CDS (II) 2023 General Knowledge · Question 3 of 120

Correct — A, At infinity. The radius of curvature is 50 cm, and for a spherical mirror the focal length is half the radius, so f = 25 cm. The object has been placed 25 cm from the mirror — that is, exactly at the principal focus. Light leaving a point at the focus of a concave mirror comes back off the surface as a parallel beam, and a parallel beam never converges, so the image is formed at infinity. The mirror formula says the same thing in symbols. Taking the usual sign convention with distances measured from the pole against the incoming light, u = −25 cm and f = −25 cm, and 1/v = 1/f − 1/u = (−1/25) − (−1/25) = 0, so v is infinite. The image is real, inverted and enormously magnified, which in practice means there is no usable image on a screen at all — the arrangement is the one used the other way round, to turn a small source into a searchlight beam.

• (b)At 50 cm — This is the centre of curvature, and it is where the image would form if the object had also been kept at 50 cm. Placing the object at C gives an image at C, the same size and inverted — but the object here is at 25 cm, not 50.
• (c)At 25 cm — The tempting symmetry — object at 25 cm, so image at 25 cm. It is wrong because an object placed exactly at the focus is the one position that produces no finite image; the reflected rays leave parallel to one another.
• (d)At 75 cm — No standard object position for this mirror sends the image to 75 cm. It looks plausible only if the radius and the object distance are added, which is not a step in any mirror relation.

A concave mirror converges light. Its focal length is half its radius of curvature, f = R/2, and the position and nature of the image depend entirely on where the object sits relative to the pole P, the focus F and the centre of curvature C. Beyond C the image is real, inverted and diminished and lies between F and C; at C it is real, inverted and the same size and lies at C; between C and F it is real, inverted and magnified and lies beyond C; at F the reflected rays are parallel and the image is at infinity; and inside F the image turns virtual, erect and magnified, behind the mirror.

The examiner has hidden one extra step in an otherwise routine item: the number given is the radius, not the focal length, and a candidate who reads 50 cm as f will place the object inside the focus and reach for a virtual image. Halve the radius first. Once f = 25 cm is on the page, notice that the object distance is the same 25 cm and the answer is a definition rather than a calculation — object at focus, image at infinity. It is worth carrying the reverse statement too, because it is the reason concave mirrors are used in torches, headlights and solar cookers: put the source at the focus and you get a parallel beam out.

• For a spherical mirror of small aperture the focal length is half the radius of curvature, f = R/2.
• An object placed at the principal focus of a concave mirror gives an image at infinity, real, inverted and highly magnified.
• An object at the centre of curvature gives an image at the centre of curvature, real, inverted and of the same size.
• An object between the pole and the focus gives the only virtual image a concave mirror can produce — erect and magnified, which is why shaving and dentists' mirrors are concave.
• The mirror formula is 1/v + 1/u = 1/f, with distances measured from the pole and the incoming light taken as the negative direction.

• Using the radius of curvature in the mirror formula in place of the focal length.
• Assuming an object at the focus gives an image at the focus; it is the one position that gives no finite image.
• Dropping the sign convention and treating u and f as positive numbers, which produces the right magnitude only by accident.

Either as a one-step numerical like this one, or as a which-position-gives-which-image match involving the pole, the focus and the centre of curvature.

• practice — not a real PYQ

  The focal length of a concave mirror is 15 cm. Where should an object be placed so that the image is formed at the centre of curvature and is of the same size as the object?

  • (a)At 15 cm from the mirror
  • (b)At 30 cm from the mirror
  • (c)At 45 cm from the mirror
  • (d)At 7·5 cm from the mirror

  Answer(b) At 30 cm from the mirror — the centre of curvature is at R = 2f = 30 cm, and an object at C gives an image at C of the same size, inverted and real.

• practice — not a real PYQ

  In a torch, the bulb is placed at the principal focus of a concave reflector. What is the purpose of doing this?

  • (a)To obtain a converging beam of light
  • (b)To obtain a parallel beam of light
  • (c)To form a real image of the bulb on a screen
  • (d)To reduce the brightness of the beam

  Answer(b) To obtain a parallel beam of light — rays from a source at the focus of a concave mirror are reflected parallel to the principal axis, which is what lets a torch throw light a long way.