The angle of diffraction in a plane diffraction grating is not a single fixed value. It represents the angle at which light of a specific wavelength is diffracted by the grating, relative to the path of the incident light. This angle is dependent on several factors, including the wavelength of the light, the spacing between the lines (slits) on the grating, and the order of the diffraction maximum being observed.
Understanding Diffraction Angles
When light passes through a diffraction grating, it interferes constructively at specific angles, creating bright spots or lines known as diffraction maxima. These maxima occur at angles where the path difference between waves from adjacent slits is an integer multiple of the wavelength.
The relationship between the angle of diffraction ($\theta$), the grating spacing ($d$), the wavelength ($\lambda$), and the order of the maximum ($m$) is given by the grating equation:
$d \sin(\theta) = m\lambda$
Where:
- $d$ is the distance between adjacent slits on the grating.
- $\theta$ is the angle of diffraction for the $m$-th order maximum.
- $m$ is the order of the maximum (an integer: 0 for the central maximum, 1 for the first order, 2 for the second order, etc.).
- $\lambda$ is the wavelength of the incident light.
From this equation, it's clear that the angle $\theta$ is not constant but varies based on $m$ and $\lambda$ for a given grating ($d$).
Example: A Specific Angle of Diffraction
While there isn't one universal angle of diffraction for all plane gratings, we can provide specific examples based on given conditions.
According to the provided information:
The angle of diffraction of the second order maximum of wavelength 5×10⁻⁵ cm is 30° in the case of a plane transmission grating.
This tells us that under these particular conditions:
- Wavelength ($\lambda$): 5 × 10⁻⁵ cm (which is 500 nm)
- Order of maximum ($m$): 2 (second order)
- Angle of Diffraction ($\theta$): 30°
This specific example demonstrates that an angle of diffraction of 30 degrees is possible for the second-order maximum of light with a wavelength of 5 × 10⁻⁵ cm when using a plane transmission grating. The grating spacing ($d$) can be calculated from the grating equation using this information:
$d \sin(30°) = 2 \times 5 \times 10^{-5} \text{ cm}$
$d \times 0.5 = 10 \times 10^{-5} \text{ cm}$
$d = 20 \times 10^{-5} \text{ cm} = 2 \times 10^{-4} \text{ cm}$
So, this specific angle of 30° occurs with a grating spacing of 2 × 10⁻⁴ cm (or 2000 nm), corresponding to 5000 lines per cm.
Key Factors Influencing the Angle
The angle of diffraction is determined by the interplay of:
- Wavelength ($\lambda$): Longer wavelengths diffract at larger angles for the same order.
- Order ($m$): Higher orders of maxima occur at larger angles.
- Grating Spacing ($d$): A smaller spacing between lines (a finer grating) causes light to diffract at larger angles.
Summary Table
| Factor | Symbol | Influence on Angle ($\theta$) (assuming others constant) | Example Value (from reference) |
|---|---|---|---|
| Wavelength | $\lambda$ | Increases $\theta$ | 5 × 10⁻⁵ cm |
| Order of Maximum | $m$ | Increases $\theta$ | 2 (second order) |
| Grating Spacing | $d$ | Decreases $\theta$ (finer grating increases angle) | Calculated as 2 × 10⁻⁴ cm |
| Angle of Diffraction | $\theta$ | Result of factors | 30° |
In conclusion, "the" angle of diffraction is not a single number; it's a variable that describes where diffracted light appears. However, under specific conditions, such as for the second-order maximum of light with a wavelength of 5 × 10⁻⁵ cm, the angle of diffraction can be 30 degrees as illustrated by the given example of a plane transmission grating.
