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# Lecture 24: Interference ## Interference ### Superposition of waves - When 2 or more waves overlap in the same region of space, the displacement of the resulting wave is the sum of the individual displacements of the waves. - The waves can be EM waves or any other kind of wave. - If the waves a...

# Lecture 24: Interference ## Interference ### Superposition of waves - When 2 or more waves overlap in the same region of space, the displacement of the resulting wave is the sum of the individual displacements of the waves. - The waves can be EM waves or any other kind of wave. - If the waves are in phase, they interfere constructively. - If the waves are out of phase, they interfere destructively. - Constructive interference: the amplitude of the resulting wave is greater than that of either individual wave. - Destructive interference: the amplitude of the resulting wave is less than that of either individual wave. ### Two-source interference - Consider two sources of waves that are in phase and have the same wavelength $\lambda$. - The waves interfere constructively if the path difference between the two waves is an integer multiple of the wavelength: $\qquad$ $\Delta r = n\lambda, \quad n = 0, 1, 2,... \quad$ (constructive interference) - The waves interfere destructively if the path difference between the two waves is a half-integer multiple of the wavelength: $\qquad$ $\Delta r = (n + \frac{1}{2})\lambda, \quad n = 0, 1, 2,... \quad$ (destructive interference) ### Young's double-slit experiment - In Young's double-slit experiment, light is shone through two narrow slits that are separated by a distance $d$. - The light from the two slits interferes, producing a pattern of bright and dark fringes on a screen that is a distance $L$ away. - The angle $\theta$ to the $m$th bright fringe is given by $\qquad$ $d \sin \theta_m = m \lambda, \quad m = 0, 1, 2,...$ - If $\theta$ is small, then $\sin \theta \approx \theta$, and $\qquad$ $\theta_m \approx \frac{m \lambda}{d}$ - The distance $y_m$ from the center of the screen to the $m$th bright fringe is $\qquad$ $y_m = L \tan \theta_m \approx L \theta_m \approx \frac{m \lambda L}{d}$ ### Intensity in double-slit interference - The intensity at any point on the screen is $\qquad$ $I = I_{max} \cos^2 (\frac{\pi d \sin \theta}{\lambda})$ - The average intensity over one cycle is $\qquad$ $ = \frac{1}{2} I_{max}$ ### Interference in thin films - A thin film is a layer of material with a thickness of the order of the wavelength of light. - When light is incident on a thin film, it is partially reflected from the top surface and partially reflected from the bottom surface. - The two reflected waves interfere. - The interference can be constructive or destructive, depending on the thickness of the film, the angle of incidence, and the indices of refraction of the film and the surrounding media. - If the film has index of refraction $n_2$ and is surrounded by media with indices of refraction $n_1$ and $n_3$, then: $\qquad$ $2t = m \lambda / n_2$ (constructive if one phase change) $\qquad$ $2t = (m + \frac{1}{2}) \lambda / n_2$ (destructive if one phase change) - The phase change occurs when light reflects from a medium with a higher index of refraction. ### Problem solving strategy for thin-film interference 1. **Identify the thin film**: Determine the material that constitutes the thin film and its index of refraction ($n_2$). 2. **Identify the surrounding media**: Determine the materials surrounding the thin film and their indices of refraction ($n_1$ and $n_3$). 3. **Determine if there are any phase changes**: A phase change of $\pi$ radians (or 180 degrees) occurs when light reflects from a medium with a higher index of refraction. 4. **Apply the conditions for constructive and destructive interference**: Use the appropriate equations, considering the number of phase changes: - Constructive interference: $2t = m \lambda / n_2$ (if one phase change) - Destructive interference: $2t = (m + \frac{1}{2}) \lambda / n_2$ (if one phase change) 5. **Solve for the desired quantity**: Use the given information and the equations to solve for the unknown variable, such as the thickness of the film or the wavelength of light.