Final answer:
The focal length of a lens increases as the radius of curvature increases and decreases with an increasing refractive index; these relationships are governed by the lens maker's formula.
Step-by-step explanation:
The relationship between focal length f, radius of curvature, and refractive index is given by the lens maker's formula, which is an essential concept in optics, a branch of physics. The focal length of a lens is influenced by both its radius of curvature and its refractive index. Specifically, the focal length f is inversely proportional to the radius of curvature; meaning that as the radius of curvature increases, the focal length increases.
On the other hand, the refractive index indicates how much a material bends light. A higher refractive index means the material bends light more, resulting in a shorter focal length for a lens made of that material. Therefore, as the refractive index increases, the focal length f decreases.