Derivation : Law of Reflection and Refraction by Huygens Principle - Param Himalaya

Derivation : Law of Reflection and Refraction by Huygens Principle - Param Himalaya

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 : 

Derivation of Laws of Refraction (Snell's law) from Huygens' Principle

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 : 

Derivation of Laws of Refraction (Snell's law) from Huygens' Principle

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

Derivation of Laws of Reflection (Snell's law) 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

$AC \sin i = AC \sin r$

$\sin i = \sin r$

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