Q.A plane wavefront is incident on a concave mirror of radius of curvature $R$. The radius of the refracted wavefront will be : (A) $2R$ (B) $R$ (C) $\dfrac{R}{2}$ (D) $\dfrac{R}{4}$

Concept: Reflection of a plane wavefront by a spherical mirror — the reflected wavefront’s radius of curvature equals the mirror’s focal length.

A plane wavefront has infinite radius of curvature. After reflection from a concave mirror, it converges to the focus. The focal length of a concave mirror is $f = \frac{R}{2}$. …

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 $R$:

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 $f = R/2$.

Step-by-Step Reasoning

1. Understand the incident wavefront.

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

2. 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:

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

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

Therefore, the image distance for an incident plane wave is:

4. Interpret the reflected wavefront. …