Problem 97 - Physics for Scientists and Engineers with Modern Physics

Question

A child has a near point of \(15 \mathrm{~cm}\). What is the maximum magnification the child can obtain using an \(8.5-\mathrm{cm}\) -focallength magnifier? What magnification can a normal eye obtain with the same lens? Which person sees more detail?

Step-by-Step Solution

Answer

The normal eye, with a magnification of 3.94, sees more detail than the child.

Step 1: Understanding the Problem

The problem involves calculating the maximum magnification for a magnifier with a given focal length both for a child and for a normal eye. We must also compare which person can see more detail based on their respective maximum magnifications.

Step 2: Maximum Magnification Formula

The maximum magnification (M) a lens can provide is given by the formula: \[ M = \left( \frac{D}{f} \right) + 1 \] where \(D\) is the near point of the eye (for a child it's 15 cm, for a normal eye it's 25 cm), and \(f\) is the focal length of the lens (8.5 cm in this case).

Step 3: Calculate Maximum Magnification for Child

Substitute the child’s near point (15 cm) into the formula: \[ M = \left( \frac{15}{8.5} \right) + 1 \] Calculate to find the child’s maximum magnification.

Step 4: Compute the Child’s Magnification

After the substitution: \[ M = 1.76 + 1 = 2.76 \] Hence, the maximum magnification the child can achieve is 2.76.

Step 5: Calculate Maximum Magnification for Normal Eye

Substitute the normal near point (25 cm): \[ M = \left( \frac{25}{8.5} \right) + 1 \] Calculate to determine the magnification for a normal eye.

Step 6: Compute the Normal Eye’s Magnification

After the substitution: \[ M = 2.94 + 1 = 3.94 \] Thus, the maximum magnification for a normal eye is 3.94.

Step 7: Compare the Magnifications

The normal eye achieves a higher magnification (3.94) compared to the child (2.76). Thus, the normal eye can see more detail.

Key Concepts

Near Point

The near point of an eye refers to the closest distance at which the eye can focus on an object clearly. It varies from person to person and can also change with age. For a child, the near point is often shorter compared to adults, typically around 15 cm.
The near point is crucial when calculating magnification because it directly impacts the size of the image perceived through a lens. A shorter near point allows the eye to focus on closer images naturally.

• For children: Near point is approximately 15 cm.
• For adults (normal eyes): Near point is typically 25 cm.

This concept is integral when using visual aids like magnifying glasses, as it determines how effectively one can see high-magnification images.

Focal Length

The focal length of a lens is the distance between the lens and the point where it converges light to form a clear image. In magnifying lenses, the focal length influences how much the light is bent and thus how much the object appears magnified.
In this exercise, the lens has a focal length of 8.5 cm. This relatively short focal length suggests that the lens is designed to magnify images that are relatively close to the viewer.

• Shorter focal length leads to higher magnifying power.
• The magnification achieved by any lens depends significantly on its focal length.

Choosing the right focal length can significantly enhance the effectiveness of a lens for different viewers, reflecting their need to see more detail.

Lens Formula

The lens formula links the near point, focal length, and the magnification capability of a lens. It is a crucial equation in optics that helps in calculating the magnification a lens can provide:
\[ M = \left( \frac{D}{f} \right) + 1 \]
Here, \( M \) is the magnification, \( D \) is the near point, and \( f \) is the focal length. The lens formula helps us predict how much an object will grow in size when viewed through the lens.

• The magnification is directly proportional to the near point and inversely proportional to the focal length.
• Understanding this formula allows us to calculate and compare magnifications for different users.

This formula is essential for anyone interested in optics to understand how lenses can enhance vision.

Maximum Magnification

Maximum magnification refers to the highest possible size increase of an object as perceived through a lens. It is determined by the specific near point of the observer and the focal length of the lens.
In the given problem, a child achieves a maximum magnification of 2.76, while a normal eye can attain 3.94. It highlights how the same lens can perform differently based on individual visual characteristics.

• Higher near point (common in adults) leads to greater magnification.
• Optimal choice of lens and understanding personal near point assures the best visual enhancement.

By comparing different users' maximum magnifications with the same lens, we learn that finer details can be observed by the normal eye, owing to its ability to achieve higher magnification.