Correct Option: (b) \(\frac{R}{2}\)
Step 1 (Formula / Governing Law & Given Data): [0.25 Mark]
A plane wave consists of wavefronts that are parallel planes. When these wavefronts are incident on a concave mirror, they converge at the principal focus \(F\) after reflection. The distance of the principal focus from the pole of the mirror is the focal length \(f\), which is related to the radius of curvature \(R\) by the mirror formula:
$$f = \frac{R}{2}$$Step 2 (Step-by-Step Mathematical Working): [0.5 Mark]
- The incident plane wave can be considered as coming from an object at infinity (\(u = \infty\)).
- According to the mirror formula: $$\frac{1}{v} + \frac{1}{u} = \frac{1}{f}$$
- Substituting \(u = \infty\): $$\frac{1}{v} + \frac{1}{\infty} = \frac{1}{f}$$ $$\frac{1}{v} + 0 = \frac{1}{f} \implies v = f$$
- Since \(f = \frac{R}{2}\), the reflected rays converge at a distance of \(\frac{R}{2}\) from the mirror. The reflected wavefronts are spherical, centered at this focal point.
Step 3 (Final Concluded Answer): [0.25 Mark]
The radius of the reflected spherical wave is equal to the focal length of the concave mirror:
$$r = \frac{R}{2}$$💡 EL10 Concept Insight:
Think of a plane wave as light rays coming from very far away (like the sun); when they hit a curved mirror, they all bounce off and meet at a single point called the focus. Since the distance from the mirror to that focus is exactly half the radius of the mirror's curve, the reflected wave forms a sphere with that same radius.