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Explain the derivation of Snell's Law.
Snell's Law
Snell’s law, also called as law of refraction or Snell’s Descartes. It is defined as "the ratio of the sine of the angles of incidence to the sine of the angle of refraction is a constant, for the given pair of media".
The formula is expressed as:
$\frac{\sin\ i}{\sin\ r}=\mu =constant$ $=refractive\ index\phantom{\rule{0ex}{0ex}}$
where, i = angle of incidence,
r = angle of refraction, and
$(\mu)$ = the constant value, called the refractive index of the second medium with respect to the first.
So, it can also be given as-
$\frac{Sin{\theta }_{1}}{Sin{\theta }_{2}}=\frac{{n}_{2}}{{n}_{1}}\ or \frac{Sin{\theta }_{1}}{Sin{\theta }_{2}}=\frac{{v}_{2}}{{v}_{1}}$
${n}_{1}\ and\ {n}_{2}=refractive\ indices\ of\ two\ different\ media$
$\theta_{ 1} =incident \ angle$
$\theta_{ 2} =refracted \ angle$
${v}_{1}\ and\ {v}_{2}=phase\ velocities\ of\ two\ different\ media$
Derivation
Basically, Snell’s law formula is derived from Fermat’s principle.
Fermat’s principle states that "the light travels in the shortest path and it has less travelling time".
Now we consider a light ray travelling from point P to point Q in media with different indices of refraction, as shown in the figure. The time to travel between the two points is the distance in each medium divided by the phase velocity (speed of light in that medium).
Phase velocities in the two medium are represented as-
${v}_{1}=\frac{c}{{n}_{1}}\ and\ {v}_{2}=\frac{c}{{n}_{2}}$
Here, 'c' represents the speed of light in vacuum.
${v}_{1}\ and\ {v}_{2}=phase\ velocities\ of\ two\ different\ media$
${n}_{1}\ and\ {n}_{2}=refractive\ indices\ of\ two\ different\ media$
Let's assume that T be the time required by the light to travel from P through point O to point Q.
$T=[\frac{(\sqrt[]{{a}^{2}+{x}^{2}})}{{v}_{1}}]+[\frac{(\sqrt[]{{b}^{2}+(l-x{)}^{2}})}{{v}_{2}}]$
$T=[\frac{(\sqrt[]{{a}^{2}+{x}^{2}})}{{v}_{1}}]+[\frac{(\sqrt[]{{b}^{2}+{l}^{2}-2lx+{x}^{2}})}{{v}_{2}}]$
where $a$, $b$, $l$ and $x$ are as denoted in the figure given below, $x$ being the varying parameter.
To minimize the time we set the derivative of the time with respect to $x$ equal to
zero. We also use the definition of the sine as opposite side over hypotenuse to
relate the lengths to the angles of incidence and reflection.
$\frac{dT}{dx}=\frac{x}{{v}_{1}\sqrt{{x}^{2}+{a}^{2}}}+\frac{-(l-x)}{{v}_{2}\sqrt{(l-x{)}^{2}+{b}^{2}}}=0$ (stationary point)
Note that, $\frac{x}{\sqrt{{x}^{2}+{a}^{2}}}=sin{\theta }_{1}$ and $\frac{x}{\sqrt{(l-x{)}^{2}+{b}^{2}}}=sin{\theta }_{2}$
$\frac{dT}{dx}=\frac{\sin{\theta }_{1}}{{v}_{1}}-\frac{\sin{\theta }_{2}}{{v}_{2}}=0$
$\frac{\sin{\theta }_{1}}{{v}_{1}}=\frac{\sin{\theta }_{2}}{{v}_{2}}$
By substituting the phase velocity equation, we get-
$\frac{{n}_{1}\sin{\theta }_{1}}{c}=\frac{{n}_{2}\sin{\theta }_{2}}{c}$ [$\because{v}_{1}=\frac{c}{{n}_{1}}\ and\ {v}_{2}=\frac{c}{{n}_{2}}$]
${n}_{1}\sin{\theta }_{1}={n}_{2}\sin{\theta }_{2}$ .............. This is the final derived Snell’s law equation.
Note:- Calculas is used to derive the Snell's Law. (not in the syllabus of class-10th)
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