Problem 94 - University Physics with Modern Physics

Question

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 (\(\mu\)m) is typical near the center of the eye. We shall model the eye as a sphere 2.50 cm in diameter with a single thin lens at the front and the retina at the rear, with light receptor cells 5.0 \(\mu\)m in diameter. (a) What is the smallest object you can resolve at a near point of 25 cm? (b) What angle is subtended by this object at the eye? Express your answer in units of minutes (1\(^\circ\) = 60 min), 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 resolvable object is 0.50 mm; it subtends an angle of 6.88 minutes at the eye.

Step 1: Understand the Eye Model

The eye is modeled as a sphere with a diameter of 2.50 cm. The retina is at the back, and the receptors have a diameter of 5.0 \( \mu \)m. The near point, where the image form, is 25 cm from the eye.

Step 2: Calculate the Smallest Image Diameter

Since the image must be at least as large as a retinal cell to resolve details, we consider the smallest image size possible. That size is 5.0 \( \mu \)m on the retina.

Step 3: Proportion of Image to Object Size Using Lens Formula

To find the smallest resolvable object, use proportion based on similar triangles. The diameter of the "eye" (2.50 cm) is to the retinal cell size (5.0 \( \mu \)m) as the viewing distance (25 cm) is to the size of the smallest object on the retina. Thus, set up the proportion: \( \frac{5.0 \times 10^{-6}}{2.5 \times 10^{-2}} = \frac{x}{25} \) and solve for \( x \), the size of the smallest object.

Step 4: Solve for Smallest Resolvable Object Size

Solve the proportion: \( x = \frac{5.0 \times 10^{-6} \times 25}{2.5 \times 10^{-2}} \). This simplifies to \( x = 5.0 \times 10^{-3} \). Therefore, the smallest object you can resolve is 0.50 mm in diameter.

Step 5: Calculate Subtended Angle at the Eye

Use the formula for the subtended angle \( \theta \approx \frac{d}{D} \), where \( d = 0.50 \) mm and \( D = 250 \) mm (converted 25 cm to mm): \( \theta \approx \frac{0.50}{250} \) radians.

Step 6: Convert Angle from Radians to Minutes

First convert the angle to degrees: radians to degrees \( \theta \approx \frac{0.50}{250} \times \frac{180}{\pi} \approx 0.1146 \) degrees. Convert degrees to minutes by multiplying by 60 \( \times 60 \), to get \( 6.88 \) minutes.

Step 7: Compare with Typical Experimental Value

The calculated subtended angle is 6.88 minutes, which is larger than the typical experimental value of about 1.0 minute. This suggests visual limitations beyond our simplified model.

Key Concepts

Resolution

Resolution is all about how our eyes can distinguish between two points or objects. It determines the smallest detail we can see without them blurring together.
In optics, this is crucial when examining how well we can see, understand images, or even read small text.

• Resolution depends on many factors like light conditions and the quality of the optics.
• In this exercise, it's limited by the size of the retinal cells within your eye.

The problem asks us to find the smallest object that can be resolved at a typical viewing distance. To do that, we use the size of light receptor cells. They act as the fundamental building block for seeing clear objects.
This problem simplifies by focusing solely on the physical size of these cells, letting us estimate the smallest detectable object at a specific distance.

Retinal Cells

Retinal cells are the tiny building blocks on the back of your eye, vital for seeing anything in detail.
There are two types: rods and cones, each handling different aspects of vision.

• Cones allow us to see color and fine detail but need brighter light.
• Rods help us see in low light, but give less detail and only in grayscale.

For this problem, the typical size of retinal cells is approximated to 5 \( \mu \)m in diameter.
This size is crucial because it directly limits how small something can be and still be seen clearly without merging with other details.

Subtended Angle

The subtended angle helps describe how large an object appears from a certain distance.
It's not about the actual size of the object, but how big it seems to us when we look at it.

• Imagine holding a small coin near your eye, it looks huge! But far away, it seems tiny.
• In optics, it provides a way to measure perceived size rather than real size.

In the exercise, we calculate this angle for the smallest resolvable object at 25 cm viewing distance.
First, we compute it in radians before converting it to other units like degrees or minutes to compare with experimental values.

Eye Model

The eye model used in this exercise is a simplified version of a real eye.
It treats the eye as a sphere with a single lens and doesn't take into account complexities like the cornea or lens thickness.

• In this model, the diameter is taken as 2.50 cm.
• This simplification is used to focus attention on how the retinal cells and image size interact.

With this, we can set up proportions using distances and diameters to find how small an object can be for us to still see it clearly.
The lens formula, even simplified here, allows for a foundational look at how sight works regarding the geometry of the eye and distances.