Problem 28 - College Physics

Question

\(\cdot\) You want to view an insect 2.00 \(\mathrm{mm}\) in length through a magnifier. If the insect is to be at the focal point of the magnifier, what focal length will give the image of the insect an angular size of 0.025 radian?

Step-by-Step Solution

Answer

The focal length should be 80 mm.

Step 1: Understand the Problem

You want to use a magnifier to view an insect, and you need to find the focal length that will result in the insect's image having an angular size of 0.025 radians.

Step 2: Identify the Formula

The angular size \( \theta \) of an object viewed through a magnifier when at the focal point is given by \( \theta = \frac{h}{f} \) where \( h \) is the actual height of the object (insect) and \( f \) is the focal length of the lens.

Step 3: Rearrange the Formula

To solve for the focal length \( f \), rearrange the formula: \( f = \frac{h}{\theta} \).

Step 4: Substitute Values

Substitute the given values into the formula. The insect's length \( h \) is 2.00 mm, which must be converted to meters, so \( h = 0.002 \) meters. The angular size \( \theta \) is 0.025 radians. Substituting these values gives: \[ f = \frac{0.002}{0.025} \]

Step 5: Calculate the Focal Length

Calculate the value of \( f \) by performing the division: \[ f = 0.08 \, \text{meters} \].

Step 6: Convert to Desired Units

If needed, convert the focal length to millimeters since the result in meters might not be practical or expected for lens specifications. Thus, \[ f = 80 \, \text{mm} \].

Key Concepts

Magnification

Magnification is a critical concept in optics, especially when dealing with lenses and magnifiers. It refers to how much larger or smaller an image appears compared to its actual size. When you're using a magnifier, you're essentially enhancing your ability to see small objects by increasing the object's apparent size.
The magnification factor is defined as the ratio of the image size to the object size. It helps in determining how much a lens can enlarge the image of an object.

• For a magnifying lens, magnification is positive, meaning the image is erect and larger than the object.
• Magnification can be calculated using various methods, depending on the type of lens and application.

By adjusting the lens's properties, you can achieve different levels of magnification to observe tiny details more clearly.

Focal length

The focal length of a lens is the distance from the lens's surface to its focal point, where light rays converge. It is a crucial property of any lens as it determines how the lens bends and focuses light.
The focal length is inversely related to the lens's power, meaning lenses with shorter focal lengths have stronger focusing abilities. This impacts the size and clarity of the image you observe.

• Lenses with shorter focal lengths have a wider field of view and provide greater magnification.
• Focal length can be adjusted by changing the curvature of the lens, its thickness, or the material it's made from.

In our exercise, knowing the desired angular size and the object's height allows us to determine the focal length needed to achieve the desired image.

Angular size

Angular size refers to how large an object appears to an observer. It's measured in radians and is particularly important in situations where the actual size isn't as critical as the size perceived through a lens.
In optics, angular size helps in analyzing how objects of different physical dimensions appear under specific viewing conditions. This is crucial when using optical instruments like telescopes or magnifying glasses.

• Angular size can be estimated by the formula \( \theta = \frac{h}{f} \), where \( h \) is the object height and \( f \) is the focal length.
• The wider the angle, the larger the object appears.

In the example, the goal is to achieve an angular size of 0.025 radians for a clear and detailed view of the insect.

Lens formula

The lens formula is a fundamental concept in optics that relates the object distance, image distance, and focal length of a lens. It is generally expressed in the form \( \frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i} \), where \( f \) is the focal length, \( d_o \) is the object distance, and \( d_i \) is the image distance.
This formula helps in determining various properties of the image formed by a lens, such as position, nature, and size.

• The lens formula is applicable to both concave and convex lenses, though the sign conventions might differ.
• Understanding the lens formula is critical for designing optical systems and troubleshooting optical equipment.

In our specific problem, the focal length calculation helped in setting the parameters for magnifying the insect effectively.