

Lesson 1: How Do We Know Light Behaves as a Wave? 
Lesson 3: Mathematics of TwoPoint Source InterferenceYoung's ExperimentThe previous section of Lesson 3 discussed Thomas Young's effort to derive an equation relating the wavelength of a light source to reliably measured distances associated with a twopoint source light interference pattern. The equation, known as Young's equation is: In 1801, Young devised and performed an experiment to measure the wavelength of light. As discussed in the previous part of this lesson, it was important that the two sources of light which form the pattern be coherent. The difficulty confronting Young was that the usual light sources of the day (candles, lanterns, etc.) could not serve as coherent light sources. Young's method involved using sunlight that entered the room through a pinhole in a window shutter. A mirror was used to direct the pinhole beam horizontally across the room. To obtain two sources of light, Young used a small paper card to break the single pinhole beam into two beams, with part of the beam passing by the left side of the card and part of the beam passing by the right side of the card. Since these two beams emerged from the same source  the sun  they could be considered coming from two coherent sources. Light waves from these two sources (the left side and the right side of the card) would interfere. The interference pattern was then projected onto a screen where measurements could be made to determine the wavelength of light. Today's classroom version of the same experiment is typically performed using a laser beam as the source. Rather than using a note card to split the single beam into two coherent beams, a carboncoated glass slide with two closelyspaced etched slits is used. The slide with its slits is most commonly purchased from a manufacturer who provides a measured value for the slit separation distance  the d value in Young's equation. Light from the laser beam diffracts through the slits and emerges as two separate coherent waves. The interference pattern is then projected onto a screen where reliable measurements can be made of L and y for a given bright spot with order value m. Knowing these four values allows a student to determine the value of the wavelength of the original light source.
To illustrate some typical results from this experiment and the subsequent analysis, consider the sample data provided below for d, y, L and m.
(Note: AN_{0} = central antinode and AN_{4} = fourth antinode) The determination of the wavelength demands that the above values for d, y, L and m be substituted into Young's equation. Careful inspection of the units of measurement is always advisable. The sample data here reveal that each measured quantity is recorded with a different unit. Before substituting these measured values into the above equation, it is important to give some thought to the treatment of units. One means of resolving the issue of nonuniform units is to simply pick a unit of length and to convert all quantities to that unit. If doing so, one might want to pick a unit that one of the data values already has so that there is one less conversion. A wise choice is to chose the meter as the unit to which all other measured values are converted. Since there are 1000 millimeters in 1 meter, the 0.250 mm is equivalent to 0.000250 meter. And since there are 100 centimeters in 1 meter, the 10.2 cm is equivalent to 0.102 m. Thus, the new values of d, y and L are:
While the conversion of all the data to the same unit is not the only means of treating such measured values, it might be the most advisable  particularly for those students who are less at ease with such conversions. Now that the issue regarding the units of measurement have been resolved, substitution of the measured values into Young's equation can be performed. = 6.52 x 10^{7} m As is evident here, the wavelength of visible light is rather small. For this reason wavelength is often expressed using the unit nanometer, where 1 meter is equivalent to 10^{9} nanometers. Multiplying by 10^{9} will convert the wavelength from meters to nanometers (abbreviated nm).

y = 
d = 
m = 
L = 
b. An interference pattern is produced when light is incident upon two slits which are 50.0 micrometers apart. The perpendicular distance from the midpoint between the slits to the screen is 7.65 m. The distance between the two thirdorder antinodes on opposite sides of the pattern is 32.9 cm.
y = 
d = 
m = 
L = 
c. The fourth nodal line on an interference pattern is 8.4 cm from the first antinodal line when the screen is placed 235 cm from the slits. The slits are separated by 0.25 mm.
y = 
d = 
m = 
L = 
d. Two sources separated by 0.500 mm produce an interference pattern 525 cm away. The fifth and the second antinodal line on the same side of the pattern are separated by 98 mm.
y = 
d = 
m = 
L = 
e. Two slits that are 0.200 mm apart produce an interference pattern on a screen such that the central maximum and the 10th bright band are distanced by an amount equal to onetenth the distance from the slits to the screen.
y = 
d = 
m = 
L = 
f. The fifth antinodal line and the second nodal line on the opposite side of an interference pattern are separated by a distance of 32.1 cm when the slits are 6.5 m from the screen. The slits are separated by 25.0 micrometers.
y = 
d = 
m = 
L = 
g. If two slits 0.100 mm apart are separated from a screen by a distance of 300 mm, then the firstorder minimum will be 1 cm from the central maximum.
y = 
d = 
m = 
L = 
h. Consecutive bright bands on an interference pattern are 3.5 cm apart when the slide containing the slits are 10.0 m from the screen. The slit separation distance is 0.050 mm.
y = 
d = 
m = 
L = 
Lesson 3: Mathematics of TwoPoint Source Interference
Other Resources: Physics Home Page  Multimedia Physics Studios  Shockwave Physics Studios 
Minds On Physics Internet Modules  The Review Session
© Tom Henderson
19962007