Refraction at Spherical Surfaces & Lens Maker Refraction at Spherical Surfaces & Lens Maker Curved glass surfaces bend light in a precise, predictable way. A single spherical interface (like one side of a lens) already focuses or defocuses rays. Put two such surfaces back-to-back and you get a thin lens whose power depends on the material’s refractive index and the curvatures of its two faces. This lesson builds the chain cleanly: start with how one spherical surface refracts, obtain its formula using small-angle geometry, and then apply the same logic twice to reach the thin lens formula and lens maker’s relation. The key is to keep a consistent sign convention and stay in the paraxial world (rays close to the axis and making small angles). With those guardrails, the math becomes simple and robust. You will also see how the surrounding medium changes the focal length: a lens in air behaves differently from the same lens under water. That is because it is the relative refractive index between glass and the medium that sets the “bending strength.” We will translate these ideas into quick problem-solvers: find image position from a single curved interface, compute a lens’s focal length from its radii, get image distance from object distance and focal length, and combine lens powers. Along the way, we will call out the classic traps: mixing up the signs of R 1 and R 2 , confusing mirror sign conventions with lens conventions, and applying lens formulas beyond their valid limits (large angles, thick lenses, or misaligned optics). By the end, you should be able to read a lens’s geometry and medium and immediately know how light will focus. remember Everyday feel: A droplet of water on a page magnifies letters because its curved surfaces act like a biconvex lens. The focusing strength depends on water’s refractive index and how curved each surface is. Sign convention and approximations We use the Cartesian sign convention for refraction: light travels left to right. Distances are measured from the vertex/pole of the surface or from the optical centre of the lens along the principal axis. Quantities to the right (direction of incident light travel) are positive; to the left are negative. Thus an object placed on the left has u < 0 . A real image on the right has v > 0 . Radius of curvature R is positive if the centre of curvature lies to the right of the vertex and negative otherwise. We also assume paraxial rays: angles are small, sin and tan of small angles are approximately the angles in radians. Surfaces are spherical and smooth, and the media are homogeneous and isotropic. Within these bounds, the formulas that follow are accurate and easy to apply. tip Boundaries: Paraxial approximation only (small angles). Surfaces are spherical. Media have uniform refractive indices. For thick lenses or large apertures, principal planes and aberrations matter and thin-lens formulas no longer hold. Refractive index ( n ) Ratio that measures how much light slows in a medium compared to vacuum. Larger n means stronger bending for a given angle at an interface. Spherical surface A part of a sphere acting as a refracting boundary between two media of refractive indices n 1 and n 2 . Signed distance from the surface’s vertex to the centre of curvature. Positive if the centre is to the right (direction of light travel), negative if to the left. Radius of curvature ( R ) Pole/Vertex ( P ) The point where the principal axis meets the refracting surface. Principal axis The straight line through the pole and the centre of curvature; the reference line for measuring u , v , and R . Rays close to the principal axis and making small angles so that and (in radians). Paraxial rays Optical centre ( O ) For a thin lens, the point on the principal axis such that a ray through it emerges undeviated (ideally). Thin lens A lens whose thickness is small compared to its radii of curvature. It can be treated as two refracting spherical surfaces at the same position. Next we write the refraction formula for a single spherical surface. Keep the sign convention at hand. After we get one surface working, we will apply it twice to get the thin lens results and lens maker relation. Refraction at a single spherical surface Single-surface refraction Refraction from medium n 1 to n 2 at a spherical surface of radius R (Cartesian sign convention). This relation links the object distance, image distance, and curvature of the surface for light refracting at a single interface. This relation connects the object distance u , image distance v , and curvature R when light goes from n 1 to n 2 . Remember: u < 0 for a real object to the left, and