Question

A person can not see clearly an object kept at a distance beyond of \(100 \mathrm{~cm}\). Find the nature and the power of lens to be used for seeing clearly the object at infinity. \((1)-1 D\) (2) \(+1 D\) (3) \(-3 D\) (4) \(3 \mathrm{D}\)

Short answer

The correct lens is a converging lens with power +1 D (option 2).

Step-by-step solution

- Understanding the Problem

The person's far point is at 100 cm, meaning they cannot see objects clearly beyond this distance. To correct their vision, a lens is needed to help them see objects at infinity.

- Identify the Far Point

The far point is the distance at which the person can see clearly, given as 100 cm or 1 meter.

- Use Lens Formula

The lens formula for determining the power of the lens required to make the far point at infinity is: \[P = \frac{1}{f} = \frac{1}{d_{far}}\]

- Calculate Power

Substitute the far point distance into the formula: \[P = \frac{1}{1 \text{ meter}} = +1 \text{ diopter}\]

- Determine the Nature of the Lens

A positive power indicates that a converging lens (convex lens) is needed to correct the vision.

Key concepts

Lens Formula

To solve many problems involving lenses, we use the lens formula. This formula relates the object distance \(u\), the image distance \(v\), and the focal length \(f\) of the lens. The lens formula is:

\[ \frac{1}{f} = \frac{1}{v} + \frac{1}{u} \]

When dealing with corrective lenses, we often use a simplified version of this formula:

\[ P = \frac{1}{f} \]

Here, \(P\) represents the power of the lens in diopters (D), and \(f\) is the focal length in meters. This formula is essential for understanding how different lenses either converge or diverge light to correct vision. For instance, in the context of the given exercise, the objective was to find the necessary lens power to enable a person to see distant objects sharply.

Diopter Calculation

The concept of diopters is used to measure the optical power of a lens. The formula is:

\[ P = \frac{1}{f \text{(in meters)}} \]

In our exercise, the person's far point is at 100 cm, or 1 meter. This means the focal length \(f\) of the corrective lens should be 1 meter. Plugging this into the formula gives us:

\[ P = \frac{1}{1} = +1 \text{D} \]

This calculation shows that the lens needed is \( +1 \text{diopter}\), indicating the lens's optical power necessary to correct the vision. A positive diopter value typically points to the use of a convex lens, which helps the eye focus on distant objects properly.

Convex Lens

A convex lens, also known as a converging lens, is thicker in the middle than at the edges. This shape causes light rays entering the lens to bend toward each other and converge at a point known as the focal point.

Convex lenses are widely used in various optical devices like magnifying glasses, cameras, and corrective lenses for people with farsightedness (hyperopia). These lenses help correct vision by converging light rays before they hit the eye, thereby assisting the eye in focusing on objects that are far away.

In our exercise, the lens required has a power of \( +1 \text{D}\). This indicates a convex lens because it needs to converge light rays to create a clear image of distant objects on the person's retina. So, whenever you see a positive diopter value in vision correction, think of a convex or converging lens, which aids in focusing on distant objects by bending the light rays inward.