Problem 54 - Physics for Scientists and Engineers with Modern Physics

Question

(II) Sherlock Holmes is using an 8.80 -cm-focal-length lens as his magnifying glass. To obtain maximum magnification, where must the object be placed (assume a normal eye), and what will be the magnification?

Step-by-Step Solution

Answer

Place the object 8.80 cm from the lens; magnification is approximately 29.41.

Step 1: Understand the problem

We are given an 8.80-cm-focal-length lens, and we need to find out where the object must be placed to achieve maximum magnification and also determine the magnification.

Step 2: Identify the formula for maximum magnification

To achieve maximum magnification with a magnifying glass, the object should be placed at the focal point of the lens. The formula for the magnification (M) of the lens is given by:\[ M = \left( \frac{250\, \text{cm}}{f} \right) + 1 \]where \( f \) is the focal length of the lens and 250 cm is the near point distance of a normal human eye.

Step 3: Substitute the given values into the magnification formula

Substitute the given focal length of 8.80 cm into the formula \[ M = \left( \frac{250\, \text{cm}}{8.80\, \text{cm}} \right) + 1 \]Now, calculate the magnification \( M \).

Step 4: Calculate the Magnification

Perform the calculation:\[ M = \left( \frac{250}{8.80} \right) + 1 = 28.41 + 1 = 29.41 \]Thus, the magnification is approximately 29.41.

Step 5: Determine the object placement

For maximum magnification using a convex lens, the object must be placed at the focal point. Since the focal length is 8.80 cm, the object should be placed 8.80 cm from the lens.

Key Concepts

Magnification

Magnification refers to how much larger or smaller an image appears compared to the object's actual size. In optics, it is a critical factor that defines how powerful a lens or other optical device can make an object appear. When using a lens such as a magnifying glass, we often seek to maximize this magnification to better observe small details.

• An object appears at its largest when it is at the focal point of the lens.
• For a magnifying glass with a convex lens, practical magnification can be found using the formula:

\[ M = \left( \frac{250\, \text{cm}}{f} \right) + 1 \]Where:

• \( M \) is the magnification,
• 250 cm is the average near point for a normal human eye,
• \( f \) is the focal length of the lens in centimeters.

Each number involved in the equation plays a part in the resultant quality of magnification, revealing how much closer an object appears with the lens.

Focal Length

The focal length of a lens is the distance between the center of the lens and its focal point—the spot where light rays converge to form the sharpest image. It is a fundamental property in lenses and deeply influences their optical qualities. When using a magnifying glass, the focal length plays a pivotal role in how well the lens can magnify an object.

• Shorter focal lengths are associated with greater magnifying power.
• In our exercise, the lens has a focal length of 8.80 cm.
• Placing an object exactly at this distance from the lens focal point allows the lens to achieve maximum magnification.

Understanding focal length helps in selecting the correct lens for optical tasks and affects how lenses are applied in real-world situations.

Convex Lens

A convex lens is a transparent optical device curved outward on both sides. These types of lenses are used to converge light rays to a point called the focal point and can magnify images, making them larger in appearance. Convex lenses are widely used in tools such as magnifying glasses, telescopic devices, and cameras because of this ability.

• They gather light and focus it internally.
• Convex lenses are essential for applications requiring image enlargement.
• Our current exercise involves using a convex lens to obtain maximum magnification, a practical illustration of its real-world use.

Whether in simple magnification tasks or more complex photographic equipment, convex lenses offer fundamental optical properties useful across numerous applications.

Lens Formula

The lens formula links the physical quantities related to a lens, specifically the object distance (\( u \)), the image distance (\( v \)), and the focal length (\( f \)). It provides a concise way to calculate these relationships and helps determine the properties of images formed by lenses.The lens formula is written as:\[ \frac{1}{f} = \frac{1}{v} + \frac{1}{u} \]

• \( f \) represents the focal length of the lens.
• \( v \) denotes the distance from the lens to the image.
• \( u \) is the distance from the lens to the object.

Using this formula, one can determine either the position of an object, the location of its image, or the lens's required power to achieve a desired focal length. It is a fundamental equation used in various optical calculations.