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# Snell's Law Project (Term Paper Sample)

Instructions:

The task was to investigate the Snell\'s Law and verify it mathematically. This was a physics term paper. source..

Content:

Name
Instructor
Course
Date
Snell's Law Project
Snell's Law is also called the Law of Refraction. It explains the relationship between the angles of refraction and incidence for a wave hitting on an interface between 2 media with different refraction indices. The objective of the project was to verify Snell's Law using a mathematical model.
left188788 In this case,
n1 & n2 are the refractive indices of media 1& 2 in that order.
While Î¸1 & Î¸2 represent incident & refracted angles respctively.
(#1) Assume we have a light source at the point A (-1,3) and the x-axis is a mirrored surface.
A ray of light travels from A toward the origin and ends up at B (1, âˆš3).
For a perfect mirror, i=r. In the above diagram, i=r=30â°
Tan (30â°) = oppadj= 1âˆš3
(#2) Optimization
Assume the light source is at A (0, 2). If the ray bounces off the x-axis and end up at B (4, 1)
Let x be the coordinate on the x-axis
Since I =R, we can find the total distance
Travelled by the light wave.
Let X be the point at which the incident
Ray hits the x-axis before bouncing off.
OX = âˆšx2 + 22
BX = âˆš(4-x)2 + 12
* The total distance is a function of x (0â‰¤xâ‰¥4)
fx= âˆšx2 + 22 + âˆš(4-x)2 + 12 = âˆšx2 + 4 + âˆšx2 - 8x + 17
f' (x)=xâˆš(x2-4) + x-4âˆš(x2-8x+17 = 0â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦.. (i)
Rearranging the equation (i) we get.
xâˆš(x2-4) = x-4âˆš(x2-8x+17
Cross-multiplying the equation and squaring both sides to eliminate the root sign, we get
x2(x2-8x+17)-(4-x)2(x2-4)
x4- 8x3+ 17x2=(16-8x+x2) (x2-4) = 16x2âˆ’64 âˆ’ 8x3 âˆ’ 32x+ 4x4
17x2= 20x2 âˆ’ 32x + 64
3x2 âˆ’ 32x + 64 = 0â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦ (ii)
This is a quadratic equation which can be solved using the quadratic formula,
x=-bÂ±b2-4ac2a, where a=3, b=-32, c= 64
Substituting for a, b, & c in the above equation, we obtain
x1,2 = 32 Â± 166 =
x1 =8 & x2 = 83
x=83
*
369697020447000-335009430543500
180892323116800-255460550355500 ...

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