Derivation : Law of Reflection and Refraction by Huygens Principle (Snell's law) - Param Himalaya
Derivation of Laws of Refraction (Snell's law) from Huygens' Principle
(i) Plane wavefront refracted in denser medium :
AB is an incident wavefront striking the interface XY at point A. Refractive index of medium II is greater than refractive index of medium I, $n_2>n_1$.
Let $v_1$ be speed of light in medium I and $v_2$ be the speed in medium II ($v_2 < v_1$). According to Huygens' principle, every point on incident wavefront AB acts as a source of disturbance. Time in which wavelet reaches from B to C is given by
$t = \frac{BC}{v_1} \quad \text{i.e.,} \quad BC = v_1 t$
Draw AD = $v_2 t$ to get a point on the secondary spherical wavefront originating from point A on the incident wavefront. Join CD. Therefore, CD is the refracted wavefront.
$\frac{BC}{AD} = \frac{v_1}{v_2} \quad \dots (i)$
In $\triangle BAC, \quad BC = AC \sin i$
and In $\triangle ACD, \quad AD= AC \sin r$
Substituting these values in eqn. (i), we get,
$\frac{\sin i}{\sin r}= \frac{v_1}{v_2} =\frac{v_1/c}{v_2/c}$
$\frac{\sin i}{\sin r}= \frac{c/v_2}{c/v_1}$
$\frac{c}{v_1}= n_1$
and $\frac{c}{v_2} = n_2$
$\frac{\sin i}{\sin r}= \frac{n_2}{n_1}$
or $n_1 \sin i = n_2 \sin r$
which is Snell's law of refraction
(ii) Plane Wavefront Refracted in rarer medium :
Consider a plane wavefront AB incident on an interface XY of medium I and medium II. Let $n_1$ and $n_2$ be the refractive indices of these media such that $n_2 < n_1$ and $v_2 > v_1$
According to Huygens' principle, every point on incident wavefront acts as a source of new wavelets.
Let wavelet reach from B to C in time $t$ with velocity $v_1$. Then $BC = v_1 t$.
Let velocity of light in rarer medium be $v_2$, then in time $t$, the distance travelled by wavelet in rarer medium is given by
$AD = v_2 t$
Therefore
$\frac{BC}{AD}= \frac{v_1}{v_2}$
In $\triangle ABC, \quad BC = AC \sin i$ and
In } $\triangle ACD, \quad AD = AC \sin r$
Substituting these values in eqn. (i), we get,
$\frac{\sin i}{\sin r}= \frac{v_1}{v_2}$
But $\frac{v_1}{v_2} = \frac{c}{v_2} \times \frac{v_1}{c}$
$\frac{v_1}{v_2} = \frac{n_1}{n_1}$
So $\frac{\sin i}{\sin r} = \frac{n_2}{n_1}$
$n_1 \sin i = n_2 \sin r$
which is Snell's law of refraction.
Derivation of Laws of Reflection from Huygens' Principle
Consider a reflecting surface XY on which a plane wavefront AB is incident at A.
According to Huygens' principle, each point on incident wavefront AB acts as a source of new disturbance. If $v$ is speed of light and $t$ is time taken by wavefront to reach point C from point B, then distance.
Draw a sphere of radius BC with point A as the centre.
Draw CD tangent to this sphere.
Then
$AD = BC = vt$
Right angled triangles ADC and ABC are congruent.
From $\triangle ABC$,
$BC = AC \sin i$
From $\triangle ADC$,
$AD = AC \sin r$
Since
$BC = AD$
or
$AC \sin i = AC \sin r$
$\sin i = \sin r$