Which one of the following statements about the aperture of a convex lens is correct?
- (a)It is equal to its radius of curvature.
- (b)It is equal to its focal length.
- (c)It is independent of its radius of curvature.
- (d)It is equal to half of its focal length.
Correct — C, It is independent of its radius of curvature. The aperture of a lens is the effective diameter of the circular area through which light is allowed to pass — a measure of how wide the lens is, or of the width of any stop placed in front of it. The radius of curvature is a different quantity altogether: it describes how sharply each face of the lens is bowed, and together with the refractive index of the glass it fixes the focal length through the lens-maker's formula. The two can be varied one at a time. Grind two lenses to the same curvature and cut one narrower than the other, and they have the same focal length and different apertures; stop down a single lens with a ring of card and the aperture falls while nothing about the glass has changed. That independence is what makes the aperture matter in its own right — it governs how much light is gathered and how bright the image is, not where the image forms.
- (a)It is equal to its radius of curvature. — These are not even the same kind of measurement. The radius of curvature belongs to the sphere each face is cut from, and a single lens has two of them, one for each face; the aperture is the width of the piece of glass actually used.
- (b)It is equal to its focal length. — The focal length follows from the radii of curvature and the refractive index by the lens-maker's formula, and a wide lens and a narrow one cut to the same curvature share it. Two lenses can therefore have the same focal length and quite different apertures.
- (d)It is equal to half of its focal length. — This borrows a relation from the wrong instrument. For a spherical mirror the focal length is half the radius of curvature — a real formula, about a mirror, connecting focal length to curvature and saying nothing about aperture.
Three quantities describe a lens and each does a different job. The radii of curvature of the two faces, with the refractive index, set the focal length and so decide where the image forms. The focal length, inverted and measured in metres, gives the power in dioptres. The aperture is the working diameter, and it decides how much light reaches the image and how bright that image is. A camera lens marked with an f-number is quoting the focal length divided by the aperture, which is exactly why the two are named separately.
The item rewards knowing what a word means rather than any calculation, and three of the four options try to force an equality between quantities that measure different things. One purist's caveat is worth stating plainly: a lens cannot be made wider than the sphere its surfaces are cut from, so a very strongly curved surface does set an upper limit on how large the aperture can be. That is a physical ceiling, not a relation — within the ceiling the aperture is free, and the option that says it is independent of the radius of curvature is the one the key wants.
- The aperture of a lens is the effective diameter of the area through which light passes.
- The radius of curvature and the refractive index fix the focal length through the lens-maker's formula.
- Two lenses of the same curvature but different widths have the same focal length and different apertures.
- Aperture governs light-gathering and image brightness; focal length governs where the image forms.
- For a spherical mirror the focal length is half the radius of curvature — a mirror relation, and no statement about aperture.
Two of these describe the shape of the surfaces; the third describes how much of the lens is used.
- Carrying the mirror relation f equals R by 2 across to a lens.
- Assuming a bigger aperture means a shorter focal length.
- Reading aperture as the hole in a diaphragm only; for a bare lens it is the width of the glass itself.
As a definition item like this one, or through the lens-maker's formula and the power of a lens in a short numerical.
Which one of the following statements regarding lenses is not correct?
- (a) A convex lens produces both real and virtual images.
- (b) A concave lens produces both real and virtual images.
- (c) A convex lens can produce images equal, greater and smaller than the size of the object.
- (d) A concave lens always produces images smaller than the size of the object.
Answer(b) A concave lens produces both real and virtual images.
The same testing style on the same instrument — four sentences about a lens, one of which quietly asserts something that is not true. There the false claim is that a diverging lens can make a real image of a real object; here it is that the width of the glass is tied to the curvature of its faces.
- practice — not a real PYQ
The aperture of a lens principally determines which one of the following?
- (a)The position of the image
- (b)The brightness of the image
- (c)The sign of the magnification
- (d)The refractive index of the glass
Answer(b) The brightness of the image — the aperture fixes how much light is gathered, while the focal length fixes where the image forms.
- practice — not a real PYQ
For a spherical mirror, the focal length f and the radius of curvature R are related as
- (a)f = R
- (b)f = R/2
- (c)f = 2R
- (d)f = R/4
Answer(b) f = R/2 — a mirror relation, which is exactly the formula that tempts candidates into the wrong options on a lens-aperture question.