Name: Waves and Their Phases
Uploaded: 2019-07-19
Duration: 6 min 51 s
Description: In this video, we will learn how two or more waves can be studied in terms of their phases. Equations of two waves and graphs of two waves. Learn more!

Question

Derive Snell's law of refraction using Huygen's principle.

Accepted Answer

There are two laws of refraction of light. The first is also called the Snell's law.

Snell's law is actually the relationship between the angle of incidence, the angle of refraction and the refractive indices of the two media. According to Snell's law:

$\frac{sini}{sinr} = μ$ (constant called refractive index), where, $i$ is the angle of incidence, $r$ is the angle of refraction and $μ$ is the refractive index.

Let $XY$ be a plane surface separating air from a denser medium. $PA$ is the incident plane wavefront and $P' A'$ is the refracted wavefront.

$∠ NAL = i$ (angle of incidence) and

$∠ N' AA' = r$ (angle of refraction).

When the disturbance from the point $P$ , on incident wavefront just reaches the point $P'$ on the surface $XY$ , the secondary wavelet from point $A$ on the surface will grow in to a sphere (inside the denser medium)of radius $AA'$ such that:

$\frac{AA'}{v} = \frac{PP'}{c}$ or

$AA' = \frac{v}{c} \times PP'$

Consider any point $Q$ on the incident wavefront. Suppose that when disturbance from point $P$ reaches point $P'$ , the disturbance from the point $Q$ reaches point $Q'$ via $K$ . Time taken by light to go from $Q$ to $Q'$ will be:

$t = \frac{QK}{c} + \frac{KQ'}{v}$ or

$t = \frac{AK (sini)}{c} + \frac{KP' (sinr)}{v}$ or

$t = \frac{AK (sini)}{c} + \frac{(AP' - AK) sinr}{v}$ or

$t = \frac{AP' (sinr)}{v} + AK (\frac{sini}{c} - \frac{sinr}{v})$

The rays from different points on the incident wavefront will take the same time to reach the corresponding points on the refracted wavefront, if $t$ is independent of $AK$ . It follows that:

$\frac{\sin (i)}{c} - \frac{\sin (r)}{v} = 0$ or

$\frac{sini}{sinr} = \frac{c}{v}$

But $\frac{c}{v} = μ$ (constant) is called the refractive index of the denser medium with respect to air.

Thus, $\frac{sini}{sinr} = μ$ (constant).

This verifies the Snell's law.

Derive Snell's law of refraction using Huygen's principle.

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