R is positive if the centre of curvature lies to the right. Special cases: if u - (object at infinity), the image distance v equals the surface focal length. If R (a plane interface), the right side becomes zero and we get n 2/v = n 1/u . Geometry is set with pole P, centre of curvature C, principal axis PC. Small-angle relations convert heights to angles. Signs follow Cartesian convention. Angles measured from the normal. Linearized Snell’s law. Group angle terms. Use signed distances. Final single-surface refraction formula. Refraction at a spherical surface: n 2 ( 1 v ) - n 1 ( 1 u ) = n 2 - n 1 R Small angles (paraxial approximation): sinθ ≈ tanθ ≈ θ (radians) Spherical surface separating homogeneous media of refractive indices n1 and n2 Cartesian sign convention n 2 ( 1 v ) - n 1 ( 1 u ) = n 2 - n 1 R Define the focal distances for the surface: if the object is at infinity in medium n 1 , the image forms at v = f 2 = n 2 R n 2 - n 1 . If the image is at infinity in medium n 2 , the object must be at u = f 1 = n 1 R n 2 - n 1 . Note that f 1 and f 2 refer to focusing from either side and have signs as per the convention. Surface focal distances Focal distances of a single spherical refracting surface (object at infinity on one side or the other). cm Image distance v (cm) For fixed n1, n2, R, plotting v vs u from the single-surface formula yields a branch resembling a rectangular hyperbola with asymptotes u = 0 and v = constant (the surface focal distance). Geometric behavior of image distance v versus object distance u for a convex refracting surface from air to glass. n1 control n2 control control cm Object distance u (cm) custom -∞ Object at infinity → image at v = f2 f2 -R Representative finite object computed v Use n2(1/v) − n1(1/u) = (n2 − n1)/R with the stated signs. cm Image distance v and nature of image Air to glass refraction at a convex spherical surface. easy n1 (air) = 1.0 n2 (glass) = 1.5 Object distance u = −30 cm Radius of curvature R = +10 cm (centre to the right) The positive v indicates a real image to the right of the surface. A proper ray diagram would show rays bending toward the normal inside glass and converging beyond the surface. From two surfaces to a thin lens A thin lens is just two spherical surfaces very close together. Apply the single-surface formula at the first surface to get an intermediate image, then let that act as the object for the second surface. When the separation (thickness) is negligible, the distances add neatly and yield the thin lens formula. The same method, by placing the object at infinity, yields the lens maker relation that ties the focal length to the material refractive index and surface curvatures. 1 v - 1 u = 1 f Thin lens (negligible thickness) in a uniform medium Paraxial rays and Cartesian sign convention Same surrounding medium on both sides of the lens Thin lens formula: 1 v - 1 u = 1 f Valid for thin lenses in a uniform surrounding medium (usually air). Thin lens formula This equation links the object and image distances to the lens's focal length under the thin lens approximation. Transverse magnification of a thin lens is m = h i h o = v u . With the Cartesian convention, u<0 for a real object, so a real image on the right gives v>0 and m<0 (inverted). For a virtual image on the left, v<0 and m>0 (erect). Sign of m indicates orientation: negative for inverted, positive for erect. Magnification Use the ratio of image to object dimensions to determine the image's size and orientation. Lens power is P = 1/f (with f in metres) in dioptres (D). A converging lens has f>0 and P>0 ; a diverging lens has f<0 and P<0 . Powers add for thin lenses in contact. Use f in metres to get P in dioptres. Power of a lens The power defines the lens strength, which is then calculated using the material properties in the subsequent lens maker formula. Lens maker formula The lens maker relation tells you how f depends on the material and the two radii. In a medium of refractive index n (usually air n=1 ) and lens material of index n L , the lens focuses because light speeds differ across the boundary. With the standard sign convention ( R 1 is the radius of the first surface encountered by the light; R 2 is for the second), a biconvex lens with light from the left typically has R 1>0 and R 2<0 . Thin lens in air (n = 1) Paraxial approximation Spherical surfaces with radii R1 and R2 Lens maker (air): 1 f = (n L - 1) ( 1 R 1 - 1 R 2 ) 1 f = (n L - 1) ( 1 R 1 - 1 R 2 ) For a thin lens in air (surrounding index ≈ 1). Lens maker (air) This formula relates the focal length of a thin lens to its material index and the radii of its two curved