Wave Optics

To understand the relationship between the speeds of light in two different media and their refractive indices, we start by recalling Snell's Law and the definition of refractive index. The refractive index of a medium is given by the ratio of the speed of light in vacuum to the speed of light in that medium . Mathematically, this is represented as:

For two different media with refractive indices and and speeds of light and respectively, the relationship between these can be expressed as:

Dividing these two equations, we get:

Rearranging the equation gives:

To solve for the focal length of a convex lens when the object and image distances are both equal to , we use the lens formula:

where:

= focal length of the lens

= image distance

= object distance

Given that and , we substitute these values into the lens formula:

Taking the reciprocal to solve for :

Thus, the focal length of the convex lens is .

Farsightedness, also known as hypermetropia, is a common refractive error of the eye where distant objects can be seen clearly, but close ones do not come into proper focus. This condition occurs when the eyeball is too short or the refractive power of the eye is not sufficient, causing light rays to focus behind the retina instead of directly on it. Hypermetropia can be corrected using convex lenses, which help in converging the light rays onto the retina for clear vision of nearby objects. Other options like myopia refer to nearsightedness, presbyopia is age-related farsightedness due to stiffening of the lens, and cataract is a clouding of the lens, but neither relates directly to farsightedness as hypermetropia does.

To understand the relationship among the speed of light wave ( ), wavelength ( ), and frequency ( ), we utilize the fundamental equation of wave mechanics:

This equation indicates that the speed of a wave is the product of its frequency and wavelength. Therefore, to express wavelength in terms of frequency and speed, rearrange the equation:

From the options given:

The correct relationship, according to wave mechanics, is: