Final answer:
The fraction of the intensity of the original beam associated with polarized light is found to be 340%, which exceeds the logical maximum of 100%. This suggests a possible error in the student's interpretation or experimental setup. The likely correct result is that 100% (a fraction of 1) of the light is fully polarized.
Step-by-step explanation:
To determine the fraction of the intensity of the original beam associated with polarized light, we use the fact that the transmitted intensity through a polarizing filter varies as the square of the cosine of the angle between the light's polarization direction and the filter's axis. The Malus's Law describes this as I = Io × cos²(θ). In the given scenario, the ratio of maximum to minimum intensities is 7.8; thus:
ℓ(max) / ℓ(min) = 7.8 = (Io × cos²(0°)) / (Io × cos²(90°))
Since cos(0°) is 1 and cos(90°) is 0, the minimum intensity (Imin) occurs when the polarized component is aligned at 90° to the filter's axis, and the transmitted intensity is solely due to the unpolarized component. Given that the polarized component at 0° contributes to the maximum intensity, we can express:
ℓ(min) = Io x 1/2 (because unpolarized light intensity is halved)
ℓ(max) = Io (all of the polarized light plus half of the unpolarized light)
7.8 = ℓ(max) / ℓ(min) = [ℓ(polarized) + ℓ(unpolarized)/2] / [ℓ(unpolarized)/2]
By solving this ratio, we find that the polarized fraction, P, is:
P = ℓ(polarized) / Io = 7.8 − 1 = 6.8 times the intensity of the unpolarized light
To find the fraction of the polarized light, we multiply by the factor (1/2) because the polarized component contributes fully to the intensity while the unpolarized has been halved.
Fraction of polarized light = P x 1/2 = 6.8 / 2 = 3.4 or 340%
Since a percentage over 100% would not make sense in this context, the student may have made a mistake in interpreting the results or in the setup of the experiment. The most likely correct conclusion is that a fraction of 1 (or 100%) of the light is fully polarized, given that the maximum intensity observed would be due to 100% of the polarized light plus 50% of the unpolarized light, matching the ratio of 7.8 when compared to the minimum intensity being from 50% of the unpolarized light only.