Using Huygens principle, explain reflection of a plane wave by a plane reflecting surface.

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Q. Using Huygens principle, explain reflection of a plane wave by a plane reflecting surface.

Gujarat GsebGSEB Higher Secondary Certificate (HSC) Examination 2024Subjective· 2mImportance★★★★★

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Huygens' principle treats each point of an advancing wavefront as a source of secondary wavelets; applying this to a plane wave striking a plane mirror geometrically proves the law of reflection.

Let a plane wavefront AB be incident on a reflecting surface MN at angle i, with B touching the surface first (or A, depending on orientation) while the other end (say A) is still approaching.

In the time τ taken for the ray from B to reach the surface at C (BC = vτ), the wavelet from A would have already started emitting secondary wavelets from the surface, spreading out as a hemisphere of radius vτ (since A struck the surface earlier and reflected). The new reflected wavefront CE is drawn as a common tangent to all such secondary wavelets.

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CBSE 2023Set 55/4/11 markMCQ
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›Reveal solutionSolution
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.Using Huygens principle, explain reflection of a plane wave by a plane reflecting surface.

Concept understanding — Reflection of Light Using Huygens' Principle

Concept and Intuition

Step-by-Step Reasoning

Q.Using Huygens principle, explain reflection of a plane wave by a plane reflecting surface.

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