Results

__Brewster's angle results:__

Amount of water | Lamp height (corrected) (cm) | Horizontal sample dist. from lamp (corrected) (cm) | Index of refraction |

1 cup | 111.4 | 172.8 | 1.55 |

1/2 cup | 111.4 | 157.8 | 1.42 |

1/4 cup | 111.4 | 152.8 | 1.37 |

There were some minor details for which we needed to adjust our final lamp
height and horizontal sample distance in order to have the most accurate
measurement of Brewster's angle. Others doing this experiment should be
careful to perform similar corrections if necessary. (1) The height
of the sample's surface was subtracted from the true lamp height in order to
have the height of the lamp relative to the surface of the sample. **
Careful!** Make sure that the lamp height measures the vertical distance
from the source of the rays to the location at which the light hits the sample.
(2) Additionally, half of the horizontal length of the sample was added to
the original horizontal distance from the lamp to the sample because our beams
were incident upon the center of the sample. **Careful**! Make
sure that your horizontal sample distance represents the horizontal distance
from the lamp to the location where the light actually touches the sample.
Had we not made these corrections, we would not have obtained the most accurate value of
Brewster's angle.

__Gel Lens results:__

Assuming that our gel lenses are thin lenses in air, we can apply the lens-maker equation:

In our case, we only have one radius of curvature since the other side can be assumed to be flat (the water meniscus had negligible curvature for wide watch glasses). This equation therefore allows us to connect our measurement of n and f to obtain an R. Moreover, it also allows us to make sense of the inversely proportional relation clearly present between the index of refraction and the focal length.

Amount of water | Concentration | n (from above) | Focal Length | R=(n-1)*f |

1 cup | 0.25 | 1.55 | 15.03 | 8.27cm |

1/2 cup | 0.5 | 1.42 | 14.3 | 6.01cm |

1/4 cup | 1 | 1.37 | 13.92 | 5.15cm |

Average |
6.47cm |

We also calculated R by measuring the dimensions of the watch glass.

By finding the height and diameter of the watch glass, we build a 2 by 2 system of equations that can be easily solved by a number of methods (such as simple substitution) with R and θ as unknowns. Namely, R sinθ = diameter/2 and R (1-cosθ) = height.

Using a micrometer, we measured a
51.28mm diameter and a 5.33mm height. After the calculations, the R for
these values is 64.12mm, or **6.412cm**,
which is close to the 6.47cm value found from the index of refraction and focal
length measurements. Whether the difference in these values is due to the
gel lenses not being exactly spherical or thin lenses or due to experimental/human error is
up for debate.