Question

A ray parallel to the principal axis is incident at \(30^\circ\) from the normal on a concave mirror having radius of curvature \(R\). The point on the principal axis where rays are focused is \(O\) such that \(PQ\) is:



 

• A. \(\dfrac{R}{2}\)
• B. \(\dfrac{R}{\sqrt{3}}\)
• C. \(\dfrac{2\sqrt{R}-R}{\sqrt{2}}\)
• D. \(R\left(1-\dfrac{1}{\sqrt{3}}\right)\)

Hint:

For oblique incidence on spherical mirrors, focal length depends on angle: \[ f_{\theta} = \frac{R}{2}\cos\theta \] Use geometry instead of paraxial approximation.

Answer 1

The Correct Option is D

Step 1: For a ray parallel to the principal axis, reflection occurs such that angle of incidence equals angle of reflection.
Step 2: Geometry of the concave mirror gives focal shift due to oblique incidence.
Step 3: Using mirror geometry: \[ PQ = R\left(1-\cos 30^\circ\right) \]
Step 4: Substitute \(\cos 30^\circ = \frac{\sqrt{3}}{2}\): \[ PQ = R\left(1-\frac{\sqrt{3}}{2}\right) = R\left(1-\frac{1}{\sqrt{3}}\right) \]