What is the formula for double slit fringes?
2 Answers
The formula for the interference fringes produced in a double-slit experiment is derived from the principles of wave interference. When light passes through two closely spaced slits, the waves emanating from each slit interfere with each other, producing a pattern of bright and dark fringes on a screen placed at a distance.
### Key Variables:
- **λ** = Wavelength of the light used
- **d** = Distance between the two slits
- **L** = Distance from the slits to the screen where the fringes are observed
- **m** = Order of the fringe (an integer: 0, ±1, ±2, …)
- **y** = Position of the m-th fringe on the screen (measured from the central maximum)
- **θ** = Angle at which the m-th order fringe occurs, relative to the central axis
### Formula for Fringe Spacing:
The fringe spacing, \( \Delta y \), is the distance between two adjacent bright or dark fringes on the screen, measured from one maximum (or minimum) to the next. The formula for the position of the m-th order bright fringe is given by:
\[
y_m = \frac{m \lambda L}{d}
\]
Where:
- \( m \) is the fringe order (m = 0 corresponds to the central maximum, m = ±1 corresponds to the first order maxima, and so on),
- \( \lambda \) is the wavelength of the light,
- \( L \) is the distance from the slits to the screen,
- \( d \) is the slit separation.
### Condition for Constructive Interference (Bright Fringes):
Bright fringes occur when the path difference between the two waves from the slits is an integer multiple of the wavelength, i.e.,
\[
\Delta \text{Path} = m\lambda \quad \text{(for maxima)}
\]
Thus, for bright fringes, the angle θ or the position on the screen y satisfies:
\[
\sin \theta = \frac{m \lambda}{d} \quad \text{(for maxima)}
\]
or
\[
y_m = \frac{m \lambda L}{d} \quad \text{(on the screen)}
\]
### Condition for Destructive Interference (Dark Fringes):
Dark fringes occur when the path difference is an odd multiple of half the wavelength, i.e.,
\[
\Delta \text{Path} = \left(m + \frac{1}{2}\right) \lambda \quad \text{(for minima)}
\]
This means that the dark fringes are located at:
\[
\sin \theta = \frac{(m + 1/2) \lambda}{d} \quad \text{(for minima)}
\]
or
\[
y_m = \frac{(m + 1/2) \lambda L}{d} \quad \text{(on the screen)}
\]
### Summary of Key Points:
1. **Fringe Spacing:** The distance between adjacent bright (or dark) fringes on the screen is determined by the formula:
\[
\Delta y = \frac{\lambda L}{d}
\]
where \( \lambda \) is the wavelength of the light, \( L \) is the distance from the slits to the screen, and \( d \) is the distance between the slits.
2. **Bright Fringe Position:** The m-th order bright fringe occurs at:
\[
y_m = \frac{m \lambda L}{d}
\]
3. **Dark Fringe Position:** The m-th order dark fringe occurs at:
\[
y_m = \frac{(m + 1/2) \lambda L}{d}
\]
### Example:
If light of wavelength \( \lambda = 500 \, \text{nm} \) (or \( 500 \times 10^{-9} \, \text{m} \)) passes through two slits that are separated by \( d = 0.2 \, \text{mm} \) (or \( 0.2 \times 10^{-3} \, \text{m} \)), and the screen is located \( L = 2 \, \text{m} \) away from the slits, the distance between adjacent bright fringes is:
\[
\Delta y = \frac{(500 \times 10^{-9} \, \text{m})(2 \, \text{m})}{0.2 \times 10^{-3} \, \text{m}} = 5 \, \text{mm}
\]
This means that the distance between each consecutive bright fringe will be 5 mm on the screen.
This formula and the conditions for constructive and destructive interference are fundamental to understanding the diffraction and interference patterns observed in double-slit experiments.
### Key Variables:
- **λ** = Wavelength of the light used
- **d** = Distance between the two slits
- **L** = Distance from the slits to the screen where the fringes are observed
- **m** = Order of the fringe (an integer: 0, ±1, ±2, …)
- **y** = Position of the m-th fringe on the screen (measured from the central maximum)
- **θ** = Angle at which the m-th order fringe occurs, relative to the central axis
### Formula for Fringe Spacing:
The fringe spacing, \( \Delta y \), is the distance between two adjacent bright or dark fringes on the screen, measured from one maximum (or minimum) to the next. The formula for the position of the m-th order bright fringe is given by:
\[
y_m = \frac{m \lambda L}{d}
\]
Where:
- \( m \) is the fringe order (m = 0 corresponds to the central maximum, m = ±1 corresponds to the first order maxima, and so on),
- \( \lambda \) is the wavelength of the light,
- \( L \) is the distance from the slits to the screen,
- \( d \) is the slit separation.
