What is the formula for Young's double-slit fringes?
1 Answer
Young's double-slit experiment is a fundamental demonstration of the wave nature of light. In this experiment, light passing through two slits creates an interference pattern on a screen. The pattern consists of bright and dark fringes. To understand and predict the positions of these fringes, we use a mathematical formula derived from the principle of interference.
### Formula for Young's Double-Slit Fringes:
The condition for constructive and destructive interference, which determines the positions of the bright and dark fringes, is as follows:
1. **Constructive interference (bright fringes)** occurs when the path difference between the light from the two slits is an integer multiple of the wavelength, i.e.,
\[
\Delta x = m\lambda
\]
where:
- \( m \) is an integer (0, ±1, ±2, ±3, ...) corresponding to the order of the fringe (central maximum is \( m = 0 \)),
- \( \lambda \) is the wavelength of the light,
- \( \Delta x \) is the path difference between the two slits for constructive interference.
2. **Destructive interference (dark fringes)** occurs when the path difference is an odd multiple of half the wavelength, i.e.,
\[
\Delta x = (m + \frac{1}{2})\lambda
\]
where \( m \) is again an integer (0, ±1, ±2, ±3, ...).
### Position of the Fringes on the Screen:
The formula for the position \( y_m \) of the \( m \)-th fringe (where \( m = 0, ±1, ±2, ±3, \dots \)) on a screen is derived based on the geometry of the setup. The setup typically consists of two slits separated by a distance \( d \), and the screen is located at a distance \( L \) from the slits (with \( L \) being much larger than \( d \)).
The position \( y_m \) of the \( m \)-th bright fringe (constructive interference) is given by:
\[
y_m = \frac{m \lambda L}{d}
\]
where:
- \( y_m \) is the distance from the central maximum (the \( m = 0 \) fringe) to the \( m \)-th bright fringe on the screen,
- \( \lambda \) is the wavelength of the light,
- \( L \) is the distance from the slits to the screen,
- \( d \) is the separation between the two slits,
- \( m \) is the fringe order (central maximum is \( m = 0 \), the first bright fringe is \( m = ±1 \), the second is \( m = ±2 \), and so on).
### Explanation of the Terms:
- **Wavelength (\( \lambda \))**: This is the distance between two consecutive peaks (or troughs) of the wave. The wavelength determines how much the wave "oscillates" over a given distance and plays a central role in the spacing of the fringes.
- **Slit Separation (\( d \))**: The distance between the two slits influences the diffraction pattern. If the slits are closer together, the fringes will spread out more; if they are farther apart, the fringes will be closer together.
- **Screen Distance (\( L \))**: The distance between the slits and the screen determines how large the interference pattern will appear. A larger \( L \) means the fringes will be spaced farther apart on the screen.
- **Fringe Order (\( m \))**: The integer \( m \) identifies which fringe is being considered. The central maximum corresponds to \( m = 0 \), and the other maxima correspond to \( m = ±1, ±2, \dots \).
### Example:
Let's say you have the following setup:
- Wavelength of light: \( \lambda = 500 \, \text{nm} = 5 \times 10^{-7} \, \text{m} \),
- Slit separation: \( d = 0.1 \, \text{mm} = 1 \times 10^{-4} \, \text{m} \),
- Distance to screen: \( L = 2 \, \text{m} \).
To find the position of the first bright fringe (i.e., \( m = 1 \)):
\[
y_1 = \frac{1 \times (5 \times 10^{-7}) \times 2}{1 \times 10^{-4}} = 1 \times 10^{-2} \, \text{m} = 1 \, \text{cm}.
\]
This means the first bright fringe will appear 1 cm away from the central maximum on the screen.
### Conclusion:
The formula \( y_m = \frac{m \lambda L}{d} \) allows us to calculate the positions of the bright fringes in Young's double-slit experiment. It illustrates how the fringe spacing is influenced by the wavelength of light, the separation between the slits, and the distance to the screen.
### Formula for Young's Double-Slit Fringes:
The condition for constructive and destructive interference, which determines the positions of the bright and dark fringes, is as follows:
1. **Constructive interference (bright fringes)** occurs when the path difference between the light from the two slits is an integer multiple of the wavelength, i.e.,
\[
\Delta x = m\lambda
\]
where:
- \( m \) is an integer (0, ±1, ±2, ±3, ...) corresponding to the order of the fringe (central maximum is \( m = 0 \)),
- \( \lambda \) is the wavelength of the light,
- \( \Delta x \) is the path difference between the two slits for constructive interference.
2. **Destructive interference (dark fringes)** occurs when the path difference is an odd multiple of half the wavelength, i.e.,
\[
\Delta x = (m + \frac{1}{2})\lambda
\]
where \( m \) is again an integer (0, ±1, ±2, ±3, ...).
### Position of the Fringes on the Screen:
The formula for the position \( y_m \) of the \( m \)-th fringe (where \( m = 0, ±1, ±2, ±3, \dots \)) on a screen is derived based on the geometry of the setup. The setup typically consists of two slits separated by a distance \( d \), and the screen is located at a distance \( L \) from the slits (with \( L \) being much larger than \( d \)).
The position \( y_m \) of the \( m \)-th bright fringe (constructive interference) is given by:
\[
y_m = \frac{m \lambda L}{d}
\]
where:
- \( y_m \) is the distance from the central maximum (the \( m = 0 \) fringe) to the \( m \)-th bright fringe on the screen,
- \( \lambda \) is the wavelength of the light,
- \( L \) is the distance from the slits to the screen,
- \( d \) is the separation between the two slits,
- \( m \) is the fringe order (central maximum is \( m = 0 \), the first bright fringe is \( m = ±1 \), the second is \( m = ±2 \), and so on).
### Explanation of the Terms:
- **Wavelength (\( \lambda \))**: This is the distance between two consecutive peaks (or troughs) of the wave. The wavelength determines how much the wave "oscillates" over a given distance and plays a central role in the spacing of the fringes.
- **Slit Separation (\( d \))**: The distance between the two slits influences the diffraction pattern. If the slits are closer together, the fringes will spread out more; if they are farther apart, the fringes will be closer together.
- **Screen Distance (\( L \))**: The distance between the slits and the screen determines how large the interference pattern will appear. A larger \( L \) means the fringes will be spaced farther apart on the screen.
- **Fringe Order (\( m \))**: The integer \( m \) identifies which fringe is being considered. The central maximum corresponds to \( m = 0 \), and the other maxima correspond to \( m = ±1, ±2, \dots \).
### Example:
Let's say you have the following setup:
- Wavelength of light: \( \lambda = 500 \, \text{nm} = 5 \times 10^{-7} \, \text{m} \),
- Slit separation: \( d = 0.1 \, \text{mm} = 1 \times 10^{-4} \, \text{m} \),
- Distance to screen: \( L = 2 \, \text{m} \).
To find the position of the first bright fringe (i.e., \( m = 1 \)):
\[
y_1 = \frac{1 \times (5 \times 10^{-7}) \times 2}{1 \times 10^{-4}} = 1 \times 10^{-2} \, \text{m} = 1 \, \text{cm}.
\]
This means the first bright fringe will appear 1 cm away from the central maximum on the screen.
### Conclusion:
The formula \( y_m = \frac{m \lambda L}{d} \) allows us to calculate the positions of the bright fringes in Young's double-slit experiment. It illustrates how the fringe spacing is influenced by the wavelength of light, the separation between the slits, and the distance to the screen.