Given below are two statements: one is labelled as Assertion (A) and the other is labelled as Reason (R).
Assertion (A): A concave mirror of small aperture, cut from a hollow sphere of diameter 60 cm, brings a beam of light parallel to its principal axis to a focus 15 cm from its pole.
Reason (R): The focal length of a spherical mirror of small aperture is half of its radius of curvature.
In the light of the above statements, choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → Both (A) and (R) are true and (R) is the correct explanation of (A).
Assertion (A) A concave mirror of small aperture, cut from a hollow sphere of diameter 60 cm, brings a beam of light parallel to its principal axis to a focus 15 cm from its pole. — True. The mirror is part of the sphere, so its radius of curvature is half the diameter, \( R=30\,\mathrm{cm} \). A parallel paraxial beam (rays close to the axis) is reflected to the principal focus F. This point lies at a distance \( f=R/2 \), that is 15 cm, from the pole.
Reason (R) The focal length of a spherical mirror of small aperture is half of its radius of curvature. — True. For rays close to the axis, the angles are small. The geometry of reflection then gives \( f=R/2 \).
Why R explains A?
The point where the parallel beam meets is the principal focus. Its distance from the pole is the focal length.
Reason (R) gives this distance as half the radius of curvature, \( 30\,\mathrm{cm}/2=15\,\mathrm{cm} \). This is the value stated in (A).
NCERT (2026-27):
Focal Length of Spherical Mirrors – “… We assume that the rays are paraxial, i.e., they are incident at points close to the pole P of the mirror and make small angles with the principal axis. The reflected rays converge at a point F on the principal axis of a concave mirror …. The point F is called the principal focus of the mirror. …”
Focal Length of Spherical Mirrors – “The distance between the focus F and the pole P of the mirror is called the focal length of the mirror, denoted by f. We now show that f = R/2, where R is the radius of curvature of the mirror. …”