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Learning Introductory Physics with Activities

Paul J. Emigh, Rebecka Tumblin, Kathryn Hadley, Danielle Skinner

Section 8.8 Two-Slit Interference

A diagram showing the double-slit experiment and the interference pattern that is shown on the screen. Two slits sit on the right side. Plane waves hit the two slits from the left side. As they exist the slits, the wave front becomes spherical, and multiple concentric circles are shown to represent how the light wave behaves. The interference pattern has seven evenly distributed bright spots. The center is labelled m = 0, and the rest are labelled from m = 1 to m = 3 above and below the center.
Figure 8.8.1. Interference pattern produced by light traveling through two adjacent slits. The central maxima is at a value of \(m = 0\text{.}\)
The Double-Slit Experiment was originally conducted by Thomas Young in 1801 to show that light does indeed behave like a wave. Just like water waves interacting with each other, as light travels through two slits, the crests and troughs of the light waves will interact with each other and produce an interference pattern on the screen behind the slits. In fact, the general rules for interference are the same as you saw in Section 7.6.

Definition 8.8.2. Constructive Interference.

Constructive Interference, which leads to a very bright spot on a screen, occurs when \(\Delta D = m \lambda\) where \(m = 0, 1, 2, ...\text{.}\)

Definition 8.8.3. Destructive Interference.

Destructive Interference, which leads to a dark spot on a screen, occurs when
\begin{equation*} \Delta D = (m + \frac{1}{2}) \lambda \end{equation*}
where \(m = 0, 1, 2, ...\text{.}\)
The spacing of the interference pattern can be determined by looking at the path length difference of the rays exiting the slits.

Definition 8.8.4. Path Length Difference.

For a screen that is far away from two sources of light, the Path Length Difference can be approximated as
\begin{equation*} \Delta D = d \sin \theta \end{equation*}
where \(d\) is the distance between slits and \(\theta\) is the angle between the center line and the path to a spot on the screen.
On the left side, there is a diagram of the souble-slit experiment, with the slits sitting on the left a distance L away from a screen. A dashed line extends from the center of the slits to the screen at an angle theta sub m and hits a point y sub m on the screen on the right. A small circle covers the double-slit experiment on the right. On the right side, there is a zoomed-in picture of the double-slit set up inside a circle. Two slits sit at a distance d away from each other. A ray extends from each slit to the right side of the circle at an angle theta. The rays are labelled "r 1" and "r 2". Another line extends from the top slit to the bottom ray at an angle theta, indicating the path length difference.
Figure 8.8.5. Left: A diagram of the double slit experiment. The slits are separated from the screen by a distance of \(L\text{,}\) and the angle to the mth bright spot is denoted by \(\theta_m\text{.}\) The vertical distance to the mth bright spot is \(y_m\text{.}\) Right: A zoomed-in view of the double slits. The slits are separated by a distance \(d\text{.}\) Two different rays, \(r_1\) and \(r_2\) travel towards the screen at an angle \(\theta\text{,}\) making the path length difference between the two rays equal to \(d \sin\theta\text{.}\)

Exercises Two-Slit Interference Activities

1. Two-Slit Interference Simulation.

Play with the light interference simulation for a few minutes. Click on the "Slits" tab. On the right hand side, increase the number of slits to two.
Write down some observations. What happens when you move the slit width to its minimum and maximum values? What happens as you increase the slit separation? What happens when you increase and decrease the frequency?

2. The Small Angle Approximation.

In the video, you learned about using the small-angle approximation for interference. Use the small-angle approximation to write an equation relating \(d\text{,}\) \(y\text{,}\) \(L\text{,}\) and \(\lambda\) for points of maximum constructive interference.

3. Measure the Wavelength.

You are conducting the double slit experiment and you have a light source with an unknown wavelength. Your slits are separated by a distance of \(0.5\) mm and your screen is 3.4 m. You measure the third bright fringe to be at a position of 4.7 mm to the right of the center line on the screen. What is the wavelength of light?
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