Refraction of Light at Curved Surfaces — Telangana (SSC) Class 10 Science Solutions (Free)

Refraction is the bending of light as it passes from one transparent medium into another of different optical density, caused by a change in the speed of light between the two media. Snell's Law: n₁ sinθ₁ = n₂ sinθ₂ where n₁ and n₂ are the refractive indices of the first and second medium, and θ₁ and θ₂ are the angles of incidence and refraction (both measured from the normal — the line perpendicular to the surface at the point of incidence). Why it bends toward the normal in a denser medium: Light travels more slowly in an optically denser medium (higher n). When a light ray enters a denser medium at an angle, the part of the wavefront that arrives first slows down before the rest of it does, which effectively pivots the direction of travel toward the normal — the same way a car's front-left wheel dropping onto softer ground while the rest is still on hard ground turns the car's path toward that side. Conversely, light speeds up when passing into a rarer (less dense) medium, so it bends AWAY from the normal.

A lens is a transparent optical device with at least one curved surface, which refracts light in a predictable, useful way as it passes through. Convex lens (converging lens): thicker at the centre than at the edges. Parallel rays of light passing through it are bent inward and meet (converge) at a single point called the principal focus. Depending on where the object is placed, it can form either a real image (light rays actually meet, can be captured on a screen) or a virtual image (only appears to originate from a point, cannot be captured on a screen). Concave lens (diverging lens): thinner at the centre than at the edges. Parallel rays passing through it spread out (diverge) as if they all originated from a single point behind the lens. A concave lens used alone ALWAYS forms a virtual, erect, and diminished (smaller) image, regardless of where the object is placed — this is a key fact examiners often test directly. By convention (the New Cartesian Sign Convention, used throughout this chapter): a convex lens has positive focal length, a concave lens has negative focal length.

Using the New Cartesian Sign Convention: distances are measured from the optical centre, with the object always placed on the left so its distance u is taken as NEGATIVE. Given: f = +20 cm (convex lens, positive), u = −40 cm Step 1: Apply the lens formula 1/v − 1/u = 1/f 1/v = 1/f + 1/u = 1/20 + 1/(−40) = 2/40 − 1/40 = 1/40 v = +40 cm Step 2: Find the magnification m = v/u = 40/(−40) = −1 Step 3: Interpret the result v is positive → the image forms on the opposite side of the lens from the object → REAL image. m is negative → the image is INVERTED. |m| = 1 → the image is the SAME SIZE as the object. This is a standard result worth remembering: when an object is placed at 2F (twice the focal length) of a convex lens, the image forms at 2F on the other side — real, inverted, and exactly the same size as the object.

Focal length (f): the distance between the optical centre of a lens and its principal focus — the point where parallel rays converge (convex) or appear to diverge from (concave). Power (P): a measure of how strongly a lens converges or diverges light, defined as the reciprocal of the focal length measured in METRES: P = 1/f (measured in dioptres, D) A convex lens, having positive focal length, always has positive power (it converges light). A concave lens, having negative focal length, always has negative power (it diverges light). A lens with a shorter focal length has a HIGHER power — it bends light more strongly, which is why very powerful corrective spectacle lenses look noticeably more curved than mild ones. Given: f = 25 cm = 0.25 m P = 1/f = 1/0.25 = +4 D The power of the lens is +4 dioptres (positive, confirming it converges light as expected for a convex lens).

Given: f = −15 cm (concave lens, negative by convention), u = −30 cm (object distance, negative) Step 1: Apply the lens formula 1/v − 1/u = 1/f 1/v = 1/f + 1/u = 1/(−15) + 1/(−30) = −2/30 − 1/30 = −3/30 = −1/10 v = −10 cm Step 2: Find the magnification m = v/u = (−10)/(−30) = +1/3 Step 3: Interpret v is negative → the image forms on the SAME side as the object → VIRTUAL image. m is positive → the image is ERECT. |m| = 1/3 (less than 1) → the image is DIMINISHED (smaller than the object). This confirms the general rule for concave lenses: wherever the object is placed, the image is always virtual, erect and diminished, forming between the object and the lens.

A convex lens is thicker at its centre than at its edges. When a ray of light travels through glass (or any denser medium), it slows down and bends toward the normal at each curved surface it crosses. Because a convex lens is shaped like two curved surfaces bulging outward, a ray passing anywhere through it — except exactly along the centre — gets bent toward the principal axis. Rays entering parallel to the axis, from far away, all bend inward enough to meet at one common point: the principal focus. This convergence of rays to a point is why it is called a converging lens. Practical uses of this converging property: - Magnifying glasses use a convex lens held close to an object to produce an enlarged, virtual image. - Camera lenses use convex lenses to converge light from a scene onto a real image on the film or sensor. - The human eye's own lens is convex, converging light onto the retina — which is exactly why a convex lens is prescribed to correct hypermetropia (long-sightedness), where the eye's natural focusing power is too weak.

The lens maker's formula relates a lens's focal length to the physical properties of the lens itself — the material it is made of and the curvature of its two surfaces — rather than to any object or image: 1/f = (n − 1) [1/R₁ − 1/R₂] Where: - f = focal length of the lens - n = refractive index of the lens material relative to the surrounding medium (for ordinary glass in air, n ≈ 1.5) - R₁ = radius of curvature of the first surface (the one light meets first) - R₂ = radius of curvature of the second surface Sign convention for R: a radius is positive if its centre of curvature lies on the outgoing (right) side of the lens, and negative if it lies on the incoming (left) side — following the same New Cartesian convention used for object and image distances. This formula is what lets manufacturers calculate the exact curvature to grind into a lens surface in order to achieve a required focal length and power — essential for producing prescription spectacles to a precise specification.

When thin lenses are placed in contact, their POWERS simply add (this is a major reason power, not focal length, is used when combining lenses). Given: convex lens f₁ = +20 cm = +0.20 m, concave lens f₂ = −40 cm = −0.40 m Step 1: Find each lens's power P₁ = 1/f₁ = 1/0.20 = +5 D P₂ = 1/f₂ = 1/(−0.40) = −2.5 D Step 2: Add the powers P = P₁ + P₂ = 5 + (−2.5) = +2.5 D Step 3: Find the combined focal length f = 1/P = 1/2.5 = 0.4 m = 40 cm The combination has a power of +2.5 D and a focal length of +40 cm — since the net power is positive, the combination as a whole still behaves as a CONVERGING system, just weaker than the convex lens alone.