The wavelength of light can be precisely measured using a plane diffraction grating by observing the pattern of light produced and applying the fundamental diffraction grating equation.
A plane diffraction grating is a device containing a large number of equally spaced parallel lines or slits ruled on a flat surface, typically glass or plastic. When light passes through these slits, it diffracts and interferes, creating a pattern of bright spots (maxima) on a screen placed some distance away. The angle at which these bright spots appear depends on the wavelength of the light and the spacing between the slits on the grating.
The Diffraction Grating Equation
The relationship between the wavelength of light and the diffraction pattern is given by the equation:
d sin θ = nλ
Where:
- d is the distance between the slits (the grating element). To find d read the information on the packaging, it will say how many lines per mm. For example, if it says 500 lines/mm, then d = 1 mm / 500 lines = 0.002 mm = 2 x 10⁻⁶ meters.
- θ is the angle to the normal made by the maximum. This is the angle between the original direction of the light and the direction of the bright spot on the screen.
- n is the order of the maximum. The central bright spot (directly in line with the light source) is the 0th order (n=0). The first bright spots on either side are the 1st order (n=1), the next are the 2nd order (n=2), and so on.
- λ is the wavelength of the light being measured.
By rearranging this equation, we can solve for the wavelength: λ = (d sin θ) / n.
Step-by-Step Measurement Process
Measuring the wavelength of light using a diffraction grating involves setting up the experiment and taking specific measurements:
- Set up the Apparatus: Place the light source (e.g., a laser or a spectral lamp) in front of the diffraction grating. Position a screen some distance behind the grating, perpendicular to the path of the light.
- Observe the Pattern: When the light passes through the grating, you will see a series of bright spots (diffraction maxima) on the screen. The central spot is the brightest (n=0), with dimmer spots appearing symmetrically on either side.
- Determine the Grating Element (d): As mentioned in the reference, find the number of lines per millimeter (or inch) specified on the grating's packaging. Convert this value to the distance per line in meters to get 'd'.
- Example: If the grating has 600 lines/mm,
d = 1 mm / 600 = (1 x 10⁻³ m) / 600 ≈ 1.67 x 10⁻⁶ meters.
- Example: If the grating has 600 lines/mm,
- Identify the Order (n): Choose which bright spot you will measure. The first bright spot away from the center is n=1, the second is n=2, and so on. It's often best to measure a higher order (like n=2 or n=3) for better accuracy.
- Measure Distances on the Screen:
- Measure the distance between the grating and the screen. Let's call this distance 'L'.
- Measure the distance from the central bright spot (n=0) to the chosen bright spot of order 'n'. Let's call this distance 'y'. It's often more accurate to measure the distance between the +n and -n spots and divide by two.
- Calculate the Angle (θ): The angle θ can be found using trigonometry from the measured distances L and y. Consider the right triangle formed by the grating, the center spot, and the chosen bright spot of order n.
tan θ = opposite / adjacent = y / L- Calculate θ by taking the inverse tangent:
θ = arctan(y / L).
- Substitute Values into the Equation: Plug the values for d, n, and the calculated θ into the formula
λ = (d sin θ) / nand calculate the wavelength λ.
Example Calculation
Let's say:
- Grating has 500 lines/mm, so
d = (1 x 10⁻³ m) / 500 = 2 x 10⁻⁶ m. - You measure the first-order maximum (n=1).
- The distance from the grating to the screen (L) is 0.5 meters.
- The distance from the central spot to the first-order spot (y) is 0.2 meters.
- Calculate θ:
tan θ = y / L = 0.2 m / 0.5 m = 0.4
θ = arctan(0.4) ≈ 21.8° - Calculate λ:
λ = (d sin θ) / n
λ = (2 x 10⁻⁶ m * sin(21.8°)) / 1
λ ≈ (2 x 10⁻⁶ m * 0.371) / 1
λ ≈ 0.742 x 10⁻⁶ m = 742 nanometers (nm)
This result would suggest the light source is emitting red light.
Required Equipment
To perform this measurement, you will need:
- A coherent light source (like a laser pointer or a spectral lamp)
- A plane diffraction grating with a known number of lines per unit length
- A screen or a wall to observe the pattern
- A meter stick or measuring tape to measure distances L and y
- A calculator with trigonometric functions
By carefully performing these steps and measurements, the wavelength of the light can be accurately determined using the diffraction grating equation.
