Problem 30 - College Physics

Question

\(\cdot\) A thin lens with a focal length of 6.00 \(\mathrm{cm}\) is used as a sim-ple magnifier. (a) What angular magnification is obtainable with the lens if the object is at the focal point? (b) When an object is examined through the lens, how close can it be brought to the lens? Assume that the image viewed by the eye is at infinity and that the lens is very close to the eye.

Step-by-Step Solution

Answer

Maximum angular magnification is 4.17. Object can be as close as 6.00 cm from the lens.

Step 1: Understanding Angular Magnification Formula

The angular magnification for a simple magnifier is given by the formula \( M = \frac{25\, \text{cm}}{f} \), where \( f \) is the focal length of the lens in centimeters. Here, \( f = 6.00 \) cm.

Step 2: Calculating Angular Magnification

Substitute \( f = 6.00 \) cm into the formula: \( M = \frac{25\, \text{cm}}{6\, \text{cm}} = \frac{25}{6} \approx 4.17 \). Thus, the angular magnification is approximately 4.17.

Step 3: Object Location Relative to Lens

The object must be at the focal point of the lens for the image viewed to be at infinity through a simple magnifier. This occurs because the rays diverge from the focal point such that they exit parallel through the lens. Hence, the object should be placed at 6.00 cm from the lens.

Step 4: Comparison with Eye's Near Point

Assume the lens is very close to the eye and the image is at infinity for maximum comfort. Since the object is at the focal point and the calculated focal length is 6.00 cm, 6.00 cm is how close the object can be brought to the lens while allowing the image to be focused at infinity.

Key Concepts

Thin Lens Formula

In geometric optics, the thin lens formula is a fundamental equation that relates the object distance (\(d_o\)), the image distance (\(d_i\)), and the focal length (\(f\)) of a lens. The equation is expressed as:

• \(\frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i}\)

This formula is crucial for predicting how a lens will form an image depending on the position of the object in front of it. It applies to both converging (convex) and diverging (concave) lenses. However, for a simple magnifier, we often consider scenarios where the object is placed at the focal point to achieve specific outcomes, like viewing images at infinity. Understanding this relationship helps in designing optical instruments and predicting their behavior in practical applications.

Focal Length

The focal length (\(f\)) of a lens is the distance from the lens to the point where parallel rays of light converge or appear to converge. This value is a crucial determinant of a lens's power to magnify or converge light.
For instance, a shorter focal length, like our example of 6.00 cm, means the lens has a higher optical power and can magnify objects more.
In the context of a simple magnifier, placing an object at the focal length from the lens results in a virtual image that appears at infinity, which is ideal for comfortable viewing with minimal eye strain. Moreover, the focal length is a fixed property of the lens and is defined by its shape and the refractive index of the material from which it is made.

Angular Magnification

Angular magnification for a simple magnifier indicates how much larger an object appears when viewed through the lens compared to viewing with the naked eye at the nearest comfortable distance (usually 25 cm for the human eye).
The formula for angular magnification (\(M\)) is given by:

• \(M = \frac{25\,\text{cm}}{f}\)

Where \(f\) is the focal length of the lens.
For a lens with a 6.00 cm focal length, this value is calculated as 4.17. This means the object appears approximately 4.17 times larger when observed through the lens.
This capability makes simple magnifiers extremely useful tools for activities requiring detailed visual inspection, such as reading small print or examining fine details in objects.

Simple Magnifier

A simple magnifier is a single convex lens used to magnify objects, making them easier to see. This device operates by creating a virtual image of an object when it is placed at the focal point of the lens.
In our scenario, assuming the lens is placed very close to the eye and the viewed image is at infinity, this setup minimizes eye strain by allowing the object to be viewed comfortably.
Additionally, by positioning the object at the focal length from the lens (6.00 cm in our example), parallel rays exit the lens, making the optics appear to come from infinity.

• This arrangement is ideal for prolonged use as it is least taxing on the eyes.
• The simple design of these magnifiers also makes them affordable and easy to use educational tools.