Derive laws of reflection of light using Huygens' principle.

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Q. Derive laws of reflection of light using Huygens' principle.

Maharashtra MsbshseMaharashtra HSC (MSBSHSE) Board 2023Subjective· 3mImportance★★★★★

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Huygens' principle applied to a plane wavefront hitting a reflecting surface.

Let a plane wavefront $AB$ be incident on reflecting surface $MN$ at angle of incidence $i$ , with $A$ striking the surface first. While the secondary wavelet from $A$ travels to a point on $MN$ where $B$ 's wavelet arrives (taking time $t$ ), $B$ travels a distance $BC = vt$ to reach the surface at $C$ . By Huygens' principle, $A$ acts as a new secondary source and, in the same time $t$ , its wavelet spreads out to radius $AD = vt = BC$ .

The reflected wavefront is the tangent plane $CD$ from the common surface point $C$ to the wavelet from $A$ . In right triangles $ABC$ and $ADC$ :

$BC = AD = vt$ (equal radii)
$AC$ is common …

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CBSE 2023Set 55/4/11 markMCQ
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A plane wavefront has infinite radius of curvature. After reflection from a concave mirror of radius $R$ , the wavefront becomes spherical with radius equal to half the mirror's radius of curvature, i.e., $\frac{R}{2}$ . The correct option is (C).

Concept and Intuition

A wavefront is a surface of constant phase. For a plane wavefront, the rays are parallel, meaning the wavefront has an infinite radius of curvature — it is flat. When such a wavefront strikes a concave mirror, the mirror converges the parallel rays to its focus. The reflected wavefront is therefore spherical, converging to the focal point.

The key relationship: for a spherical mirror, the focal length $f$ is half the radius of curvature $R$ :
$f = \frac{R}{2}$
The reflected wavefront's radius of curvature is exactly the distance from the mirror to the point where the rays converge — which is the focal point for an incident plane wave. So the reflected wavefront has radius $f = R/2$ .

Watch out

A common mistake is to think the reflected wavefront's radius equals the mirror's radius $R$ . That would be true only if the incident wavefront itself originated from the centre of curvature. Here, the incident wavefront is plane, so the reflected wavefront converges to the focus, not the centre.

Step-by-Step Reasoning

Understand the incident wavefront.

A plane wavefront means all rays are parallel to the principal axis. The wavefront has infinite radius of curvature: $R_{\text{incident}} = \infty$ .

Recall the mirror equation in terms of wavefront curvature.

For a spherical mirror, the relationship between object distance $u$ , image distance $v$ , and focal length $f$ is:

$\frac{1}{v} + \frac{1}{u} = \frac{1}{f}$
For a plane wavefront, the object is at infinity: $u = \infty$ , so $\frac{1}{u} = 0$ . Thus:
$\frac{1}{v} = \frac{1}{f} \quad \Rightarrow \quad v = f$

Relate focal length to radius of curvature. For any spherical mirror:

$f = \frac{R}{2}$
Therefore, the image distance for an incident plane wave is:
$v = \frac{R}{2}$

Interpret the reflected wavefront. …
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Q.Derive laws of reflection of light using Huygens' principle.

Concept understanding — Reflection of Light Using Huygens' Principle

Concept and Intuition

Step-by-Step Reasoning

Q.Derive laws of reflection of light using Huygens' principle.

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