surfaces. In a surrounding medium of refractive index n m 1 (e.g., water), the relative index matters. Replace (n L - 1) by (n L/n m - 1) . The focal length increases in water since n L/n m is closer to 1 than n L/1 . Lens maker (general medium): 1 f m = ( n L n m - 1 ) ( 1 R 1 - 1 R 2 ) Thin lens in a uniform medium of index n m Paraxial approximation and Cartesian convention 1 f m = ( n L n m - 1 ) ( 1 R 1 - 1 R 2 ) Typical signs to keep straight: with light from the left, a biconvex lens has R 1>0 (centre of the first surface to the right) and R 2<0 (centre of the second surface to the left). A biconcave lens flips those signs: R 1<0 , R 2>0 . Plane surfaces correspond to |R| . neet-alert Classic trap: Writing m = -v/u for lenses. Correct: for thin lenses with Cartesian sign convention, use m = v/u . The minus sign belongs to mirrors in this convention. Quantity Positive when Biconvex (light left→right) Biconcave (light left→right) u (object distance) To the right of vertex (direction of light); real object on left has u < 0 Usually negative Usually negative v (image distance) To the right of vertex Real image: v > 0; Virtual: v < 0 Virtual image (often): v < 0 R1 Centre of curvature to the right R1 > 0 R1 < 0 R2 Centre of curvature to the right R2 < 0 R2 > 0 Converging lens f > 0 f < 0 Same sign as 1/f P > 0 P < 0 Sign convention quick-sheet With the formulas and signs established, we now compute a focal length from given radii and index, then place an object and find the image using the thin lens formula. This two-step flow is standard in NEET-style numericals. n L = 1.50 (glass), air outside R1 = +20 cm, R2 = −20 cm u = −30 cm medium Biconvex lens in air: find f, then v for a given u. Focal length f and image distance v Use lens maker (air): 1/f = ( n L − 1)(1/R1 − 1/R2). Then 1/v − 1/u = 1/f. cm Because |u| > f , the lens produces a real image to the right. The negative magnification indicates inversion. The image is larger in magnitude since |v| > |u| . Lens material n L = 1.60 Water outside n m = 1.33 R1 = +10 cm, R2 = −15 cm u = −20 cm hard Focal length in water f m and image distance v A lens immersed in water: find effective focal length and image position. Use 1/ f m = ( n L / n m − 1)(1/R1 − 1/R2), then thin lens formula with f m . cm For thin lenses in contact: 1/F = 1/f1 + 1/f2. Equivalent to adding powers: P total = P1 + P2. Effective focal length of two thin lenses sharing an axis without separation: 1/Feq = 1/f1 + 1/f2. Using lens power P=1/f (metres), combinations become quick: P total = P 1 + P 2 + . This is common in optometry and in instrument design where multiple thin elements act together. Plane limit: When R in the single-surface formula, n 2/v = n 1/u . This is the small-angle refraction law at a plane interface and leads to the apparent depth relation for near-normal viewing. remember The thin lens formula uses m = −v/u like mirrors. For thin lenses with Cartesian signs, m = v/u . The negative sign is for the mirror magnification with the same convention. For a biconvex lens, both radii are positive. With light from left to right, R 1>0 (first surface), but R 2<0 (second surface). Mixing this up flips the predicted focal length. “First face forward, second face backward”: For light from left, a biconvex lens has its first centre to the right (R1 > 0) and the second to the left (R2 < 0). Check these boundaries before applying formulas Are angles small (paraxial)? If not, expect deviations and aberrations. Is the lens thin? If not, use principal planes or thick-lens treatment. Is the surrounding medium uniform on both sides? If not, use the general lens maker form with nL/nm. Are R 1 and R 2 signs chosen with the correct direction of incident light? Fix the direction of incident light and write the sign convention on the diagram. Assign signs to u , v , R (or R 1 , R 2 ) and note n 1 , n 2 . Single surface: use n 2(1/v) - n 1(1/u) = (n 2 - n 1)/R ; solve for the unknown and interpret the sign of v . Lens maker: compute f from R 1 , R 2 , and n L (or n L/n m ). Thin lens: use 1/v - 1/u = 1/f and then m = v/u for size and orientation. Problem-solving flow (spherical refraction and lenses) Refractive index contrast (nL − 1) none custom nL = 1 → no focusing typical glass 0.5 Common crown glass in air slope × 0.5 1/f (m−1) m−1 For fixed R1 and R2, 1/f varies linearly with (nL − 1) with slope (1/R1 − 1/R2). Lens power grows linearly with refractive index contrast for fixed curvatures. R1 control R2 control 2D PLOT 1/f vs (n − 1) for a thin lens invf = slope nm1 nm1 invf slope 1/R₁ − 1/R₂ When lenses are separated by a distance d (not in contact), the equivalent power