Lesson 7: Lenses

Rule 1: Convex and concave lenses — A convex lens (बहिर्वक्र भिंग) is thicker at the centre than at the edges; a concave lens (अंतर्वक्र भिंग) is thinner at the centre. A ray is refracted twice in a lens, once on entering and once on leaving.

• Easy : Reading glass is thick in the middle, so it is convex; the lens in a door peep hole is thin in the middle, so it is concave.
• Medium : Check by touch: run a finger across the surface; if the middle bulges out, it is convex.
• Exam-type : A thick-at-the-centre lens held in sunlight focuses the rays to a bright spot; a thin-at-the-centre lens only spreads them.

Rule 2: Terms related to lenses — Centres of curvature (C₁, C₂) are the centres of the spheres that form the two surfaces, and R₁, R₂ are the radii of curvature. The principal axis (मुख्य अक्ष) is the line through both centres. The optical centre (O) (प्रकाशीय केंद्र): a ray through it passes without changing its path. The principal focus (F) (मुख्य नाभी): in a convex lens, rays parallel to the axis converge at F (converging lens); in a concave lens they spread out and seem to come from F (divergent lens). Focal length (f) (नाभीय अंतर) is the distance OF. The image of a distant object forms near the focus.

In the figure : In the figure convex and concave lenses.

• Easy : If the focal length of a convex lens is 15 cm, parallel rays meet at 15 cm from the optical centre.
• Medium : Sun's rays are almost parallel, so a convex lens focuses them at F; the distance from the lens to the bright spot is the focal length.
• Exam-type : Why is a convex lens called converging and a concave lens divergent? One brings parallel rays together, the other spreads them apart.

Spot the difference

Rule 1: Rules of ray diagrams — Three rays are used for a convex lens: (1) a ray parallel to the principal axis passes through the focus; (2) a ray through the focus comes out parallel to the axis; (3) a ray through the optical centre passes straight. Any two of them are enough to find the image.

In the figure : In the figure a ray diagram for a convex lens.

• Easy : Draw a parallel ray and a ray through O from the top of the object; where they cross is the top of the image.
• Medium : Draw the ray through the focus from the same point as a third check; it must meet the others at the same point.
• Exam-type : If the three rays do not meet at one point, the diagram is wrong; redraw with a ruler.

Rule 2: Images in a convex lens — Object at infinity: point image at F₂ (real, inverted). Beyond 2F₁: smaller, real, inverted image between F₂ and 2F₂. At 2F₁: same size, real, inverted image at 2F₂. Between F₁ and 2F₁: larger, real, inverted image beyond 2F₂. At F₁: very large, real, inverted image at infinity. Between F₁ and O: very large, virtual and erect image on the same side as the object.

In the figure : In the figure the image for different object positions.

• Easy : A camera photographs a distant tree: the object is beyond 2F, so the image is small, real and inverted.
• Medium : A projector puts the slide between F and 2F, so the picture on the screen is large, real and inverted; hence slides are loaded upside down.
• Exam-type : A magnifying glass works when the object is closer than F: the image is virtual, erect and magnified, so we cannot catch it on a screen.

Rule 3: Concave lens image — A concave lens always forms a virtual, erect and smaller image between O and F₁, wherever the object is. For an object at infinity, the image is a point at F₁.

• Easy : A door peep hole lets you see a small, erect view of the person outside.
• Medium : Unlike a convex lens, the image of a concave lens does not depend much on object distance: it is always virtual, erect and smaller.
• Exam-type : Name one case where a convex lens gives a virtual image: object between F and O.

Spot the difference

Rule 1: Sign convention and formula — In the Cartesian convention, O is the origin and the principal axis is the X-axis. The object is on the left. Distances to the right of O are positive, to the left negative; heights above the axis are positive, below negative. f is positive for a convex lens and negative for a concave lens. Lens formula: 1/v − 1/u = 1/f. Magnification (विवर्धन) M = h₂/h₁ = v/u. A negative M means the image is real and inverted.

In the figure : In the figure using the lens formula step by step.

• Easy : Easy: u = −40 cm, f = +20 cm. 1/v = 1/20 − 1/40 = 1/40, so v = +40 cm and M = 40/(−40) = −1. The image is real, inverted and the same size.
• Medium : Medium: u = −30 cm, f = +10 cm, h₁ = 6 cm. 1/v = 1/10 − 1/30 = 2/30, so v = +15 cm; M = 15/(−30) = −0.5; h₂ = −0.5 × 6 = −3 cm (3 cm, real, inverted, smaller).
• Exam-type : Exam: a concave lens has f = −15 cm and the object is at u = −30 cm. 1/v = 1/f + 1/u = −1/15 − 1/30 = −3/30, so v = −10 cm; M = (−10)/(−30) = +1/3. The image is virtual, erect and smaller, on the same side as the object.

