When defined as an angle at which an incident

When light is incident on a transparent solid material, one
part of it gets reflected and another part gets refracted. If light is incident
on the interface between the two media such that there is 90° angle between the
reflected and refracted rays, the reflected light will be linearly polarized.

Brewster’s angle, or the
polarizing angle, is defined as an angle at which an incident beam of
unpolarized light is reflected after complete polarization.1

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Background

In his studies on polarized light, Brewster discovered that when light
strikes a reflective surface at a certain angle, the light reflected from that
surface is plane-polarized. He elucidated a simple relationship between the
incident angle of the light beam and the refractive index of the reflecting
material. When the angle between the incident beam and the refracted beam
equals 90 degrees, the reflected light becomes polarized. This rule is often
used to determine the refractive index of materials that are opaque or
available only in small quantities.2

Brewster’s Angle and Polarized Light

When considering the incidence of non-polarized light on a flat
insulating surface, there is a unique angle at which the reflected light waves
are all polarized into a single plane. This angle is commonly referred to
as Brewster’s angle, and can be easily calculated utilizing the
following equation for a beam of light traveling through air:

n = sin (?i)/sin
(?r) = sin (?i)/sin (?90-i) = tan (?i)

Where n is the refractive index of the medium from
which the light is reflected, ?i is the angle of
incidence, and ?r is the angle of refraction. By
examining the equation, it becomes obvious that the refractive index of an
unknown specimen can be determined by the Brewster angle. This feature is
particularly useful in the case of opaque materials that have high absorption
coefficients for transmitted light, rendering the usual Snell’s law formula
inapplicable. Determining the amount of polarization
through reflection techniques also eases the search for the polarizing axis of
a sheet of polarizing film that is not marked.

For water (refractive index of 1.333) and glass (refractive index of
1.515 the critical (Brewster’s) angles are 53 and 57, degrees, respectively.

Hypothesis/Theory:

“When a beam of unpolarized light reflects
from a surface at the Brewster angle, the reflected beam will be polarized
along a direction parallel to the surface. The Brewster angle is the angle of
incidence that results in a 90° angle between the reflected and refracted
beams.”3

It is hypothesised that as the refractive index changes it will affect
Brewster’s angle.

Design and Methodology:

Equipment;

