ChemPhys 173/273

Unit 5: Light and Color

Problem Set A

Overview:

Problem Set A targets your ability to interpret and analyze two-point source interference patterns for water waves, sound waves and (primarily) light waves. There are 20 problems on the set which focus upon the proper use of two primary equations pertaining to two-point source interference patterns.

The diagram at the right depicts an interference pattern produced by two periodic waves (water waves, sound waves, light waves, etc.). The crests are denoted by the thick lines; and the troughs are denoted by the thin lines. Constructive interference occurs wherever a thick line meets a thick line or a thin line meets a thin line; this type of interference results in the formation of an anti-node. The anti-nodes are denoted by a red dot. Destructive interference occurs wherever a thick line meets a thin line; this type of interference results in the formation of a node. The nodes are denoted by a blue dot. The pattern is a standing wave pattern, characterized by the presence of nodes and anti-nodes which are "standing still" - i.e., always located at the same position on the medium. The anti-nodes (points where the waves interfere constructively) seem to be located along lines - creatively called anti-nodal lines. The nodes also fall along lines - called nodal lines. The two-point source interference pattern is characterized by a pattern of alternating nodal and anti-nodal lines. The central line in the pattern - the line which bisects the line segment which is drawn between the two sources is an anti-nodal line. This "central anti-nodal line" is a point where the waves from each source reinforce each other by means of constructive interference.

The nodal and anti-nodal points in the interference pattern result whn light from two different sources travel two different distances to the same point in the pattern in such a manner that destructive or constructive interference occurs at that point. The fact that the two waves travel two different distances means that there is a path difference - a difference in distance traveled by the two waves from the source. If the path difference is equal to some whole number of wavelengths, then a crest from one source will meet a crest from the other source (or a trough will meet a trough) and constructive interference will occur at that point. If the path difference is a half-number of wavelengths, then a crest from one source will meet a trough from the other source and destructive interference will occur at that point. This truth can be expressed mathematically by the following equation

PD = m •

m = 0, 1, 2, 3, ... for anti-nodal lines

m = 0.5, 1.5, 2.5, 3.5 ... for nodal lines

where PD is the path difference, is the wavelength, and m is the so-called order number of the line that the particular point lies along. Since the path difference is the difference in distance traveled from a source to the point on the line (nodal or anti-nodal), the equation above is sometimes written as

PD = | S1P - S2P | = m •

where S1P is the distance from source 1 to the point P and S2P is the distance from source 2 to point P. This equation will be used with regularity for problems 1 -6 of the Problem Set A.

The diagram above depicts the two-dimensional pattern of water waves, sound waves or light waves spreading out over two-dimensional space to create a pattern of nodal and anti-nodal lines. If the sources creating the pattern are monochromatic and coherent light sources then each nodal line would be representative of a collection of dark spots (destructive interference) and each anti-nodal line would be representative of a collection of bright spots (constructive interference). If this pattern were projected onto a screen at a given distance L from the sources, then there would be an appearance of alternating dark and bright spots on the screen. This is depicted in the diagram below.

In the early 19th century, physicist Thomas Young noted that there would be some observable and measurable distances evident in the pattern which would be mathematically related to the wavelength of the laser light. Young's famous equation, shown below, related these measurable quantities to the wavelength of light ().

where d is the separation distance between the sources, L is the distance from the sources to the screen, m is the order number of the nodal or anti-nodal line and y is the distance between the central anti-nodal line and a point on the mth-order nodal or anti-nodal line. The m in the above equation is the same m of the path difference equation and will be a whole number for anti-nodal lines and a half-number for nodal line. Once a point on a nodal or anti-nodal line is selected, m is determined and the measurement of y, d and L can be made. Once made, the wavelength of the light waves can be ca;ci;ated. The above equation is the main equation used in the last 14 problems of problem set A.

To be successful on this problem set, you will have to master the use of the above equations. Additionally, you will need to be able to:

• recognize the relationship between wave speed, wavelength and wave frequency: v = f • .
• read a physical description of a two-point source interference situation and appropriately extract numerical information from it.
• give attention to units and have a plan for converting units in such a manner that the unknown quantity is expressed in the requested units. You should know that: 102 cm = 1 m; 103 mm = 1 m; 109 nm = 1 m; and 1010 Angstroms = 1 m.
• know the meaning of the variables (, y, d, m, L, PD) in the two equations above.
• understand that the spacing between dark spots on the screen is regular and repeating such that the distance between the central anti-nodal position and the third anti-nodal position is roughly three times the distance between the central anti-nodal position and the first anti-nodal position.
• understand the order numbering system used to describe a projected pattern.
• interpret the varying language which is used to describe a pattern (e.g., bright spot, bright fringe, reinforcement point, anti-nodal position, central maximum, first order bright fringe, first brght band)
• practice the habits of a good problem-solver.

The following pages from The Physics Classroom tutorial are highly recommended for understanding two-point source interference patterns and Young's equation.

Anatomy of a Two-Point Interference Pattern | The Path Difference Equation |

View Sample Problem Set.

 Problem Description Audio Link 1 Calculation of the wavelength of water waves from path difference information. 2 Calculation of the wavelength of water waves from path difference information. 3 Calculation of the distance from a point on a nodal line to a source for a two-point source water wave interference pattern. 4 Calculation of the distance from a point on a nodal line to a source for a two-point source sound wave interference pattern. 5 Calculation of the distance from a point on a nodal line to a source for a two-point source radio (light) wave interference pattern. 6 Analysis of a destructive interference phenomenon resulting from radio (light) waves traversing two paths from one source and interfering at the receiver location in a home. 7 Determination of the wavelength of light from information about a two-point source interference pattern projected onto a screen. 8 Determination of the spacing between adjacent bright spots on a two-point source interference pattern projected onto a screen. 9 Determination of the distance from the central maximum of a two-point source interference pattern to a distant bright spot of the pattern. 10 Referring to the previous problem; determination of the distance between two dark spots of a two-point source interference pattern. 11 Analysis of a two-point source interference pattern to determine the distance between the central bright spot and the first bright spot of the pattern. 12 Analysis of a two-point source interference pattern to determine the distance between two distant dark bands of the pattern. 13 Determination of the slit separation distance which is responsible for creating a two-point source interference pattern projected onto a screen. 14 Referring to the previous problem; prediction of the effect of an alteration in the slit separation distance upon the spacing between the central maximum and the first order bright spot. 15 Analysis of a two-point source sound interference pattern to determine the distance between sources. 16 Determination of the distance from the sources to a screen which is responsible for creating a two-point source interference pattern. 17 Analysis of a two-point source sound interference pattern to determine the distance from the sources to a row of students who are observing the pattern of audio nodes and antinodes. 18 Analysis of a two-point source interference pattern to determine the wavelength of light. 19 Analysis of a two-point source interference pattern to determine the number of bright bands between the central maximum and the bright band at a particular location on the pattern. 20 Determination of the slit separation distance which is responsible for creating a two-point source interference pattern projected onto a screen.

Audio Help for Problem: 1 || 2 || 3 || 4 || 5 || 6 || 7 || 8 || 9 || 10 || 11 || 12 || 13 || 14 || 15 || 16 || 17 || 18 || 19 || 20

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