What is the formula for single slit diffraction?
2 Answers
Single slit diffraction refers to the pattern of light that forms when light passes through a narrow slit and spreads out, forming a series of dark and bright bands or fringes on a screen. This phenomenon occurs due to the wave nature of light, and the interference between light waves from different parts of the slit.
The formula for the angular positions of the **dark fringes** (minima) in a single slit diffraction pattern is given by:
\[
a \sin \theta = m \lambda
\]
where:
- \(a\) is the width of the slit,
- \(\theta\) is the angle from the central maximum to the \(m\)-th minima,
- \(m\) is the order of the minima (where \(m = \pm 1, \pm 2, \pm 3, \dots\)),
- \(\lambda\) is the wavelength of the light.
### Explanation:
- The diffraction pattern consists of a central bright fringe (maximum) and alternating dark and bright fringes on either side.
- The **dark fringes** occur at angles where the path difference between light from different parts of the slit is an integer multiple of the wavelength, leading to destructive interference.
- The central maximum is at \(m = 0\), and the first dark fringe occurs when \(m = \pm 1\).
The **bright fringes** (maxima) in the pattern don't have a simple, straightforward formula like the dark fringes. However, they occur between the dark fringes and are typically weaker and broader compared to the central maximum.
### For the central maximum:
- It always occurs at \(\theta = 0\) (i.e., straight ahead of the slit).
### Derivation:
- Consider a slit of width \(a\) with light passing through it. The light from different parts of the slit will interfere, and the path difference between light from the top and bottom of the slit can be used to determine the conditions for constructive or destructive interference.
- Destructive interference occurs when the path difference is an integer multiple of the wavelength, resulting in dark fringes.
For the **first minimum** (where \(m = \pm 1\)):
\[
a \sin \theta = \pm \lambda
\]
For the **second minimum** (where \(m = \pm 2\)):
\[
a \sin \theta = \pm 2\lambda
\]
and so on.
The angles at which these minima occur are crucial for determining the diffraction pattern on a screen at a given distance from the slit.
In practice, the angular separation between minima (or the width of the central maximum) can also be calculated if the distance to the screen and the slit width are known.
The formula for the angular positions of the **dark fringes** (minima) in a single slit diffraction pattern is given by:
\[
a \sin \theta = m \lambda
\]
where:
- \(a\) is the width of the slit,
- \(\theta\) is the angle from the central maximum to the \(m\)-th minima,
- \(m\) is the order of the minima (where \(m = \pm 1, \pm 2, \pm 3, \dots\)),
- \(\lambda\) is the wavelength of the light.
### Explanation:
- The diffraction pattern consists of a central bright fringe (maximum) and alternating dark and bright fringes on either side.
- The **dark fringes** occur at angles where the path difference between light from different parts of the slit is an integer multiple of the wavelength, leading to destructive interference.
- The central maximum is at \(m = 0\), and the first dark fringe occurs when \(m = \pm 1\).
The **bright fringes** (maxima) in the pattern don't have a simple, straightforward formula like the dark fringes. However, they occur between the dark fringes and are typically weaker and broader compared to the central maximum.
### For the central maximum:
- It always occurs at \(\theta = 0\) (i.e., straight ahead of the slit).
### Derivation:
- Consider a slit of width \(a\) with light passing through it. The light from different parts of the slit will interfere, and the path difference between light from the top and bottom of the slit can be used to determine the conditions for constructive or destructive interference.
- Destructive interference occurs when the path difference is an integer multiple of the wavelength, resulting in dark fringes.
For the **first minimum** (where \(m = \pm 1\)):
\[
a \sin \theta = \pm \lambda
\]
For the **second minimum** (where \(m = \pm 2\)):
\[
a \sin \theta = \pm 2\lambda
\]
and so on.
The angles at which these minima occur are crucial for determining the diffraction pattern on a screen at a given distance from the slit.
In practice, the angular separation between minima (or the width of the central maximum) can also be calculated if the distance to the screen and the slit width are known.
Single slit diffraction occurs when light passes through a narrow slit and spreads out, creating a pattern of bright and dark fringes on a screen. The diffraction pattern results from the interference of light waves coming from different parts of the slit.
### Formula for Single Slit Diffraction:
The main formula used to determine the angles of the dark fringes in a single-slit diffraction pattern is:
\[
a \sin \theta = m \lambda
\]
Where:
- \(a\) is the width of the slit (the slit aperture),
- \(\theta\) is the angle relative to the original direction of the light where the m-th dark fringe appears,
- \(m\) is the order of the dark fringe (with \(m = \pm 1, \pm 2, \pm 3, \dots\)),
- \(\lambda\) is the wavelength of the light.
### Explanation of the Formula:
1. **Dark Fringes (Minima)**: The formula above applies to the positions of the dark fringes (or minima) in the diffraction pattern. These occur where the path difference between light waves from different parts of the slit results in destructive interference.
2. **Order of the Fringe (\(m\))**: The index \(m\) represents the order of the dark fringe. For the first dark fringe on one side of the central maximum, \(m = \pm 1\), for the second dark fringe \(m = \pm 2\), and so on. Note that the central maximum itself (i.e., \(m = 0\)) is not included in this formula as it corresponds to the brightest point.