·        
Laser wavelength: 630-680 nm, power: < 1mw ·         Light intensity sensor ·         Protractor ·         G-Clamp ·         Polarizers ·         Wooden Blocks (to use to raise sensor) ·         Different transparent materials Ø  Lead glass Ø  Lactose Ø  Fructose Ø  Glucose Ø  Perspex Ø  Albumen Set up: In figure 1 below is the set-up of the experiment.                                                                           The apparatus mainly consists of a laser (pointer, in this case) in line with the transparent slab. 2 polarisers in front of the laser as to only have the s-wave passing through to the slab. Then a power meter or some form of light intensity sensor to the left of the slab, from the lasers point of view. In-between the meter and the transparent slab would be another polariser to, again, ensure only the s-wave is passing through as this experiment is based off this specific wave in the reflection.                                                                                                            This apparatus set up involves the same equipment to the above experiment. However instead of having the laser on the same desk (work top) it is suspended as to allow the laser light t bounce off of the surface of the liquid below. The light intensity sensor would also move accordingly as to continue recording the intensity of the light reflected. Variables:                                                 Independent variable(s): the refractive index Dependent Variable(s): Brewster's angle Controlled variable(s);   Procedure: 1. Transparent slabs of different refractive index values are used. 2. The prisms are placed to read 0° on the scale (protractor). 3.  The laser is placed directly in front of the transparent slab. There are to be 2 polarisers in front of the beam as well as in front of the light sensor to test for the s-waves. Then the laser is switched on and the glass slab is perpendicular to the laser beam. 4. A white screen is placed on the right hand side parallel to the of the laser beam. 5. The prism is rotated in the clockwise direction and the reflected ray from the front surface and refracted-reflected ray from back surface of the glass slab are observed on the screen. 6. The more intense (brighter) spot is due to the reflected ray (first spot from the left) and the other one is the reflected-refracted ray from the back surface of the glass slab. Any data recorded would be from the first more intense spot. 7. As the prism glass slab is rotated further, variation in the intensity of the reflected ray is noticed. This indicates that the laser has been placed properly with its 's' component normal to the table. 8. The light sensor provided along with the instrument is now clipped to the screen and connected to the light intensity meter. The sensor is positioned exactly under the reflected spot (the first spot) and relative intensity of the spot is recorded.  9. The prism is rotated in steps of 5° until 50° and then rotated in steps of 1° until 60°. This is to make sure that the drastic change in intensity is blatantly obvious when nearing Brewster's angle. 10. Recording of the intensity as the angle changes. 11. The experiment is repeated using transparent acrylic slabs (etc…). Furthermore when dealing with the liquids: 12. Clamp the laser to a large wooden board with a G-clamp. 13. Then uses a form of leverage and tilt the wooden board upwards to change the angle in which the laser hits the surface of the transparent liquids. To allow the laser light to bounce off the surface of the transparent liquid. 14. Repeat the above steps for solid transparent materials, however with the laser changing angle instead of the slabs. To be as accurate as possible one would have to use a protractor or scale of measuring angle that either shows to a significant number of decimal places or accurate as to reduce human error/parallax error. However to be as precise as possible one must continuously use the same equipment and same environment throughout the whole of the experiment as to not have random error.   Brewster's angle is governed by the well-known Snell's law of refraction n1 Sin i = n2 Sin r      Where n1 is refractive index of the first medium (air), which is equal to unity, and n2 = refractive index of the second medium (glass): n2 = n (n being the refractive index of glass with respect to air), i = angle of incidence, and r = angle of refraction. The equation, therefore, becomes         Sin i = n Sin r     When the angle between the reflected ray and refracted ray becomes 90°, the angle of incidence is called the Brewster's angle. i+ r +90° = 180° , or i +r = 90° , or  r = 90°-i Sin i = n Sin (90°-i), or Sin i = n Cos i, which gives Tan i= n Tan ?B = n,                           Or                           ?B=Tan-1n   Risk assessment:   Data analysis and discussion: Raw Data       Throughout the gathered data is has been noted that only one set of data truly reached zero on the light intensity sensor. I predict this may be due to the uncertainty of the device as well as a possible contribution of an error due to faults in my set-up and/or in the equipment itself. Analysis: A graph is drawn with the angle of rotation on the X-axis and relative light intensity on the Y-axis. Refractive index of the material is calculated by using; Tan ?B = n            at the angle of the minimum intensity/ Brewster's angle (?B). The origins of this equations is derived from the Fresnel equations. "The Fresnel equations predict that light with the p polarization (electric field polarized in the same plane as the incident ray and the surface normal) will not be reflected if the angle of incidence is Where n1 is the refractive index of the initial medium and n2 is the refractive index of the other medium." "The physical mechanism for this can be qualitatively understood from the manner in which electric dipoles in the media respond to p-polarized light. One can imagine that light incident on the surface is absorbed, and then re-radiated by oscillating electric dipoles at the interface between the two media. The polarization of freely propagating light is always perpendicular to the direction in which the light is travelling. The dipoles that produce the transmitted (refracted) light oscillate in the polarization direction of that light. These same oscillating dipoles also generate the reflected light. However, dipoles do not radiate any energy in the direction of the dipole moment. If the refracted light is p-polarized and propagates exactly perpendicular to the direction in which the light is predicted to be specularly reflected, the dipoles point along the specular reflection direction and therefore no light can be reflected."4 "With simple geometry this condition can be expressed as;                                 *where ?1 is the angle of reflection (or incidence) and ?2 is the angle of refraction* Then with Snell's law;                    One can calculate the incident angle ?1 = ?B at which no light is reflected: Therefore solving for  gives; With the previously shown tables of data gathered during the experiment, a collection of graph were made in order to not only present the data for better analysing but to show any forms of a trend. (The minimum angle at the smallest intensity is marked with an orange marker)In this graph (Graph 1) we see what is expected of the light intensity when the angle is changing. It is expected that the light intensity to decrease as we get closer to Brewster's angle as the light is becoming more polarized into the 's' plane.   The same observations can be seen in this graph (Graph 2) as well and is producing the same patterned, with light intensity decreasing the closer it is to Brewster's angle and start to increase back to its normal intensity the father it goes over the angle.                                        The same as the above description can be applied to all graphs (Graphs 3-6) as the show a similar, if not the same, trend.   