Question

A person who is farsighted can see clearly an object that is at least \(2.5 \mathrm{~m}\) away. To be able to read a book \(2.0 \cdot 10^{1} \mathrm{~cm}\) away, what kind of corrective glasses should he purchase?

Short answer

Answer: To determine the necessary lens strength in diopters, we apply the thin lens formula and use the given information about the object and image distances. The person should buy corrective glasses with a strength of -_D_ diopters, which are concave lenses used to correct farsightedness.

Step-by-step solution

Convert distances to meters

It's easier to work with all distances in meters. The object distance, \(d_o\), is the distance between the book and the person's eye, which is given as \(2.0 \cdot 10^{1} cm = 20 cm\) or \(0.2 m\). The required image distance, \(d_i\), is the minimum distance at which the person can see objects clearly, which is given as 2.5 meters.

Apply the thin lens formula

We will now apply the thin lens formula to find the focal length of the corrective lenses: \(\frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i}\) We also know that for corrective lenses, the focal length will be negative for concave lenses (which are used to correct farsightedness).

Solve for the focal length

Plug in the values of \(d_o = 0.2 m\) and \(d_i = 2.5 m\) into the formula: \(\frac{1}{f} = \frac{1}{0.2} + \frac{1}{2.5}\) To make the focal length negative, we can flip both sides of the equation: \(f = -\frac{1}{\frac{1}{0.2} + \frac{1}{2.5}}\) Calculate the value of the focal length.

Calculate diopters and find the kind of corrective glasses

To determine the necessary lens strength in diopters (D), we have to take the reciprocal of the focal length (in meters): \(D = -\frac{1}{f}\) Use the calculated negative focal length from Step 3 to determine the lens strength in diopters. The person should buy corrective glasses with this diopter value, which are concave lenses used to correct farsightedness.

Key concepts

Thin Lens Formula

Understanding how lenses focus light is crucial when discussing vision correction. The thin lens formula provides a simple way to calculate the relationship between an object's position, the image it forms through a lens, and the focal length of the lens. It is expressed as \( \frac{1}{f} = \frac{1}{d_{o}} + \frac{1}{d_{i}} \), where \(f\) is the focal length of the lens, \(d_{o}\) is the distance from the lens to the object (object distance), and \(d_{i}\) is the distance from the lens to the image (image distance).

For a person with normal vision, the image distance is typically at the retina of the eye. However, if someone is farsighted, this image forms behind the retina due to the shape of their eye. To correct this, lenses need to alter the path of incoming light so that the image falls onto the retina, making the view clear. The thin lens formula helps us determine the precise focal length required for this correction.

Diopters Calculation

When it comes to corrective lenses, the lens power is measured in diopters (D). This tells us how much the lens will bend light, with a higher value indicating a stronger lens. Diopters are calculated as the reciprocal of the focal length in meters (D = \( -\frac{1}{f} \) ).

The negative sign in the calculation for farsighted individuals is vital because it signifies that the focal length is negative, indicating the use of a diverging lens. To determine the correct strength of the lens, we simply take the reciprocal of the negative focal length obtained through the thin lens formula equation. For example, if a lens has a diopter calculation of \( -2D \), it means that the focal length is \( -0.5m \). Accurate diopter calculation is essential for crafting glasses that will properly correct the wearer's vision.

Concave Lenses

Concave lenses are pivotal in correcting farsightedness. These lenses are thinner at the center than at the edges and are also known as diverging lenses because they spread out light rays that pass through them.

This quality makes them the opposite of convex lenses, which bring light rays together. The purpose of a concave lens in corrective glasses is to diverge the light before it reaches the eye, extending the image back onto the retina. Imagining light rays spreading out may help understand why these lenses are effective for farsighted individuals whose eyes focus light behind the retina. By using concave lenses in glasses, we can adjust the focal length in such a way that the point of focus is moved forward, allowing for a clear image to form exactly on the retina.

Farsightedness Correction

Farsightedness, or hypermetropia, is a common vision impairment where distant objects can be seen more clearly than close ones. The eye's lens focuses images behind the retina instead of directly on it. To correct this, lenses need to diverge the light so that it shifts the focal point forward.

The corrective glasses for a farsighted person should have concave lenses with the right diopter strength, tailored to the individual's needs. The process involves examining the distances at which the person sees clearly and then determining the strength of the lenses required to adjust the focal point to align with the retina. The goal is to make activities such as reading a book or looking at a computer screen comfortable and focused without straining the eyes.