Problem 79 - Master Resource Book in JEE Main Physics

Question

A man's near point is \(0.5 \mathrm{~m}\) and far point is \(3 \mathrm{~m}\). Power spectacle lenses repaired for (i) reading purpose (ii) seeing distant object, respectively. (a) \(-2 \mathrm{D}\) and \(+3 \mathrm{D}\) (b) \(+2 \mathrm{D}\) and \(-3 \mathrm{D}\) (c) \(+2 \mathrm{D}\) and \(0.33 \mathrm{D}\) (d) \(-2 \mathrm{D}\) and \(+0.33 \mathrm{D}\)

Step-by-Step Solution

Answer

The answer is option (d): \\(-2 \mathrm{~D}\\) for reading and \\(+0.33 \mathrm{~D}\\) for distance vision.

Step 1: Understanding the Problem

A person's near point is the closest distance at which they can see objects clearly, while the far point is the farthest distance they can see without correction. This person's near point is 0.5 m and far point is 3 m. We need to calculate the power of the corrective lenses for both reading and distant vision.

Step 2: Calculating Lens Power for Reading

The lens needed for reading or seeing up close should correct the near point to the standard near point of 0.25 m (25 cm). Using the lens formula, \(\frac{1}{v} - \frac{1}{u} = \frac{1}{f}\), where \(v = -0.25 \mathrm{~m}\) and \(u = -0.5 \mathrm{~m}\), solve for \(f\). \[\frac{1}{f} = \frac{1}{-0.25} - \frac{1}{-0.5} = -4 + 2 = -2\, \mathrm{D}\] Thus, the power for reading is \(-2 \mathrm{~D}\).

Step 3: Calculating Lens Power for Distance Vision

To correct the far point to infinity, use the lens formula with \(v = \infty\) and \(u = -3 \mathrm{~m}\). \[\frac{1}{f} = \frac{1}{\infty} - \frac{1}{-3} = 0 + \frac{1}{3} = 0.33 \mathrm{~D}\] Therefore, the power for seeing distant objects is \(+0.33 \mathrm{~D}\).

Step 4: Matching the Results

The calculated powers for reading and distance are found in option (d), which is \(-2 \mathrm{~D}\) for reading and \(+0.33 \mathrm{~D}\) for distance vision, respectively.

Key Concepts

Lens Formula

In optics, the lens formula is crucial for understanding how lenses form images. It relates the object distance, image distance, and the focal length of a lens. The formula is represented as \(\frac{1}{v} - \frac{1}{u} = \frac{1}{f}\), where \(v\) is the image distance, \(u\) is the object distance, and \(f\) is the focal length. The lens formula helps in determining how a lens modifies light to correct or adjust the vision. In the original exercise, we used the lens formula to find the needed power of corrective lenses. Power is given by the formula \(P = \frac{1}{f}\), and it is measured in diopters (D). Knowing either the object or image distance allows us to solve for the other parameters easily. By focusing on these computations, professionals can create lenses tailored to correct refractive errors.

Corrective Lenses

Corrective lenses are designed to improve vision by adjusting the focal point of light entering the eye. They modify the path of light rays so that they properly converge on the retina, especially for those with vision issues such as nearsightedness or farsightedness. When constructing corrective lenses, we use the lens power calculated using the lens formula to ensure that the lenses adjust the focal length to meet standard vision needs. In the exercise, the target reading lens power is

• For reading: \(-2 \, \mathrm{D}\), which adjusts the near point to the more typical distance of 0.25 m.
• For distance: \(+0.33 \, \mathrm{D}\) to extend the far point to infinity, allowing distant objects to be clear.

The effectiveness of a corrective lens depends on accurate adjustments, helping people see objects at both near and far distances clearly.

Refractive Errors

Refractive errors in the eye lead to blurred vision. They occur when the shape of the eye prevents light from focusing directly on the retina. Common refractive errors include:

• Myopia (nearsightedness): Light focuses before it reaches the retina, causing distant objects to appear blurry.
• Hyperopia (farsightedness): Light focuses behind the retina, making nearby objects look blurry.
• Astigmatism: This causes a distorted image as light fails to come to a single focus point on the retina due to the irregular shape of the cornea.

Corrective lenses work by altering the focal point of incoming light rays, correcting these refractive errors. By using the lens formula, opticians adjust the lens power, helping guide light onto the correct part of the retina, thus improving visual clarity.

Near Point and Far Point

The near point and far point are key to understanding individual vision capabilities.

• The **near point** is the closest distance at which one can see an object clearly. For many, this is about 25 cm; however, in the exercise, the man's near point is 0.5 m.
• The **far point** is the farthest distance one can see clearly without correction. For someone with normal vision, this should be at infinity. In the exercise, his far point is 3 m, which requires correction to view distant objects properly.

By understanding these personal reference points, one can determine the necessary corrective interventions, such as lenses, to adjust vision to more typical standards and improve both near and distant sight.