### Condition for Constructive Interference (Bright Fringes):
Bright fringes occur when the path difference between the two waves from the slits is an integer multiple of the wavelength, i.e.,
\[
\Delta \text{Path} = m\lambda \quad \text{(for maxima)}
\]
Thus, for bright fringes, the angle θ or the position on the screen y satisfies:
\[
\sin \theta = \frac{m \lambda}{d} \quad \text{(for maxima)}
\]
or
\[
y_m = \frac{m \lambda L}{d} \quad \text{(on the screen)}
\]
### Condition for Destructive Interference (Dark Fringes):
Dark fringes occur when the path difference is an odd multiple of half the wavelength, i.e.,
\[
\Delta \text{Path} = \left(m + \frac{1}{2}\right) \lambda \quad \text{(for minima)}
\]
This means that the dark fringes are located at:
\[
\sin \theta = \frac{(m + 1/2) \lambda}{d} \quad \text{(for minima)}
\]
or
\[
y_m = \frac{(m + 1/2) \lambda L}{d} \quad \text{(on the screen)}
\]
### Summary of Key Points:
1. **Fringe Spacing:** The distance between adjacent bright (or dark) fringes on the screen is determined by the formula:
\[
\Delta y = \frac{\lambda L}{d}
\]
where \( \lambda \) is the wavelength of the light, \( L \) is the distance from the slits to the screen, and \( d \) is the distance between the slits.
2. **Bright Fringe Position:** The m-th order bright fringe occurs at:
\[
y_m = \frac{m \lambda L}{d}
\]
3. **Dark Fringe Position:** The m-th order dark fringe occurs at:
\[
y_m = \frac{(m + 1/2) \lambda L}{d}
\]
### Example:
If light of wavelength \( \lambda = 500 \, \text{nm} \) (or \( 500 \times 10^{-9} \, \text{m} \)) passes through two slits that are separated by \( d = 0.2 \, \text{mm} \) (or \( 0.2 \times 10^{-3} \, \text{m} \)), and the screen is located \( L = 2 \, \text{m} \) away from the slits, the distance between adjacent bright fringes is:
\[
\Delta y = \frac{(500 \times 10^{-9} \, \text{m})(2 \, \text{m})}{0.2 \times 10^{-3} \, \text{m}} = 5 \, \text{mm}
\]
This means that the distance between each consecutive bright fringe will be 5 mm on the screen.
This formula and the conditions for constructive and destructive interference are fundamental to understanding the diffraction and interference patterns observed in double-slit experiments.
The formula for the fringe spacing (distance between adjacent bright or dark fringes) in a double slit experiment is derived based on the interference of light waves. In this experiment, light passes through two slits and creates an interference pattern on a screen. This pattern consists of alternating bright and dark fringes.
### Key Variables:
- \( d \): The distance between the two slits (slit separation).
- \( D \): The distance from the slits to the screen where the interference pattern is observed.
- \( \lambda \): The wavelength of the light used.
- \( m \): The fringe order (an integer, \( m = 0, 1, 2, 3, \dots \), where \( m = 0 \) is the central maximum).
- \( y_m \): The position of the \( m \)-th bright or dark fringe on the screen.
### Interference Condition:
For constructive interference (bright fringes), the path difference between the two waves from the slits must be an integer multiple of the wavelength:
\[
\Delta L = m \lambda \quad (m = 0, 1, 2, 3, \dots)
\]
For destructive interference (dark fringes), the path difference must be an odd multiple of half the wavelength:
\[
\Delta L = \left(m + \frac{1}{2}\right) \lambda \quad (m = 0, 1, 2, 3, \dots)
\]
### Deriving the Fringe Spacing Formula:
In the double slit experiment, the fringe spacing depends on the angle of the interference maxima or minima, and this can be related to the geometry of the setup.
1. **Angle of the Bright Fringes:**
For constructive interference, the angle \( \theta_m \) for the \( m \)-th bright fringe (where \( m = 0, 1, 2, \dots \)) is given by the formula:
\[
\sin \theta_m = \frac{m \lambda}{d}
\]
For small angles, \( \sin \theta_m \approx \tan \theta_m \approx \theta_m \) (in radians), so:
\[
\theta_m \approx \frac{m \lambda}{d}
\]
2. **Position of the Fringes on the Screen:**
The distance \( y_m \) of the \( m \)-th fringe from the central maximum (which is located at \( m = 0 \)) on the screen at a distance \( D \) from the slits is related to the angle \( \theta_m \) by the following relationship:
\[
y_m = D \tan \theta_m \approx D \theta_m
\]
Using \( \theta_m \approx \frac{m \lambda}{d} \), we get:
\[
y_m \approx D \frac{m \lambda}{d}
\]
3. **Fringe Spacing:**
The distance between two adjacent bright fringes (or dark fringes) is the difference in their positions. The fringe spacing \( \Delta y \), which is the distance between adjacent bright fringes (or dark fringes), can be obtained by finding the difference between the positions of the \( (m+1) \)-th and \( m \)-th fringes:
\[
\Delta y = y_{m+1} - y_m
\]
Substituting the formula for \( y_m \):
\[
\Delta y = D \left( \frac{(m+1) \lambda}{d} - \frac{m \lambda}{d} \right)
\]
Simplifying:
\[
\Delta y = D \frac{\lambda}{d}
\]
### Final Formula:
Thus, the fringe spacing \( \Delta y \) is:
\[
\Delta y = \frac{D \lambda}{d}
\]
This is the formula for the spacing between adjacent bright or dark fringes in a double slit interference pattern.