Rule 2: Power of a lens — Power P = 1/f, with f in metres; the unit is the dioptre (D). A convex lens has positive power, a concave lens negative. For lenses in contact, 1/f = 1/f₁ + 1/f₂, so P = P₁ + P₂.

In the figure : In the figure power is the inverse of the focal length in metres.

• Easy : Easy: f = 50 cm = 0.5 m, so P = 1/0.5 = +2 D. A concave lens with f = −25 cm = −0.25 m has P = −4 D.
• Medium : Medium: a convex lens of +2 D is placed in contact with a concave lens of −0.5 D. P = 2 − 0.5 = +1.5 D, so f = 1/1.5 ≈ 0.67 m = 67 cm.
• Exam-type : Exam: which lens has the greater power, f = 20 cm or f = 40 cm? P = 5 D and 2.5 D; the shorter focal length gives the greater power.

Spot the difference

Rule 1: Parts and working of the eye — Light enters through the cornea (पारपटल), where most refraction takes place. The iris (बुबुळ) has the pupil (कृष्णमंडल), which contracts in bright light and widens in dim light. The convex crystalline lens forms a real, inverted image on the retina (दृष्टिपटल), whose light-sensitive cells send signals through the optic nerve to the brain. For distant objects the lens becomes flat (longer f); for near objects it becomes rounded (shorter f). The near point of a normal eye is 25 cm and the far point is infinity.

• Easy : In a dim room your pupils widen so more light enters.
• Medium : When you look from a faraway hill to a book, the lens becomes more rounded; that is accommodation.
• Exam-type : Why is the image on the retina inverted yet we see things upright? The brain interprets the signals and turns the picture upright.

Rule 2: Defects of vision — Myopia (निकटदृष्टिता / nearsightedness): near objects are seen clearly, but the image of a distant object forms in front of the retina; corrected with a concave lens (negative power). Hypermetropia (दूरदृष्टिता / farsightedness): distant objects are clear, but the image of a near object forms behind the retina; corrected with a convex lens (positive power). Presbyopia (वृद्धदृष्टिता): the focusing power decreases with age; bifocal lenses (द्विनाभी भिंगे) have a concave upper part and a convex lower part.

In the figure : In the figure eye defects and their correction.

• Easy : Easy: a student cannot read the blackboard from the back bench but can read a book easily. This is myopia; a concave lens is needed.
• Medium : Medium: spectacles of −2.5 D mean f = 1/(−2.5) = −0.4 m = −40 cm; it is a concave lens used for myopia.
• Exam-type : Exam: a person needs a +2 D lens to read at 25 cm. This is hypermetropia, and the lens is convex with f = 1/2 = 0.5 m = 50 cm.

Spot the difference

Rule 1: Microscopes — A simple microscope (magnifying glass) is a convex lens of small focal length that gives a virtual, erect, enlarged image; it is used in watch repair and for testing gems. A compound microscope has an objective (smaller cross-section and focal length) and an eyepiece, both convex lenses; magnification happens in two stages. It is used to see cells and bacteria.

In the figure : In the figure where lenses are used.

• Easy : A watchmaker uses a small magnifying glass to see the tiny parts of a watch.
• Medium : The total magnification of a compound microscope is the objective's magnification times the eyepiece's: 10× objective and 5× eyepiece give 50×.
• Exam-type : Why does a compound microscope need two lenses? One lens magnifies the object to give a first image, and the second magnifies that image again.

Rule 2: Telescope and other uses — In a telescope, the objective (large diameter and long focal length) collects the maximum light, and its image is the object for the eyepiece (small size and small focal length). Refracting telescopes use lenses; reflecting telescopes use mirrors as well. Concave lenses are used in spectacles for nearsightedness, torches, door peep holes, laser instruments (scanners, CD players), and in front of or inside the eyepiece of cameras and telescopes. Convex lenses are used in spectacles for farsightedness, cameras, projectors and spectrographs.

• Easy : The big front lens of a telescope collects more starlight, so faint stars can be seen.
• Medium : A mobile camera uses a convex lens to form a small real inverted image on the sensor, which the phone turns upright.
• Exam-type : Match: spectacles for farsightedness – convex; torch beam spreading – concave; projector – convex.

Spot the difference