# Double Slit Experiment: Explanation & Equation

Instructor: Aaron Miller

Aaron teaches physics and holds a doctorate in physics.

The double slit experiment is an idealized laboratory scenario that produces a wave interference pattern. In this article, we will discuss the basic setup of a double slit experiment and explain how the setup produces an interference pattern.

## What is Interference?

The flight attendant's voice comes over the speakers: 'As we prepare for takeoff, please turn off and stow all laptop computers and make sure your cell phones are switched to airplane mode.' While this can be an inconvenience, it is an important safety precaution that attempts to prevent wave interference effects from disrupting radio communications between the cockpit and the control tower on the ground. Wave interference may occur in circumstances when two waves are moving through the same location, causing a partial reduction or possibly complete destruction of both signals. Interfering wave sources need to be similar in frequency to have a detrimental effect. Laptop computers and cell phones produce radio waves that are close enough in frequency to the communication channels of the airplane that safety regulators do not want to take any risks that the communications could be disrupted. However, the chances of interference occurring if you accidentally forget to disable your mobile devices are quite slim because interference is also strongly dependent source location. The double slit experiment is the most rudimentary example of how an interference phenomena can be produced and demonstrates in a mathematically tractable way how interference patterns depend on both frequency and source location.

## The Double Slit Experiment Setup

The double slit experiment consists of three parts: a source of single-frequency (i.e. monochromatic) waves, an opaque screen that has two very small holes (or slits) through which the waves can pass, and a viewing screen where the waves are observed/detected after passing through the slits. The set-up is depicted in Figure 1.

In a laboratory, this kind of scenario is often demonstrated with a laser light source because lasers produce light waves in a very narrow frequency range (i.e., they are monochromatic), unlike fluorescent and incandescent light sources. We can use our eyes to see the interference pattern on the viewing screen. However, it is also common to create double slit interference using wave tanks, in which single-frequency surface water waves are created and made to interfere through two slits.

The interference pattern is observed on the viewing screen as an alternating pattern of high-intensity and low-intensity (i.e., 'bright' and 'dark' spots). The low-intensity ('dark') spots are called points of completely destructive interference. In essence, waves emerging from the slits are canceling each other out at these points on the screen -- e.g., one wave arrives on a peak while the other arrives on a trough. Next, we'll introduce an equation that predicts the locations of these destructive interference points on the viewing screen.

## The Double Slit Equations

The underlying reason that an interference pattern of 'bright' and 'dark' spots appears on the viewing screen in the double slit experiment is that waves passing through the slits travel different distances to any point on the viewing screen. This path difference may lead to the waves becoming out of phase as their individual oscillations become desynchronized. Complete desynchronization leads to destructive interference, which causes a dark spot. Between the dark spots, the waves arrive at the viewing screen in sync (or in phase) and a bright spot occurs. This is also called constructive interference.

A diagram depicting an example of the path difference is shown in Figure 2.

If a point on the viewing screen is a distance x above the center of the screen, the point is closer to the upper slit on the diagram than the lower slit. Therefore, the distance travelled by the wave from the upper slit, r2, is just a little bit smaller than the distance travelled by the wave from the lower slit, r1. This difference can be related to the wavelength of the traveling wave (i.e., the distance over which a wave pattern repeats itself) in order to predict at what points x an experimenter would observe a dark spot in the interference pattern.

The standard expression that relates x to the distance between the slits d (see Figure 2), the distance to the viewing screen L and the wavelength of the wave, conventionally represented by Greek letter lambda, is:

The variable m takes an integer value depending on which dark spot we are interested in.

The angle theta relates to x and L by the following equation:

The angle theta is the angle of inclination above the center axis of the experiment, passing directly between the two slits. If this angle is small (less than 10 degrees), then the approximation in the equation relating theta to x is valid (and can often simplify algebraic steps).

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