### Explanation:
- The fringe spacing depends on the wavelength of light \( \lambda \), the distance between the slits \( d \), and the distance to the screen \( D \).
- If the wavelength \( \lambda \) increases, the fringes will be spaced farther apart.
- If the slit separation \( d \) is smaller, the fringes will also be spaced farther apart.
- The distance \( D \) from the slits to the screen also affects the fringe spacing: the larger the screen distance, the larger the separation between fringes.
This formula applies under the assumption that the angles are small, meaning the approximation \( \sin \theta \approx \theta \) holds true.
### Key Variables:
- \( d \): The distance between the two slits (slit separation).
- \( D \): The distance from the slits to the screen where the interference pattern is observed.
- \( \lambda \): The wavelength of the light used.
- \( m \): The fringe order (an integer, \( m = 0, 1, 2, 3, \dots \), where \( m = 0 \) is the central maximum).
- \( y_m \): The position of the \( m \)-th bright or dark fringe on the screen.
### Interference Condition:
For constructive interference (bright fringes), the path difference between the two waves from the slits must be an integer multiple of the wavelength:
\[
\Delta L = m \lambda \quad (m = 0, 1, 2, 3, \dots)
\]
For destructive interference (dark fringes), the path difference must be an odd multiple of half the wavelength:
\[
\Delta L = \left(m + \frac{1}{2}\right) \lambda \quad (m = 0, 1, 2, 3, \dots)
\]
### Deriving the Fringe Spacing Formula:
In the double slit experiment, the fringe spacing depends on the angle of the interference maxima or minima, and this can be related to the geometry of the setup.
1. **Angle of the Bright Fringes:**
For constructive interference, the angle \( \theta_m \) for the \( m \)-th bright fringe (where \( m = 0, 1, 2, \dots \)) is given by the formula:
\[
\sin \theta_m = \frac{m \lambda}{d}
\]
For small angles, \( \sin \theta_m \approx \tan \theta_m \approx \theta_m \) (in radians), so:
\[
\theta_m \approx \frac{m \lambda}{d}
\]
2. **Position of the Fringes on the Screen:**
The distance \( y_m \) of the \( m \)-th fringe from the central maximum (which is located at \( m = 0 \)) on the screen at a distance \( D \) from the slits is related to the angle \( \theta_m \) by the following relationship:
\[
y_m = D \tan \theta_m \approx D \theta_m
\]
Using \( \theta_m \approx \frac{m \lambda}{d} \), we get:
\[
y_m \approx D \frac{m \lambda}{d}
\]
3. **Fringe Spacing:**
The distance between two adjacent bright fringes (or dark fringes) is the difference in their positions. The fringe spacing \( \Delta y \), which is the distance between adjacent bright fringes (or dark fringes), can be obtained by finding the difference between the positions of the \( (m+1) \)-th and \( m \)-th fringes:
\[
\Delta y = y_{m+1} - y_m
\]
Substituting the formula for \( y_m \):
\[
\Delta y = D \left( \frac{(m+1) \lambda}{d} - \frac{m \lambda}{d} \right)
\]
Simplifying:
\[
\Delta y = D \frac{\lambda}{d}
\]
### Final Formula:
Thus, the fringe spacing \( \Delta y \) is:
\[
\Delta y = \frac{D \lambda}{d}
\]
This is the formula for the spacing between adjacent bright or dark fringes in a double slit interference pattern.
### Explanation:
- The fringe spacing depends on the wavelength of light \( \lambda \), the distance between the slits \( d \), and the distance to the screen \( D \).
- If the wavelength \( \lambda \) increases, the fringes will be spaced farther apart.
- If the slit separation \( d \) is smaller, the fringes will also be spaced farther apart.
- The distance \( D \) from the slits to the screen also affects the fringe spacing: the larger the screen distance, the larger the separation between fringes.
This formula applies under the assumption that the angles are small, meaning the approximation \( \sin \theta \approx \theta \) holds true.