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

State Huygen's principle. Using the geometrical construction of secondary wavelets, explain the refraction of a plane wavefront incident at a plane surface. Hence, verify Snell's law of refraction. Illustrate with the help of diagrams, the action of convex lens.

Accepted Answer

Huygen's principle gives a method to know as to how light spreads out in the medium. Huygen's principle is based on the following postulates:

1. Each point on the primary wavefront acts as a source of secondary wavelets.

2. The new position of the wavefront at any instant is the envelope of the secondary wavelets.

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).

According to Snell's law, the ratio of the sine of angle of incidence to the sine of the angle of refraction is always constant for two given pair of media. This constant is called the refractive index. This verifies the Snell's law.

Let a plane wavefront $AB$ incidents on a double convex lens $XY$ . When the disturbance from the $O$ just reaches the point $P_{2}$ , the disturbance from the points $A$ and $B$ will cover a total distance, say $x$ given by:

$x = (\frac{O P_{1}}{c} + \frac{P_{1} P_{2}}{v}) \times c$ , where, $c$ is the velocity of light in air and $v$ is the velocity of light inside the lens. The points $X$ and $Y$ sends out the secondary wavelets of radii $(x - AX)$ and $(x - BY)$ respectively. At this instant, the point $P_{2}$ just becomes the source of secondary wavelets. The envelope we get by the joining of $A' P_{2} B'$ serves the refracted wavefront. Thus, a parallel beam of light on reflection through a convex lens gets converged.

State Huygen's principle. Using the geometrical construction of secondary wavelets, explain the refraction of a plane wavefront incident at a plane surface. Hence, verify Snell's law of refraction. Illustrate with the help of diagrams, the action of convex lens.

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