What is the equation for single slit fringe?
1 Answer
The equation for the fringe pattern produced by a **single slit diffraction** is derived from the condition for constructive and destructive interference of light passing through the slit. Here's a detailed breakdown of the equation and its components:
### Single Slit Diffraction Pattern
When light of a monochromatic wavelength \(\lambda\) passes through a single narrow slit and is observed on a screen, the light diffracts and forms a pattern of bright and dark fringes. The central maximum is the brightest, and the intensity of the fringes decreases as you move away from the center.
The position of the minima (dark fringes) and maxima (bright fringes) on the screen can be predicted using diffraction equations.
### 1. **Minima (Dark Fringes) Condition**
The minima occur where destructive interference takes place. The condition for destructive interference in single-slit diffraction is:
\[
a \sin \theta = m \lambda
\]
Where:
- \(a\) is the width of the slit.
- \(\theta\) is the angle at which the minimum occurs relative to the central axis (the straight line from the slit to the screen).
- \(m\) is an integer (\(m = \pm 1, \pm 2, \pm 3, \dots\)) representing the order of the minima.
- \(\lambda\) is the wavelength of the light.
This equation tells us that the minima occur at specific angles, where the path difference between light from different parts of the slit results in destructive interference.
### 2. **Maxima (Bright Fringes)**
The bright fringes (maxima) are not as straightforward to describe as the minima because they don't occur exactly halfway between the minima. The maxima occur at angles where the diffraction pattern has constructive interference, but the condition for their exact positions is more complex.
In general, there isn't a simple equation like for the minima to precisely determine the angles of the maxima for single-slit diffraction. However, the central maximum (zeroth-order maximum) is always at \(\theta = 0\) and is the brightest.
For higher-order maxima, we observe that they lie in between the minima but the positions aren't as easy to calculate exactly due to the overlap of different diffraction effects.
### 3. **Angular Width of Central Maximum**
The angular width of the central maximum can be determined by the positions of the first minima on either side of the central peak. Using the minima condition for \(m = \pm 1\), we get the angular position for the first minima:
\[
\theta_1 = \sin^{-1}\left(\frac{\lambda}{a}\right)
\]
Thus, the angular width of the central maximum (from one minimum to the other) is approximately:
\[
\Delta \theta = 2 \sin^{-1}\left(\frac{\lambda}{a}\right)
\]
This width increases as the slit width \(a\) decreases or as the wavelength \(\lambda\) increases.
### 4. **Linear Fringe Separation**
If you want to calculate the linear distance between two adjacent fringes on a screen, you can use the following relationship:
\[
y_m = L \tan \theta_m
\]
Where:
- \(y_m\) is the linear distance from the central maximum to the \(m\)-th order minimum or maximum on the screen.
- \(L\) is the distance from the slit to the screen.
- \(\theta_m\) is the angle corresponding to the \(m\)-th minimum (or maxima, if you can calculate them) from the central axis.
For small angles \(\theta\), \(\tan \theta \approx \sin \theta\), so you can approximate the linear position of the minima as:
\[
y_m \approx L \sin \theta_m
\]
Substituting for \(\theta_m\) from the minima condition:
\[
y_m \approx L \sin \left( \sin^{-1} \frac{m \lambda}{a} \right) = \frac{m \lambda L}{a}
\]
### Summary
The key equation for the **dark fringes** (minima) in a single-slit diffraction pattern is:
\[
a \sin \theta = m \lambda
\]
For **bright fringes**, there is no simple, exact formula, but the central maximum is always at \(\theta = 0\), and higher-order maxima lie between the minima. The width of the central maximum increases with wavelength and decreases with slit width.
### Single Slit Diffraction Pattern
When light of a monochromatic wavelength \(\lambda\) passes through a single narrow slit and is observed on a screen, the light diffracts and forms a pattern of bright and dark fringes. The central maximum is the brightest, and the intensity of the fringes decreases as you move away from the center.
The position of the minima (dark fringes) and maxima (bright fringes) on the screen can be predicted using diffraction equations.
### 1. **Minima (Dark Fringes) Condition**
The minima occur where destructive interference takes place. The condition for destructive interference in single-slit diffraction is:
\[
a \sin \theta = m \lambda
\]
Where:
- \(a\) is the width of the slit.
- \(\theta\) is the angle at which the minimum occurs relative to the central axis (the straight line from the slit to the screen).
- \(m\) is an integer (\(m = \pm 1, \pm 2, \pm 3, \dots\)) representing the order of the minima.
- \(\lambda\) is the wavelength of the light.
This equation tells us that the minima occur at specific angles, where the path difference between light from different parts of the slit results in destructive interference.
### 2. **Maxima (Bright Fringes)**
The bright fringes (maxima) are not as straightforward to describe as the minima because they don't occur exactly halfway between the minima. The maxima occur at angles where the diffraction pattern has constructive interference, but the condition for their exact positions is more complex.
In general, there isn't a simple equation like for the minima to precisely determine the angles of the maxima for single-slit diffraction. However, the central maximum (zeroth-order maximum) is always at \(\theta = 0\) and is the brightest.
For higher-order maxima, we observe that they lie in between the minima but the positions aren't as easy to calculate exactly due to the overlap of different diffraction effects.
### 3. **Angular Width of Central Maximum**
The angular width of the central maximum can be determined by the positions of the first minima on either side of the central peak. Using the minima condition for \(m = \pm 1\), we get the angular position for the first minima:
\[
\theta_1 = \sin^{-1}\left(\frac{\lambda}{a}\right)
\]
Thus, the angular width of the central maximum (from one minimum to the other) is approximately:
\[
\Delta \theta = 2 \sin^{-1}\left(\frac{\lambda}{a}\right)
\]
This width increases as the slit width \(a\) decreases or as the wavelength \(\lambda\) increases.
### 4. **Linear Fringe Separation**
If you want to calculate the linear distance between two adjacent fringes on a screen, you can use the following relationship:
\[
y_m = L \tan \theta_m
\]
Where:
- \(y_m\) is the linear distance from the central maximum to the \(m\)-th order minimum or maximum on the screen.
- \(L\) is the distance from the slit to the screen.
- \(\theta_m\) is the angle corresponding to the \(m\)-th minimum (or maxima, if you can calculate them) from the central axis.
For small angles \(\theta\), \(\tan \theta \approx \sin \theta\), so you can approximate the linear position of the minima as:
\[
y_m \approx L \sin \theta_m
\]
Substituting for \(\theta_m\) from the minima condition:
\[
y_m \approx L \sin \left( \sin^{-1} \frac{m \lambda}{a} \right) = \frac{m \lambda L}{a}
\]
### Summary
The key equation for the **dark fringes** (minima) in a single-slit diffraction pattern is:
\[
a \sin \theta = m \lambda
\]
For **bright fringes**, there is no simple, exact formula, but the central maximum is always at \(\theta = 0\), and higher-order maxima lie between the minima. The width of the central maximum increases with wavelength and decreases with slit width.