3. **Angle (\(\theta\))**: The angle \(\theta\) is measured from the central axis of the incoming light to the position of the dark fringe. It is important to note that this angle is typically small in many diffraction experiments, especially when the slit width is much larger than the wavelength of light.
### The Diffraction Pattern:
- **Central Maximum**: The central bright spot at \(m = 0\) is the most intense and occurs directly in line with the incoming light.
- **First Dark Fringe**: The first dark fringe occurs at the angle \(\theta\) where \(m = \pm 1\), the second dark fringe at \(m = \pm 2\), and so on.
- **Bright Fringes**: The bright fringes occur between the dark fringes. These bright fringes are not as intense as the central maximum and get dimmer as you move further away from the center. The exact positions of these bright fringes don't have a simple formula like the dark fringes, but they roughly lie halfway between adjacent dark fringes.
### Example:
If the slit width \(a = 0.5 \, \text{mm}\) and the wavelength of the light \(\lambda = 600 \, \text{nm}\), you can calculate the angle for the first dark fringe (\(m = 1\)) using the formula:
\[
a \sin \theta = m \lambda
\]
\[
0.5 \times 10^{-3} \sin \theta = 1 \times 600 \times 10^{-9}
\]
Solving for \(\theta\):
\[
\sin \theta = \frac{600 \times 10^{-9}}{0.5 \times 10^{-3}} = 1.2 \times 10^{-3}
\]
\[
\theta = \arcsin(1.2 \times 10^{-3}) \approx 0.0687^\circ
\]
Thus, the angle for the first dark fringe is about 0.0687°.
### Key Points:
- The single-slit diffraction pattern has a central maximum with dark fringes at specific angles, based on the slit width and the wavelength of light.
- The formula \(a \sin \theta = m \lambda\) allows you to calculate the position of dark fringes in the diffraction pattern.
- As the slit gets narrower or the wavelength gets longer, the fringes spread out more, making the diffraction pattern wider.
### Formula for Single Slit Diffraction:
The main formula used to determine the angles of the dark fringes in a single-slit diffraction pattern is:
\[
a \sin \theta = m \lambda
\]
Where:
- \(a\) is the width of the slit (the slit aperture),
- \(\theta\) is the angle relative to the original direction of the light where the m-th dark fringe appears,
- \(m\) is the order of the dark fringe (with \(m = \pm 1, \pm 2, \pm 3, \dots\)),
- \(\lambda\) is the wavelength of the light.
### Explanation of the Formula:
1. **Dark Fringes (Minima)**: The formula above applies to the positions of the dark fringes (or minima) in the diffraction pattern. These occur where the path difference between light waves from different parts of the slit results in destructive interference.
2. **Order of the Fringe (\(m\))**: The index \(m\) represents the order of the dark fringe. For the first dark fringe on one side of the central maximum, \(m = \pm 1\), for the second dark fringe \(m = \pm 2\), and so on. Note that the central maximum itself (i.e., \(m = 0\)) is not included in this formula as it corresponds to the brightest point.
3. **Angle (\(\theta\))**: The angle \(\theta\) is measured from the central axis of the incoming light to the position of the dark fringe. It is important to note that this angle is typically small in many diffraction experiments, especially when the slit width is much larger than the wavelength of light.
### The Diffraction Pattern:
- **Central Maximum**: The central bright spot at \(m = 0\) is the most intense and occurs directly in line with the incoming light.
- **First Dark Fringe**: The first dark fringe occurs at the angle \(\theta\) where \(m = \pm 1\), the second dark fringe at \(m = \pm 2\), and so on.
- **Bright Fringes**: The bright fringes occur between the dark fringes. These bright fringes are not as intense as the central maximum and get dimmer as you move further away from the center. The exact positions of these bright fringes don't have a simple formula like the dark fringes, but they roughly lie halfway between adjacent dark fringes.
### Example:
If the slit width \(a = 0.5 \, \text{mm}\) and the wavelength of the light \(\lambda = 600 \, \text{nm}\), you can calculate the angle for the first dark fringe (\(m = 1\)) using the formula:
\[
a \sin \theta = m \lambda
\]
\[
0.5 \times 10^{-3} \sin \theta = 1 \times 600 \times 10^{-9}
\]
Solving for \(\theta\):
\[
\sin \theta = \frac{600 \times 10^{-9}}{0.5 \times 10^{-3}} = 1.2 \times 10^{-3}
\]
\[
\theta = \arcsin(1.2 \times 10^{-3}) \approx 0.0687^\circ
\]
Thus, the angle for the first dark fringe is about 0.0687°.
### Key Points:
- The single-slit diffraction pattern has a central maximum with dark fringes at specific angles, based on the slit width and the wavelength of light.
- The formula \(a \sin \theta = m \lambda\) allows you to calculate the position of dark fringes in the diffraction pattern.
- As the slit gets narrower or the wavelength gets longer, the fringes spread out more, making the diffraction pattern wider.