A contact lens is in the shape of a convex meniscus (see Fig. 31.25). The inner surface is curved to fit the eye, with curvature radius \(7.80 \mathrm{mm} .\) The lens is made from plastic with refractive in\(\operatorname{dex} n=1.56 .\) If it has a \(44.4-\mathrm{cm}\) focal length, what's the curvature radius of its outer surface?

Refraction of light refers to the bending of light as it passes from one medium to another with a different refractive index, a measure of how much light slows down in a material. This concept is crucial in understanding lenses since lenses rely on refraction to bend light rays and focus them to a point.

When light enters a lens, its speed changes depending on the material of the lens. This speed change causes the light to bend or refract. The amount of bending depends on the angle at which the light enters the lens and the refractive index difference between the air (or initial medium) and the lens itself. For instance, in the exercise, the refractive index of plastic is 1.56, indicating how much steeper the light's pathway will be compared to the light traveling through air.

Using the understanding of refraction, the Lensmaker's Equation, described in the exercise, facilitates the calculation of the focal length of a lens based on its physical characteristics, such as its curvature and material.

The curvature radius of a lens surface is a measure of the degree of its curvature, meaning how 'bent' or 'curved' the lens is. This curvature greatly influences how light is refracted when passing through the lens.

A lens can have different curvatures on the two sides, leading to the terms 'radius of curvature of the first surface' and 'radius of curvature of the second surface,' respectively. For a spherical lens surface, the curvature radius is the radius of the sphere of which the lens surface is a part. In the exercise, one radius is provided (7.8 mm for the inner surface), and we are asked to find the curvature radius of the outer surface.

The sign of the curvature radius can indicate the type of lens surface: positive for surfaces that are convex (bulging outwards) and negative for concave surfaces (curving inwards). With this information, we can better understand how each lens surface contributes to the overall bending of light rays through the lens, impacting the focal point and magnification.

The focal length of a lens is the distance from the lens at which parallel rays of light converge to a single point after passing through the lens. It's a critical measure as it indicates how powerfully a lens converges or diverges light. Lenses with shorter focal lengths bend rays more strongly, bringing them to focus in a shorter distance.

In the context of the problem we have, a convex meniscus lens (like the contact lens in the exercise) is a positive lens, meaning it converges light and therefore has a positive focal length. The focal length is dependent on the refractive index of the lens material and the curvatures of the two lens surfaces.

The exercise provides the focal length (44.4 cm, which is 444 mm when converted to match the units of curvature radius), and we use the Lensmaker's Equation to find the unknown curvature radius. Understanding the focal length helps us in designing lenses with specific properties for various applications, such as in eyewear, cameras, and microscopes.