity Distribution at an angle $\theta \; :$

$$I(\theta )=I_0{\left(\frac{\sin \beta }{\beta }\right)}^2$$

• Diffraction Equation for Grating

$$d\sin \theta =n\lambda$$

• Displacement in z-direction {(z-polarized wave) travelling along the x or y axis.} $$z(x, t) = a \sin(kx - \omega t)$$
• Malus’s Law : Intensity after passing through two polaroids

$$I = I_0 \cos^2\theta$$

• Intensity through Three Polaroids when middle polaroid at angle θ: $$I=\frac{I_0}{4}{\sin }^2(2\theta )$$