Final answer:
The focal length of a thin convex lens is inversely proportional to its radii of curvature and is affected by the refractive indices of the lens material and the surrounding medium.
Step-by-step explanation:
The dependence of the focal length of a thin convex lens on its radii of curvature can be understood through the lens maker's equation. This mathematical relationship indicates that the focal length, f, is inversely proportional to the radii of curvature R1 and R2 of the lens surfaces, as well as dependent on the index of refraction of the lens material, n, and that of the surrounding medium. A plano-convex lens, with one flat surface (R₁ = ∞) and one convex surface (R₂ = -35 cm and n = 1.5), illustrates this dependence, yielding a specific focal length through the lens maker's equation.
For a lens in air, where the refractive index is typically considered to be 1.0, and assuming that the lens material has a higher index of refraction, the focal length becomes shorter as the radii of curvature decrease, making the lens more powerful. Conversely, if the radii of curvature increase, the focal length increases, resulting in a less powerful lens. This relationship is crucial in applications such as designing eyeglasses, cameras, and optical instruments where precision in focal length is required for proper functionality.