Light of wavelength 600 nm is incident normally on a diffraction grating 3.00 cm wide. If second order line is observed at an angle of 30°, what is the total number of lines on the grating?
12500
Diffraction Grating Fundamentals
A diffraction grating is an optical component with a periodic structure that diffracts light into several beams traveling in different directions. The directions of these beams depend on the spacing of the grating and the wavelength of the incident light. It's a crucial device in spectroscopy, used to separate light of different wavelengths based on their wavelengths.
Grating Equation Application
To determine the total number of lines on a diffraction grating, we first need to find the grating spacing 'd'. The fundamental principle governing diffraction gratings is described by the grating equation, which relates the angle of diffraction, the order of the spectrum, the wavelength of light, and the grating spacing.
The formula for a diffraction grating is given by:
$$\text{d} \sin \theta = \text{n} \lambda$$
- d represents the grating spacing (the distance between two adjacent lines on the grating).
- θ is the angle of diffraction for a particular order.
- n is the order of the spectrum (e.g., 1st order, 2nd order, etc.).
- λ is the wavelength of the incident light.
Wavelength and Given Parameters
Let's list the given parameters from the problem regarding the diffraction grating and incident light:
- Wavelength of light, $$\lambda = 600 \text{ nm} = 600 \times 10^{-9} \text{ m}$$
- Order of the spectrum, $$n = 2$$ (since it's the second order line)
- Angle of observation, $$\theta = 30^\circ$$
- Width of the diffraction grating, $$\text{W} = 3.00 \text{ cm} = 3.00 \times 10^{-2} \text{ m}$$
Calculating Grating Spacing
Now, we will use the diffraction grating equation to calculate the grating spacing 'd'.
$$\text{d} \sin \theta = \text{n} \lambda$$
Substitute the known values into the equation:
$$\text{d} \sin(30^\circ) = 2 \times (600 \times 10^{-9} \text{ m})$$
We know that the sine of 30 degrees is 0.5.
$$\text{d} \times 0.5 = 1200 \times 10^{-9} \text{ m}$$
To find 'd', divide both sides by 0.5:
$$\text{d} = \frac{1200 \times 10^{-9} \text{ m}}{0.5}$$
$$\text{d} = 2400 \times 10^{-9} \text{ m}$$
This can also be written in a more standard scientific notation as:
$$\text{d} = 2.4 \times 10^{-6} \text{ m}$$
Total Lines on Grating Calculation
The total number of lines (N) on the diffraction grating can be found by dividing the total width of the grating (W) by the grating spacing (d). This calculation tells us how many individual lines are present across the entire width of the grating.
The formula for the total number of lines is:
$$\text{N} = \frac{\text{W}}{\text{d}}$$
Substitute the values for 'W' and the calculated 'd':
$$\text{N} = \frac{3.00 \times 10^{-2} \text{ m}}{2.4 \times 10^{-6} \text{ m}}$$
Perform the division:
$$\text{N} = \frac{3.00}{2.4} \times 10^{-2 - (-6)}$$
$$\text{N} = 1.25 \times 10^{4}$$
$$\text{N} = 12500$$
Therefore, the total number of lines on the diffraction grating is 12500.
| Parameter | Value |
|---|---|
| Wavelength ($\lambda$) | $600 \text{ nm}$ |
| Order ($n$) | $2$ |
| Angle ($\theta$) | $30^\circ$ |
| Grating Width (W) | $3.00 \text{ cm}$ |
| Calculated Grating Spacing (d) | $2.4 \times 10^{-6} \text{ m}$ |
| Total Number of Lines (N) | $12500$ |
Important Questions from Diffraction
In the dispersion of white light by a common glass prism, which one among the following is correct?
In a double-slit experiment, when light of wavelength $\text{600 nm}$ is used, the central maximum and the second bright fringe are separated by $\text{3 mm}$ on a screen placed $\text{1.5 m}$ away. If the entire apparatus is then immersed in a liquid with a refractive index of $\text{1.5}$, what will be the angular separation between the first and fourth dark fringes?
Which one of the following statements about X-rays is not true?
When light passes from air to water, the angle of refraction is:
The primary rainbow appears after the rain is due to