The small bars on the outside of the plotted data represent the uncertainty in my data. They are small uncertainties compared to my data however they affect my results nonetheless as it does change it by 0.5 lux or degrees which can lead to come changes in the values. This can possibly lead to my data being slightly off from the 'accepted' values. However going through some of the graph the pattern is not the same, but simply similar in all the transparent material I have used for my experiment. These changes in the angle for my materials is due to the densities and refractive index of the material. These values generally indicate whether light will pass through it with little to no difficulty, which allows the light to maintain much of its original intensity. Or if the material will absorb the light forcing it to weaken thus leading to weaker readings on the light sensor.    Finally after finding Brewster's angle for each transparent material we are able to compare that to the refractive index via the equation; 1/n1*n2. This makes a graph that looks like this:    This graph then shows that there is a moderate, if not strong, correlation. At a correlation (Pearsons Moment Correlation Coefficient, r) of r ? 0.81 (to 2 significant figure) Uncertainty: Some uncertainties to my data would be from the measuring equipment, mainly the light intensity sensor and protractor, such as the light sensor and the protractor in measuring the light intensity and angle. When looking for the percentage uncertainty in my data I first have to find absolute uncertainty then the fractional uncertainty which will then be multiplied by 100 to get the percentage. 1)   This value means that of my gathered data the overall uncertainty is 0.1%, approximately, suggesting these values may be inaccurate by this much.  Also the type of light source I have is also important as the source can affect the results due to dispersion. As such using a laser light is much simpler compared to that obtained by using a spectrometer and white light. Hence at the Brewster's angle of incidence the intensity of reflected light becomes zero. Also laser lights are more efficient because more of the energy used to create the light is focused in the beam while the average white light's energy is more likely to disperse creating unwanted background light. These uncertainties have been added to my graphs as error bars which show the uncertainty of ± 0.5 for both the light censor and the protractor. Conclusion: In this experiment Brewster's law was determined through the methods of manipulating Snell's law and finding the angles in which light intensity reaches its lowest value. After there were plans to, later, compare Brewster's angle to the refractive index. The value obtained for the refractive index of my materials was through the use of the equation Tan ?B = n, then compared them to values taken from a trusted physics websites, for example; nPerspex=1.486. This gives me some data to compare my results to, in order to see how accurate they are. Therefore, when everything was correctly graphed for analysis I concluded that my experiment had mostly supported the hypothesis. The graphs showed the expected inverse parabola shape for each transparent material as well as the expected positive linear relationship between the refractive index and Brewster's angle. However this does not completely support the hypothesis as the graph between the refractive index and Brewster's angle was to have a line of best fit that should pass through the point (0,0), which it did not. There is a downward shift of the data which would show low accuracy of the light intensity.  However, it is assumed that this is mainly due to systematic errors. Nevertheless, the precision may be improved by having done some repeats and gathering more data. This would allow for an average to be taken and the precision to be heightened. Brewster's angle was identified to be qB = (56.0 ± 0.5°), which gave a value for nPerspex in agreement with the accepted value.  To conclude I have observed a very strong representation of the expected graphing trends which leads me to believe that my experiment has gone quite well as not only does the final values of my data match or at least are similar to that of the accepted values but the final graph which compares the refractive index to Brewster's angle show the correct general outline expected, but obviously not completely what was expected. This however is presumed to be the fault of error, mainly systematic.     Evaluation: Numerous sources of error within an experiment can have a significant impact on both the accuracy and precision of my results. Firstly, the light intensity sensor is something that had to be moved around when trying to read the intensity as the spot of reflection moved as well. Due to this there is error in having the sensor directly in line with the reflected beam of light and measuring its intensity. Therefore, the values recorded may not completely represent the lights intensity accurately. The light intensity sensor may further be affected due to background light. Though I was able to significantly reduce this, I wasn't able to be completely rid of it. Light was able to enter through the crack under the door. There was also light that was coming from an online computer screen near-by. Though these lights are not very strong they could still affect the light sensor and therefore affecting my data. However the light intensity sensor was not the only piece of equipment to move, as the laser itself was moved as well when recording for the liquids as well as transparent slabs were also moved. This could affects the results as well considering as the transparent slabs were moved tin order to change the angle at which the laser light was hitting it the slabs may have been moved from being in line with the light therefore effecting the intensity of the light being reflected.  When recording data for the transparent liquid and having to clamp the laser to a wooden board in order to have the light come down onto the water (Figure 2) there is the possibility the  laser was jostled thus causing a possible shift leading to the reflected light to become less powerful therefore effecting the results of intensity. Next there was the protractor that was used to measure the angle at which the transparent materials (solid or liquid) were reflecting the laser light at. This a parallax error as the values shown by the instrument can be 'under read' or 'over read', as well as the values can appear to change at different angles. Furthermore, there is the systematic error of the light intensity sensor, considering that it only presented a whole number. The values are being rounded to the nearest whole instead of presenting decimal places which would make the values more accurate. This can affect the accuracy as the uncertainty becomes bigger than if it did show values to even one or two decimal places. Furthermore, there are limitations, just, within my data. By this it is the lack of repeats, in order to minimise errors, as the data gathered for the experiment the first time was time consuming and there was a large quantity. Due to the lack of repeats an average of the data is not able to be taken and therefore outliers are not really able to be identified. To improve the accuracy and precision of my results the apparatus would be the most likely of places to start. The first improvement of the equipment used would be the light sensor. Having a specific brand of a light sensor limits not only how well it detects the intensity but also how well it picks up the light reflected. Using one of the later models that records the intensity as the light is reflected instead of placing the sensor in front of the light would not only be more accurate and reduce errors but it would make the experiment easier, allowing more time to do repeats. Next the protractor, which presented a high uncertainty, could be improved by either using a new high end turn table that has the angles already engraved into it, which will not only improve the protractor uncertainty but the problem of moving the transparent slab and causing greater uncertainty. Or to us a larger protractor that shows the angles in more depth, such as going in-between 2 whole numbered angle lines. The set-up of my apparatus would be the next step in improving my results and reducing any error. First of all, the most concerning about my set up was clamping the laser to a wooden board to give leverage to cause a reflection with the transparent liquids. I would alter this as to have the laser suspended securely so that it had a little to no chance of being jostled when there is a change in angle.