is modified. You cannot just add 1/f; a separation term appears. Use the lens combination formula with separation if needed. tip Edge cases to notice: As R 1 (first face plane), the lens maker formula reduces to 1/f = (n L - 1)(-1/R 2) (air). If both R 1 and R 2 go to infinity, the element is a parallel plate (no focusing, only lateral shift). If n L 1 , the lens loses power. Physical intuition for signs: Positive curvature means a centre to the right; for the first surface, this typically corresponds to a bulge facing the object (converging for n L > n m ). The second surface of a biconvex lens then has its centre to the left, making R 2<0 , and the minus sign in (1/R 1 - 1/R 2) effectively adds the magnitudes of both curvatures. Why focal length changes in water: Refraction depends on the ratio of speeds v vac /v medium , captured by n . When a lens sits in water, the contrast between n L and n m reduces; rays bend less at each surface, increasing f . This is why goggles need a curved faceplate designed for water. Derivation sanity checks: Dimensional analysis gives [1/f] = [1/R] , consistent with the lens maker formula. Symmetry: swapping R 1 and R 2 while reversing light direction keeps physics invariant, provided you also swap signs appropriately. Common workflow in exams: (1) Sketch and assign signs. (2) Compute f via lens maker if needed. (3) Use 1/v - 1/u = 1/f to get v . (4) Use m = v/u for size and orientation. (5) If multiple elements are in contact, add powers first to get a single equivalent f . Nature of images with a converging lens (air): |u| > f gives a real, inverted image to the right; |u| = f sends rays parallel (image at infinity); |u| < f gives a virtual, erect image to the left. The same decision tree holds in a different medium after replacing f with f m . Single-surface limiting behavior: For u - , v tends to f 2 with sign set by R and (n 2 - n 1) . For u approaching the surface from the left, the image distance approaches a finite limit unless the curvature sign causes divergence. Always check consistency with ray sketches. Error control in numericals: Keep lengths in consistent units (all cm or all m). Use 2 significant figures for final NEET answers. Do not round intermediate steps aggressively; keep at least 3–4 significant digits until the end. Interpreting negative results: A negative v for a lens means the image is on the same side as the object (virtual). A negative f means the lens is diverging (like biconcave in air). A negative m indicates inversion; positive m indicates an erect image. Design perspective: For a target focal length with a given glass, you can trade between R 1 and R 2 . Making one face flatter (large |R| ) and the other more curved keeps 1/R 1 - 1/R 2 fixed, hence the same f , but may reduce aberrations or improve manufacturability. Connection to apparent depth: At a plane interface ( R ), the single-surface law reduces to n 2/v = n 1/u . In near-normal viewing ( u vertical depth), this yields the classic apparent depth relation, bridging curved and plane refraction cases conceptually. Reality check with signs on a biconcave lens (air): R 1<0 , R 2>0 . Then (1/R 1 - 1/R 2) is negative, and with (n L - 1)>0 , we get 1/f<0 so f<0 as expected for a diverging lens. The algebra matches physical intuition. Surface versus lens focal distances: A single surface has different focal distances depending on the incident side ( f 1, f 2 ). A thin lens in a uniform medium has a single f because the two surfaces combine symmetrically under the thin-lens approximation. When to avoid the thin-lens shortcut: If lens thickness is not negligible or the medium differs on the two sides of the lens, you should either (a) use the two-surface approach carefully including the separation, or (b) invoke principal planes for a thick-lens model. Combining a positive and a negative lens in contact can yield a weak net power (telephoto or wide-angle groups). Because powers add, a small mismatch can set a precise F while other elements correct aberrations. Key terms recap Small-angle rays near the axis so that sin and tan can be approximated by the angle in radians. Paraxial rays Cartesian sign convention Left-to-right positive axis, with distances measured from the pole/optical centre; objects to the left have negative u. Lens maker formula Links lens focal length to indices and surface curvatures: 1/f = (n L - 1)(1/R 1 - 1/R 2) in air. Power (dioptre) Reciprocal of focal length in metres, P = 1/f ; positive for converging, negative for diverging lenses. Ratio of image height to object height; for thin lenses m = v/u with sign conveying orientation. Magnification