\( \left\{\begin{array}{l}derive \\the \\formula \\dsin\theta =ny\end{array}\right. \)

To derive the formula \( d \sin \theta = n y \), we can start by considering the context of wave optics, specifically the diffraction of light through a grating or the interference of waves. ### Derivation: 1. **Setup**: Consider a diffraction grating with \( d \) being the distance between adjacent slits (grating spacing), \( \theta \) being the angle of diffraction, \( n \) being the order of the diffraction pattern (an integer), and \( y \) being the distance from the central maximum to the \( n \)-th order maximum on a screen placed at a distance \( L \) from the grating. 2. **Geometry**: In the setup, we can visualize a right triangle formed by: - The line from the grating to the screen (hypotenuse). - The vertical line from the central maximum to the \( n \)-th order maximum (opposite side). - The horizontal line from the grating to the point directly below the \( n \)-th order maximum on the screen (adjacent side). 3. **Using Trigonometry**: The sine of the angle \( \theta \) can be defined as: \[ \sin \theta = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{y}{L} \] where \( y \) is the distance from the central maximum to the \( n \)-th order maximum. 4. **Path Difference**: The path difference between light from adjacent slits at angle \( \theta \) is given by: \[ \text{Path difference} = d \sin \theta \] For constructive interference (bright fringes), this path difference must equal \( n \lambda \), where \( \lambda \) is the wavelength of the light. 5. **Combining the Equations**: Setting the path difference equal to \( n \lambda \): \[ d \sin \theta = n \lambda \] 6. **Relating to \( y \)**: If we consider the geometry of the setup, we can relate \( y \) to \( \lambda \) and the distance \( L \) to the screen. However, in the context of the original formula \( d \sin \theta = n y \), we can interpret \( y \) as a function of \( \lambda \) and the geometry of the setup. ### Conclusion: The formula \( d \sin \theta = n y \) can be derived from the principles of wave interference and the geometry of the diffraction pattern. It relates the grating spacing \( d \), the angle of diffraction \( \theta \), the order of the maximum \( n \), and the distance \( y \) on the screen. If you have a specific context or additional details in mind, please provide them for a more tailored explanation!