Physical Optics Cheatsheet

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Physical Optics Principle of Superposition (উপস্থাপন নীতি) When two or more waves overlap at a point, the resultant displacement at that point is the vector sum of the individual displacements due to each wave. If $y_1$ and $y_2$ are the displacements due to two individual waves, then the resultant displacement $y$ is: $$y = y_1 + y_2$$ S₁ S₂ $y_1$ $y_2$ Phenomena Based on Superposition (উপস্থাপন নীতির উপর ভিত্তি করে দুটি ঘটনা) Interference (ব্যতিচার): Occurs when two coherent waves superpose, leading to a redistribution of energy, resulting in bright and dark fringes. Beats (বীট): Occurs when two waves of slightly different frequencies superpose, resulting in a periodic variation in sound intensity (loudness). Stationary waves (or Standing waves) and YDSE (Young's Double Slit Experiment) are also examples of interference phenomena. Coherent Source (সুসংগত উৎস) Two sources are said to be coherent if they emit light waves of the same frequency, constant phase difference, and same amplitude. The phase difference between the waves from two coherent sources remains constant over time. A single source is often used to create two coherent sources (e.g., in Young's Double Slit Experiment). Resultant Amplitude (লব্ধি বিস্তার) Consider two waves from two coherent sources $S_1$ and $S_2$ arriving at a point. Let their displacements be: $y_1 = A_1 \sin(kx - \omega t)$ $y_2 = A_2 \sin(kx - \omega t + \phi)$ According to the principle of superposition, the resultant displacement $y$ is: $$y = y_1 + y_2 = A_1 \sin(kx - \omega t) + A_2 \sin(kx - \omega t + \phi)$$ Expanding $A_2 \sin(kx - \omega t + \phi)$: $$A_2 \sin(kx - \omega t)\cos\phi + A_2 \cos(kx - \omega t)\sin\phi$$ So, $y = (A_1 + A_2 \cos\phi) \sin(kx - \omega t) + (A_2 \sin\phi) \cos(kx - \omega t)$ Let $A_1 + A_2 \cos\phi = A \cos\theta$ (1) And $A_2 \sin\phi = A \sin\theta$ (2) Then, $y = A \cos\theta \sin(kx - \omega t) + A \sin\theta \cos(kx - \omega t)$ Using the identity $\sin(A+B) = \sin A \cos B + \cos A \sin B$, we get: $$y = A \sin(kx - \omega t + \theta)$$ Where $A$ is the resultant amplitude (লব্ধি বিস্তার) and $\theta$ is the phase difference (দশা পার্থক্য) with respect to the first wave. To find the resultant amplitude $A$, square equations (1) and (2) and add them: $$(A \cos\theta)^2 + (A \sin\theta)^2 = (A_1 + A_2 \cos\phi)^2 + (A_2 \sin\phi)^2$$ $$A^2 (\cos^2\theta + \sin^2\theta) = A_1^2 + 2A_1 A_2 \cos\phi + A_2^2 \cos^2\phi + A_2^2 \sin^2\phi$$ $$A^2 = A_1^2 + A_2^2 + 2A_1 A_2 \cos\phi$$ Thus, the resultant amplitude $A$ is: $$A = \sqrt{A_1^2 + A_2^2 + 2A_1 A_2 \cos\phi}$$ To find the phase difference $\theta$, divide equation (2) by equation (1): $$\frac{A \sin\theta}{A \cos\theta} = \frac{A_2 \sin\phi}{A_1 + A_2 \cos\phi}$$ $$\tan\theta = \frac{A_2 \sin\phi}{A_1 + A_2 \cos\phi}$$ $A$ $A_2$ $A_1$ $\theta$ $\phi$ Resultant Intensity (লব্ধি তীব্রতা) The intensity $I$ of a wave is proportional to the square of its amplitude $A$. $$I = \frac{1}{2} \rho \omega^2 A^2 v$$ Where $\rho$ is density, $\omega$ is angular frequency, and $v$ is wave velocity. For a given medium and frequency, $\frac{1}{2} \rho \omega^2 v$ is constant. So, $I \propto A^2$ or $I = KA^2$, where $K$ is a constant. Therefore, for the individual waves: $I_1 = KA_1^2 \implies A_1 = \sqrt{I_1/K}$ $I_2 = KA_2^2 \implies A_2 = \sqrt{I_2/K}$ Substitute $A_1$ and $A_2$ into the resultant amplitude equation: $$A^2 = \frac{I_1}{K} + \frac{I_2}{K} + 2\sqrt{\frac{I_1}{K}}\sqrt{\frac{I_2}{K}} \cos\phi$$ Multiply by $K$ to get the resultant intensity $I = KA^2$: $$KA^2 = I_1 + I_2 + 2\sqrt{I_1 I_2} \cos\phi$$ Thus, the resultant intensity $I$ is: $$I = I_1 + I_2 + 2\sqrt{I_1 I_2} \cos\phi$$