Back To CoursePhysics 101: Help and Review
16 chapters | 195 lessons
Aaron teaches physics and holds a doctorate in physics.
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 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 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).
The variable m takes on different integer values corresponding to different dark spots on the viewing screen. For example, the dark spot corresponding to m=0 is closest to the center of the viewing screen. Setting m=0 will yield a positive x, which indicates the corresponding dark spot is slightly above the center of the screen. Other dark spots can be extrapolated from this reference point and can be seen in Figure 3.
There is one assumption that is very important when applying this equation -- it is assumed that the distance to the viewing screen L is much bigger than the distance between the slits d. Other than this restriction (which all introductory textbooks make, otherwise there is no simple relation between the parameters of the experiment) the equation applies to any kind of monochromatic wave source, implying that points of destructive interference appear on the viewing screen for any type of wave, including sound waves, light waves or radio waves.
There is a similar equation for the positions of bright spots of the interference pattern:
The angle theta is related to x and L in the same way, and again, m takes on integer values corresponding to observed bright spots. m=0 corresponds to the bright spot in the center of the viewing screen. However, this equation is not as useful in practice because we are generally more interested in the locations where a traveling wave is destroyed by interference, since this is what, for example, would cause an interruption in a plane's communications.
In this lesson, we discussed the most basic scenario where destructive interference occurs -- a double slit experiment. The 'dark' spots of the interference pattern are where an incoming signal in the form of a wave cannot be detected. In the double slit experiment, these points are predicted by an equation that we discussed how to use. The locations of the points depend on the wavelength (or frequency) of the interfering wave sources (the slits) and the distance between these sources.
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Back To CoursePhysics 101: Help and Review
16 chapters | 195 lessons
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