Problem 104 - University Physics with Modern Physics

Question

What Is the Smallest Thing We Can See? The smallest object we can resolve with our eye is limited by the size of the light receptor cells in the retina. In order for us to distinguish any detail in an object, its image cannot be any smaller than a single retinal cell. Although the size depends on the type of cell (rod or cone), a diameter of a few microns \((\mu \mathrm{m})\) is typical near the center of the eye. We shall model the eye as a sphere 2.50 \(\mathrm{cm}\) in diameter with a single thin lens at the front and the retina at the rear, with light receptor cells 5.0\(\mu \mathrm{m}\) in diameter. (a) What is the smallest object you can resolve at a near point of 25 \(\mathrm{cm}\) ? (b) What angle is subtended by this object at the eye? Express your answer in units of minutes \(\left(1^{\circ}=60 \mathrm{min}\right),\) and compare it with the typical experimental value of about 1.0 min. (Note: There are other limitations, such as the bending of light as it passes through the pupil, but we shall ignore them here.)

Step-by-Step Solution

Answer

The smallest object resolvable is 50 microns and it subtends an angle of about 0.687 minutes, close to the typical 1.0 minute.

Step 1: Understand the Problem

This task involves understanding the limits of human eye resolution based on the size of the light receptor cells in the retina. We need to find the smallest object size that can be resolved when it is at the nearest point of clear vision, which is 25 cm away, and determine the angle it subtends at the eye.

Step 2: Model the Eye as a Lens System

Model the eye as a lens system where the diameter is 2.50 cm. The retina is at the rear, with receptor cells 5.0 microns in diameter, providing the limit on image size at the retina.

Step 3: Calculate the Image Distance

Use the approximation that the lens is at the front of the eye, and the retina is at the back, so the distance from the lens to the retina is equal to the eye's diameter: 2.50 cm.

Step 4: Use the Lens Formula to Find Object Size

The lens formula is given by \(\frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i}\), where \(d_o\) is the object distance (25 cm), and \(d_i\) is the image distance (2.50 cm). For small angles, we approximate that \(\frac{h_i}{d_i} = \frac{h_o}{d_o}\). Knowing \(h_i = 5.0 \times 10^{-6}\) m (cell diameter), solve for \(h_o\).

Step 5: Calculate Object Height

For the smallest object, assume \(h_i = 5.0 \times 10^{-6}\) m. Rearrange the relation for similar triangles: \(h_o = h_i \times \frac{d_o}{d_i}\). Thus, \(h_o = 5.0 \times 10^{-6}\) m \(\times \frac{25}{2.5}\) = \(50 \times 10^{-6}\) m or 50 microns.

Step 6: Determine Angle Subtended at the Eye

Use small angle approximation, where the angle \(\theta\) in radians is \(\theta = \frac{h_o}{d_o}\). \(\theta = \frac{50 \times 10^{-6}}{0.25}\). Convert this angle from radians to degrees, then to minutes.

Step 7: Convert Radians to Degrees and Minutes

First, convert the angle from radians to degrees using the approximation \(1\) radian \(= \frac{180}{\pi}\) degrees. Then convert degrees to minutes knowing \(1^\circ = 60\) minutes. This will allow for comparison with the experimental value.

Step 8: Finalize the Calculations

Convert the calculated angle to minutes and compare with the known experimental value. Express any deviations or confirmations from typical values.

Key Concepts

Human Eye Anatomy

The human eye is an extraordinary organ capable of capturing intricate detail through its complex structure. At the front of the eye is the cornea, which helps to focus incoming light. Behind the cornea lies the lens, which fine-tunes this focus. The retina, positioned at the back of the eye, is a crucial component where images are formed. It contains specialized cells known as rods and cones, which serve as the retina's light receptors and are essential for color detection and light sensitivity.
The eye's anatomy allows it to function like a sophisticated camera. With a typical eye diameter of around 2.50 cm, it acts as a spherical optical device. This compact system efficiently processes light, enabling us to perceive images and colors in high detail.

Lens Formula

Understanding how an eye focuses involves grasping the concept of the lens formula. In simple terms, the lens formula is used to relate the distance of an object from a lens (\(d_o\)), the distance from the lens to the image formed (\(d_i\)), and the focal length of the lens (\(f\)). The lens formula is expressed as:
\[\frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i}\]
In the context of the human eye, the lens has to adjust constantly to bring objects at different distances into focus. For the exercise of resolving the smallest object at a near point of 25 cm, you use this formula to compute the size of the image on the retina. This calculation shows how any object creates an image on the retina by converging light rays through the lens system.

Retinal Light Receptor Cells

Retinal cells play a pivotal role in our visual resolution. These cells, known as rods and cones, are embedded in the retina and determine the sharpness with which we see an image. Typically, these light receptor cells measure a few microns in diameter—rods are responsible for vision at low light levels, while cones are vital for color vision and detail perception.
In this exercise, light receptor cells are modeled at about 5.0 microns in diameter, which is fundamental in determining the smallest resolvable object. If the image on the retina is smaller than a retinal cell, it can't be detected distinctly. Thus, the size of these cells essentially limits the finest detail your eyes can resolve.

Angle Subtension

Angle subtension is a critical concept in understanding how we perceive the size of objects. It refers to the angle formed between the lines extending from the observer's eye to the edges of an object. This angle implies how large or small an object appears from a particular distance.
For instance, an object at a 25 cm nearest vision distance in this exercise subtends a certain angle at the eye which can be quantified. Using the formula for small angles, the subtension angle (\(\theta\)) is calculated as:
\[\theta = \frac{h_o}{d_o}\]
where \(h_o\) is the height of the object and \(d_o\) is the distance from the eye. Understanding angle subtension is essential for quantifying perceived size, which can be converted from radians to degrees and then to minutes for precise comparison with standard observational values, such as one minute of visual arc.