Spherical mirror formula relating an object distance 'u', image distance 'v' and focal length of mirror 'f' may be applied to a plane mirror when
- (a)focal length goes to infinity.
- (b)focal length goes to zero.
- (c)image distance goes to zero.
- (d)image distance goes to infinity.
Answer
Why
Correct — A, focal length goes to infinity. A plane mirror can be treated as a spherical mirror whose surface is perfectly flat — that is, one with an infinite radius of curvature. Since the focal length is half the radius of curvature (f = R/2), an infinite radius means an infinite focal length. Putting f towards infinity in the mirror formula 1/v + 1/u = 1/f makes 1/f go to zero, which gives v = −u — exactly the plane-mirror result that the image is as far behind the mirror as the object is in front. So the formula applies to a plane mirror when the focal length goes to infinity.
Why the others are wrong
- (b)focal length goes to zero. — If the focal length went to zero, 1/f would become infinite, which does not describe a flat mirror; a plane mirror is the limit of a very large (infinite), not a vanishing, focal length.
- (c)image distance goes to zero. — Image distance going to zero would place the image at the mirror itself, which is not the general behaviour of a plane mirror (where the image lies behind the mirror, at v = −u).
- (d)image distance goes to infinity. — Image distance going to infinity means the object is at the focus, a special case for a curved mirror, not the condition that reduces the formula to the plane-mirror rule.
Concept
For a spherical mirror the focal length is related to the radius of curvature by f = R/2, and the mirror formula is 1/v + 1/u = 1/f. A plane (flat) mirror is the limiting case of a spherical mirror as its radius of curvature — and hence its focal length — becomes infinite. In that limit 1/f = 0, so the formula reduces to 1/v = −1/u, that is v = −u: the image is virtual and just as far behind the mirror as the object is in front.
Think of flattening a curved mirror: the flatter it is, the larger its radius of curvature, so R towards infinity means f towards infinity. Substituting f towards infinity (so 1/f = 0) into the mirror formula reproduces the familiar plane-mirror result, so option (a) is the condition.
Key facts
- For a spherical mirror, focal length f = R/2, where R is the radius of curvature.
- A plane mirror is the limit of a spherical mirror with infinite radius of curvature, so f goes to infinity.
- With f towards infinity, the mirror formula 1/v + 1/u = 1/f gives v = −u.
- A plane mirror forms a virtual, erect image as far behind the mirror as the object is in front.
Study next
Common traps
- Confusing f towards zero with f towards infinity — a flat mirror has an infinite (not zero) focal length.
- Forgetting f = R/2, which is why an infinite radius of curvature gives an infinite focal length.
Asked as the limiting condition that reduces the spherical-mirror formula to a plane mirror — recognise that the focal length goes to infinity.
Related PYQs
The correct relation between the radius of curvature R and focal length f of a spherical mirror is
- (a) R = f
- (b) R = 2f
- (c) R = 3f
- (d) R = 4f
Answer(b) R = 2f
Same relation — f = R/2 (that is, R = 2f). That 2020 NDA item states the spherical-mirror relation directly; this one uses its limit (R and f going to infinity) to explain why the mirror formula also fits a plane mirror.
Practice
- practice — not a real PYQ
For a spherical mirror, the focal length f and the radius of curvature R are related by
- (a)f = R
- (b)f = R/2
- (c)f = 2R
- (d)f = R/4
Answer(b) f = R/2 — the focal length is half the radius of curvature. - practice — not a real PYQ
The image formed by a plane mirror is
- (a)real and inverted
- (b)virtual and as far behind as the object is in front
- (c)real and magnified
- (d)virtual and nearer than the object
Answer(b) virtual and as far behind as the object is in front — v = −u.