Refraction at Spherical Surfaces and Lens Maker's Formula

Choose the direction of incident light; mark R signs for each surface; use relative refractive index; substitute in lens maker's formula; convert focal length to metre if power is required; interpret positive or negative f.

Diagram should show a spherical refracting surface, pole P, centre C, radius R, object O, image I, normal through C, and refracted ray. For a lens, mark first and second surfaces, R1, R2, optical centre and principal axis.

For a biconvex lens, taking both R1 and R2 as positive gives 1/f = 0, which wrongly suggests no focusing.

Refraction at a spherical surface describes image formation when light passes between two media separated by a curved refracting boundary. The lens maker's formula gives the focal length of a thin lens in terms of refractive index and radii of curvature of its two surfaces.

Refraction at a spherical surface describes image formation when light passes between two media separated by a curved refracting boundary. The lens maker's formula gives the focal length of a thin lens in terms of refractive index and radii of curvature of its two surfaces. Spherical surface formula: n2/v - n1/u = (n2 - n1)/R. Lens maker's formula for a thin lens in air: 1/f = (n - 1)(1/R1 - 1/R2). More generally, n is the relative refractive index of lens material with respect to surrounding medium. Distances R1, R2 and f are in metres for SI use; refractive index is dimensionless. A biconvex glass lens has positive power in air because its surfaces make parallel rays converge. If placed in a medium with refractive index close to glass, its focal length increases greatly because the relative refractive index decreases.

For a single spherical refracting surface, the object and image distances are measured from the pole of the surface and the radius is signed using the Cartesian convention. The relation connects n1, n2, u, v and R. A thin lens is treated as two spherical refracting surfaces close together. Combining the two refractions gives the lens maker's formula. This result explains why focal length depends on lens material, surrounding medium and curvatures of the two faces. Spherical surface formula: n2/v - n1/u = (n2 - n1)/R. Lens maker's formula for a thin lens in air: 1/f = (n - 1)(1/R1 - 1/R2). More generally, n is the relative refractive index of lens material with respect to surrounding medium. Distances R1, R2 and f are in metres for SI use; refractive index is dimensionless. For a thin biconvex glass lens in air, n = 1.5, R1 = +20 cm and R2 = -20 cm. 1/f = (1.5 - 1)(1/20 - 1/(-20)) = 0.5 x (1/20 + 1/20) = 0.5 x 1/10 = 1/20 per cm. Thus f = 20 cm = 0.20 m. The lens is converging with power +5 D. It is asked as derivation of lens maker's formula, radius sign questions, focal length numericals, and conceptual questions on a lens immersed in another medium. For a biconvex lens, taking both R1 and R2 as positive gives 1/f = 0, which wrongly suggests no focusing.