View source for Optics/Fermat's Principle

==Derivation for Law of Reflection==
The derivation of Law of [[Reflection]] using [[Fermat's principle]] is straightforward. The Law of [[Reflection]] can be derived using elementary [[calculus]] and [[trigonometry]]. The generalization of the Law of Reflection is [[Snell's law]], which is derived bellow using the same principle.

The medium that light travels through doesn't change. In order to minimize the time for light travel between to points, we should minimize the path taken.

[[Image:Lawofreflection.png|thumb|right|&theta;<sub>i</sub> = &theta;<sub>r</sub>.<br/>the angle of incidence equals the angle of reflection]]

1. Total path length of the light is given by

:<math>L=d_1+d_2\,</math>

2. Using [[Pythagorean theorem]] from [[Euclidean Geometry]] we see that

:<math>d_1=\sqrt{x^2 + a^2}\,</math> and <math>d_2=\sqrt{(l-x)^2 + b^2}\,</math>

3. When we substitute both values of d<sub>1</sub> and d<sub>2</sub> for above, we get 

:<math>L=\sqrt{x^2 + a^2} + \sqrt{(l-x)^2 + b^2}</math>

4. In order to minimize the path traveled by light, we take the first [[derivative]] of L with respect to x. 

:<math>\frac{dL}{dx}=\frac{x}{\sqrt{x^2 + a^2}} + \frac{-(l-x)}{\sqrt{(l-x)^2 + b^2}}=0</math>

5. Set both sides equal to each other.

:<math>\frac{x}{\sqrt{x^2 + a^2}} = \frac{(l-x)}{\sqrt{(l-x)^2 + b^2}}</math>

6. We can now tell that the left side is nothing but <math>\sin\theta_i</math> and the right side <math>\sin\theta_r</math>  means

:<math>\sin\theta_i=\sin\theta_r \!\ </math>

7. Taking the [[Inverse trigonometric function|inverse sine]] of both sides we see that the angle of incidense equals the angle of reflection

:<math>\theta_i=\theta_r \!\ </math>

==Derivation for Snell's Law==

The derivation of Snell's Law using [[Fermat's Priciple]] is straightforward. Snell's Law can be derived using elementary [[calculus]] and [[trigonometry]]. Snell's Law is the generalization of the above in that it does not require the medium to be the same everywhere.

[[Image:Snellslaw_diagram.png|thumb|right|Snell's law relates n<sub>1</sub>, &theta;<sub>1</sub> and n<sub>2</sub>, &theta;<sub>2</sub>.]]

To mark the speed of light in different media refractive indices named n<sub>1</sub> and n<sub>2</sub> are used. 
:<math>v_1=\frac{c}{n_1} \!\ </math>
:<math>v_2=\frac{c}{n_2}</math>

Here <math>c</math> is the [[speed of light]] in the vaccuum and <math>n_1,n_2 \ge 1 \,</math> because all materials slow down light as it travels through them. 

1. Time for the trip equals distance traveled divided by the speed.

:<math>t=\frac{d_1}{v_1}+\frac{d_2}{v_2}</math>

2. Using the  [[Pythagorean theorem]] from [[Euclidean Geometry]] we see that

:<math>\frac{d_1}{v_1}=\frac{\sqrt{x^2 + a^2}}{v_1}\,</math> and <math>\frac{d_2}{v_2}=\frac{\sqrt{b^2 + (l-x)^2}}{v_2}\,</math>

3. Substituting this result into equation (1) we get 

:<math>t=\frac{\sqrt{x^2 + a^2}}{v_1} + \frac{\sqrt{b^2 + (l-x)^2}}{v_2}</math>

4. We take the spacial derivative with respect to time. This may seem strange at first but if we then take the [[Multiplicative inverse|reciprocal]] of <math>\frac{dt}{dx}</math>,we end up with <math>\frac{dx}{dt}</math> which is what we are looking for.

:<math>\frac{dt}{dx}=\frac{x}{v_1\sqrt{x^2 + a^2}} + \frac{-(l-x)}{v_2\sqrt{(l-x)^2 + b^2}}=0</math>

5. After careful examination the above equation we see that it is nothing but

:<math>\frac{dt}{dx}=\frac{\sin\theta_1}{v_1} - \frac{\sin\theta_2}{v_2}=0</math>

6. Taking the [[Multiplicative inverse|reciprocal]] of both sides we get 

:<math>\frac{dx}{dt}=\frac{v_1}{\sin\theta_1} - \frac{v_2}{\sin\theta_2}=0</math>

7. Setting both sides equal to each other we get 

:<math>\frac{v_1}{\sin\theta_1}=\frac{v_2}{\sin\theta_2}</math>

8. Multiplying both sides by <math>\sin\theta_1\sin\theta_2</math> we get

:<math>v_1\sin\theta_2=v_2\sin\theta_1 \!\ </math>

9. Substituting <math>\frac{c}{n_1}</math> for v<sub>1</sub> and <math>\frac{c}{n_2}</math> for <math>v_2</math> we get

:<math>\frac{c}{n_1}\sin\theta_2=\frac{c}{n_2}\sin\theta_1</math>

10. Simplifying both sides we get our final result
:<math>n_1\sin\theta_1=n_2\sin\theta_2 \!\ </math>




[[Category:Optics]]