A particular eye has a focal length of \(2.0 \mathrm{cm}\) instead of the 2.2 \(\mathrm{cm}\) that would put a sharply focused image on the retina. (a) Is this eye nearsighted or farsighted? (b) What corrective lens is needed?

Nearsightedness, scientifically known as myopia, is a common vision problem where objects in the distance appear blurry, while close objects can be seen clearly. This happens because the light rays entering the eye converge at a point before they reach the retina, resulting in a blurry image of distant objects.

In people with nearsightedness, the eyeball is usually longer than normal or the cornea has too much curvature, causing light rays to focus at a point in front of the retina. Symptoms typically include difficulty seeing distant objects, squinting, and eye strain. To correct this, diverging lenses, also known as concave lenses, are used. These lenses spread out the light rays before they enter the eye, thus allowing them to focus directly on the retina and create a clear image.

It's also worth noting that myopia is a growing concern worldwide, with increased screen time being one of the factors contributing to its rise—making understanding and addressing this condition more important than ever.

Farsightedness, or hypermetropia, is the opposite condition of myopia. If you are farsighted, you may find it easy to see distant objects, but struggle with objects that are close to you. Here, the light entering the eyes focuses behind the retina, usually because the eyeball is too short or the cornea has too little curvature.

Common signs of hypermetropia include difficulty focusing on close tasks such as reading or sewing, eye strain, and headaches after prolonged periods of close work. Corrective convex lenses, or converging lenses, are typically used to treat farsightedness. These lenses bend incoming light rays inward, helping them to focus directly on the retina and create a sharp image.

To ensure daily comfort and prevent further eye strain, getting the right prescription for hypermetropia is essential. Annual eye check-ups can help detect changes in vision and update prescriptions accordingly.

The lens formula is a fundamental concept in optics that explains how lenses form images. It's described by the equation \(\frac{1}{f} = \frac{1}{v} - \frac{1}{u}\), where \(f\) is the focal length of the lens, \(v\) is the distance of the image from the lens (image distance), and \(u\) is the distance of the object from the lens (object distance).

This formula is especially crucial when determining the type of corrective lens needed for vision problems like myopia or hypermetropia. By knowing the focal length of the faulty eye and the desired image distance on the retina, one can calculate the correct focal length of the needed lens. For example, when dealing with a nearsighted eye, a negative value for the corrective lens' focal length indicates the use of a diverging lens, as it is needed to adjust the focus onto the retina.

Mastery of the lens formula enables optometrists and lens manufacturers to precisely tailor lenses to correct individual vision issues, ensuring that the wearer has clear, comfortable vision in their daily lives.