Human Hair Measurement
Purpose: To use the separation of light rays (interference) to rather accurately measure the thickness of human hair by measuring the distance between the reflected laser rays.
Materials/ tools:
Reflected Surface (whiteboard)
Taped, paper-punched-3x5 index card
Measuring Stick (LONG ruler)
Procedures:
1. Obtain a strand of human hair. Tape it across the hole of the index card.
2. Set the card parallel to the reflective surface, 1 meter apart. Set up the HeNe laser and aim it ONTO the hair strand.
3. The reflective surface would show a diffraction pattern of light. Make sure that the laser ray is orthogonal to the surface. Measure the distance of the light diffraction.
The reflected light ray (center) has trailing, fading light diffraction immediately to its left and right
Data and Uncertainties:
Variables:
L = Distance between the surface to hair strand
y = Distance between light diffraction
λ = laser's wavelength
d = hair thickness
λ = d y / L
λ = 632.8 nm = 632.8 * 10^-9 m (Since it is the standard wavelength of HeNe laser, it is assumed accurate, and has no uncertainties)
Helium Neon laser is assumed to have a fixed wavelength... whoa!
Experiment 1:
L = 1.00 +/- 0.05 m
y = 0.25 +/- 0.1 cm = 0.0025 +/- 0.001 m (large uncertainty due to difficulty measuring closely-spaced diffraction using large measuring stick)
d = λ L / y
d = 2.53 * 10^-4 m = 2.53 * 10^-1 mm = 253 μm
Experiment 2:
L = 1.02 +/- 0.05 m
y = 0.0052 +/- 0.001 m
d = 1.24 * 10^-4 m = 124 μm
Uncertainty by partial derivatives
d(y, L) = λ L / y
∂d/ ∂L = 1/ uy
∂d/ ∂y = uL * ln (y)
ud = ((∂d / ∂L * uL)^2 + (∂d / ∂y * uy)^2 ) ^(0.5)
ud = ((1/uy * uL)^2 + (uL * ln (uy))^2) ^(0.5)
Experiment 1:
ud = ((1/ 0.001 * 0.05)^2 + (0.05 * ln(0.0025) * 0.001)^2)^0.5
ud = (2500 + 0)^0.5
ud = 50
Experiment 1 = 254 μm +/- 50 m = 254 +/- 50000000 μm
This uncertainty value makes no sense. 2nd method of uncertainty will be used by adding up all the uncertainties:
Uncertainty by adding up all the uncertainties:
Experiment 1 and 2: uL = 0.05 m, uy = 0.001m, uλ = 0
ud = Σ_uncertainties = 0.051
Experiment 1:
d = 2.53 * 10^-4 +/- 0.051 m = 253 +/- 51000 μm
Experiment 2:
d = 124 +/- 51000 μm
The average human hair thickness ranges from 40 to 250 μm, according to http://wiki.answers.com/Q/What_is_the_average_thickness_of_a_human_hair
Our experiment yields value of about 120 and 250 μm, which falls within the expected values (although the uncertainties are rather outrageously substantial).
Error analysis:
One of the possible cause of errors is the measuring of the diffraction. Since the diffraction was spread very closely to each other and the most intense ray (middle ray) outshone the other rays, it was highly possible that there was an additional ray hidden under the brightest ray. Miscounting one ray could throw off the data by magnitudes of ten.
Another possible error is the position of the laser and the surface. During the experiment, it was merely being approximated that they were perpendicular to each other. If they were not truly perpendicular, then the distance, L, was not the true value.
Another yet possible error is the condition of hair itself. It is possible that the person whom hair was being experimented on used a thickening hair product, causing the hair to be thicker than usual. One of the person was a female, so the chances of hair thickener being used would be higher.
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