A diffraction grating has slit spacing d. Fringes are viewed on a screen at distance L. Find an expression for the wavelength of light that produces a first-order fringe on the viewing screen at distance L from the center of the screen.

Verified step by step guidance

1. Start by recalling the diffraction grating equation: $n\lambda \; =\; d\; sin\; \theta$, where $n$ is the order of the fringe, $\lambda$ is the wavelength of light, $d$ is the slit spacing, and $\theta$ is the angle of diffraction.
2. For the first-order fringe, set $n\; =\; 1$. The equation simplifies to $\lambda \; =\; d\; sin\; \theta$.
3. Next, relate the angle $\theta$ to the geometry of the setup. The tangent of the angle is given by $tan\; \theta \; =\; y\; /\; L$, where $y$ is the distance from the center of the screen to the first-order fringe, and $L$ is the distance from the grating to the screen.
4. For small angles, $sin\; \theta \; \approx \; tan\; \theta$. Substitute $tan\; \theta$ into the diffraction equation: $\lambda \; =\; d\; (y\; /\; L)$.
5. The final expression for the wavelength of light producing the first-order fringe is $\lambda \; =\; (d\; y)\; /\; L$. This relates the wavelength to the slit spacing, the fringe position, and the screen distance.

Key Concepts

Diffraction Grating

A diffraction grating is an optical component with a periodic structure that splits and diffracts light into several beams. The spacing between the slits, denoted as 'd', determines the angles at which constructive interference occurs, leading to the formation of bright fringes on a screen. The grating's ability to separate light into its component wavelengths is crucial for applications in spectroscopy.

Constructive Interference

Constructive interference occurs when two or more waves combine to produce a wave of greater amplitude. In the context of diffraction, this happens when the path difference between light waves from adjacent slits is an integer multiple of the wavelength. For the first-order fringe, this condition is expressed mathematically, allowing us to derive the wavelength of light based on the observed fringe position.

Wavelength and Fringe Position

The wavelength of light is the distance between successive peaks of the wave and is a key factor in determining the position of the diffraction fringes. The relationship between the wavelength, slit spacing, and the angle of diffraction is given by the grating equation, which can be rearranged to find the wavelength based on the distance to the screen and the fringe order. Understanding this relationship is essential for solving problems involving diffraction patterns.