**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

**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

where **S _{1}P** is the
distance from source 1 to the point P and

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: 10
^{2}cm = 1 m; 10^{3}mm = 1 m; 10^{9}nm = 1 m; and 10^{10}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.

**Additional Readings/Study
Aids:**

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

Young's Equation | Young's Experiment | Other Applications

View Sample Problem Set.

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