Answer to Question #177009 in Optics for Manisha Nayak

Question

State and explain fermat's principle. Using this principle derive shell's law of refraction .

Answer 1

Fermat's principle: " The actual path of propagation of light (trajectory of a light ray) is the path which can be followed by the light within the least time, in comparison with all other hypothetical paths between the same two points.''

''Above statement is the original wordings of Fermat (A famous French scientist of 17th century)"

Deduction of Snell's law of refraction from Fermat's principle:

As shown in the figure to the right, assuming the refractive index of medium 1 and medium 2 are ${\displaystyle n_{1}}$  and ${\displaystyle n_{2}}$  respectively. Light enters medium 2 from medium 1 via point O.

${\displaystyle \theta _{1}}$ is the angle of incidence, ${\displaystyle \theta _{2}}$ is the angle of refraction with respect to the normal.

The phase velocities of light in medium 1 and medium 2 are

${\displaystyle v_{1}=c/n_{1}}$  and

${\displaystyle v_{2}=c/n_{2}}$  respectively.

${\displaystyle c}$ is the speed of light in vacuum.

Let T be the time required for the light to travel from point Q through point O to point P.

${\displaystyle T={\frac {\sqrt {x^{2}+a^{2}}}{v_{1}}}+{\frac {\sqrt {b^{2}+(l-x)^{2}}}{v_{2}}}={\frac {\sqrt {x^{2}+a^{2}}}{v_{1}}}+{\frac {\sqrt {b^{2}+l^{2}-2lx+x^{2}}}{v_{2}}}}$

where a, b, l and x are as denoted in the figure above, x being the varying parameter.

To minimize it, one can differentiate :

${\displaystyle {\frac {dT}{dx}}={\frac {x}{v_{1}{\sqrt {x^{2}+a^{2}}}}}+{\frac {-(l-x)}{v_{2}{\sqrt {(l-x)^{2}+b^{2}}}}}=0}$  (stationary point)

Note that ${\displaystyle {\frac {x}{\sqrt {x^{2}+a^{2}}}}=\sin \theta _{1}}$ and ${\displaystyle {\frac {l-x}{\sqrt {(l-x)^{2}+b^{2}}}}=\sin \theta _{2}}$

Therefore,

$\Rightarrow {\displaystyle {\frac {\sin \theta _{1}}{v_{1}}}={\frac {\sin \theta _{2}}{v_{2}}}}$

$\Rightarrow{\displaystyle {\frac {n_{1}\sin \theta _{1}}{c}}={\frac {n_{2}\sin \theta _{2}}{c}}}$

$\Rightarrow{\displaystyle n_{1}\sin \theta _{1}=n_{2}\sin \theta _{2}}$

Hence, $\dfrac{sin{\theta_1}}{sin{\theta_2}}=\dfrac{n_2}{n_1}$ which is the Snell's Law.