Determining the wavelength of monochromatic light is typically done using methods that involve diffraction or interference, such as passing the light through a diffraction grating or a double slit.
The wavelength of monochromatic light can be precisely determined using principles of diffraction, often involving a diffraction grating or a multi-slit setup. When light passes through such a structure, it creates an interference pattern of bright and dark fringes on a screen. The position of these fringes depends directly on the wavelength of the light, the spacing of the grating or slits, and the distance to the screen.
Understanding Diffraction
Diffraction occurs when light waves bend around obstacles or spread out after passing through narrow openings. With multiple equally spaced openings (like a diffraction grating), the diffracted waves interfere constructively at certain angles, producing bright spots or lines (maxima) on a screen.
The Formula Explained
The angles at which these bright maxima occur are governed by the equation:
$n\lambda = D \sin \theta$
Where:
- $n$ is the order of the bright maximum (a positive integer: 1 for the first maximum away from the center, 2 for the second, and so on). The central bright spot corresponds to $n=0$.
- $\lambda$ (Lambda) is the wavelength of the monochromatic light (what you want to find).
- $D$ is the spacing between the centers of adjacent slits or lines on the diffraction grating.
- $\theta$ (Theta) is the angle between the direction to the central maximum ($n=0$) and the direction to the $n$-th order maximum.
From this formula, we can rearrange to solve for the wavelength:
$\lambda = \frac{D \sin \theta}{n}$
As noted in the provided reference, when measuring the second order maximum (where $n$ equals two), the formula becomes $\lambda = \frac{D \sin \theta}{2}$.
Here is a breakdown of the formula components:
| Symbol | Meaning | Units (SI) |
|---|---|---|
| $\lambda$ | Wavelength of light | Metres (m) |
| $D$ | Slit/Grating spacing | Metres (m) |
| $\theta$ | Angle to the $n$-th order maximum | Radians or Degrees |
| $n$ | Order of the maximum | Dimensionless |
Steps for Measurement Using a Diffraction Grating
To experimentally determine the wavelength ($\lambda$) of a monochromatic light source (like a laser) using a diffraction grating, follow these steps:
- Set up the Apparatus: Position the monochromatic light source so the beam passes through the diffraction grating and shines onto a screen placed a measured distance away.
- Measure Grating Spacing ($D$): Obtain the value for the grating spacing ($D$) from the specifications of the diffraction grating used. This is usually given in lines per millimeter or lines per inch, which must be converted to spacing in meters (e.g., if it's 300 lines/mm, $D = \frac{1 \text{ mm}}{300} = \frac{1 \times 10^{-3} \text{ m}}{300}$).
- Identify and Measure Maxima: Locate the central bright maximum ($n=0$) and at least one higher-order maximum (e.g., $n=1$, $n=2$). Measure the distance ($x$) from the center of the central maximum to the center of the $n$-th order maximum on the screen.
- Measure Distance to Screen ($L$): Measure the perpendicular distance ($L$) from the diffraction grating to the screen.
- Calculate the Angle ($\theta$): Use trigonometry to find the angle $\theta$ for the $n$-th order maximum. Assuming the screen is perpendicular to the path of the central maximum, $\tan \theta = \frac{x}{L}$. Thus, $\theta = \arctan\left(\frac{x}{L}\right)$.
- Apply the Formula: Substitute the measured values for $D$, $\theta$, and the chosen order $n$ into the formula $\lambda = \frac{D \sin \theta}{n}$ and calculate the wavelength.
For greater accuracy, measure the distance between the $+n$ and $-n$ order maxima ($2x$) and use half of that distance for $x$. Also, measure multiple orders ($n=1, 2, 3, \dots$) and calculate $\lambda$ for each, then average the results.
Practical Considerations
- Using a laser provides a good source of monochromatic light.
- Ensure the screen is perpendicular to the laser beam path.
- Accurate measurement of distances ($x$ and $L$) and knowing the grating spacing ($D$) are crucial for a precise result.
- Higher-order maxima ($n=2, 3, \dots$) spread out more, potentially making angle measurements easier, but they are also less intense.
By following these steps and utilizing the diffraction formula, specifically $\lambda = \frac{D \sin \theta}{n}$, one can determine the wavelength of monochromatic light.
