Problem 50 - Physics for Scientists and Engineers with Modern Physics

Question

(II) The closely packed cones in the fovea of the eye have a diameter of about \(2 \mu \mathrm{m}\). For the eye to discern two images on the fovea as distinct, assume that the images must be separated by at least one cone that is not excited. If these images are of two point-like objects at the eye's \(25-\mathrm{cm}\) near point, how far apart are these barely resolvable objects? Assume the diameter of the eye (cornea-to-fovea distance) is \(2.0 \mathrm{~cm} .\)

Step-by-Step Solution

Answer

The objects must be 50 µm apart to be resolvable.

Step 1: Understand the problem

The problem involves determining the distance between two point-like objects that can be resolved by the eye when viewed at the near point. The images on the fovea should be separated by at least one cone to be distinguishable. Given the diameter of a cone and the diameter of the eye, we need to find the separation of the objects at a near-point distance of 25 cm.

Step 2: Determine the separation needed on the fovea

For the eye to resolve two points, there must be at least one non-excited cone between their images. The diameter of one cone is given as \(2\, \mu \mathrm{m}\), so the required separation on the fovea is \(2\times 2 = 4\, \mu \mathrm{m}\) to account for one non-excited cone between two excited cones.

Step 3: Use geometry to relate eye diameter to image separation

Using similar triangles, the separation on the fovea \(d_f\) is proportional to the separation \(d_o\) of the objects. The ratio of these separations is the same as the ratio of the distances from the eye lens to the objects (25 cm) and the eye lens to the fovea (2 cm): \[ \frac{d_o}{d_f} = \frac{25\, \mathrm{cm}}{2\, \mathrm{cm}} \]

Step 4: Calculate the separation of the objects

Using the relation derived in Step 3, solve for \(d_o\): \[ d_o = d_f \times \frac{25}{2} = 4\, \mu \mathrm{m} \times \frac{25\, \mathrm{cm}}{2\, \mathrm{cm}} = 50\, \mu \mathrm{m} \] The barely resolvable separation of the two objects at the near point is \(50\, \mu \mathrm{m}\).

Key Concepts

Resolution of the eye

The human eye's ability to distinguish two separate points is referred to as its resolution. This resolution is crucial in our daily activities, from reading to navigating our surroundings. In the fovea, the eye's most sensitive area, cones are packed closely together. These cones are photoreceptor cells responsible for processing detailed visual information.
To discern two objects, they must stimulate separate cones on the fovea, with at least one non-stimulated cone between them. For this reason, the eye's resolution is partly determined by the size and packing of these cones. In our exercise, it is mentioned that a cone has a diameter of about 2 micrometers (\(2 \mu \mathrm{m}\)). Thus, to resolve two points, they need to be separated by about 4 micrometers on the fovea.
This leads to the concept of visual acuity, which is the clarity or sharpness of vision. An eye with high resolution can distinguish finer details. Our resolution also relies on factors like lighting, contrast, and even our eye health.

Similar triangles

Similar triangles in optics help us relate distances and sizes in a simple and efficient way. The principle of similarity states that if two triangles have the same angles, their side lengths are proportional. This property is useful when determining the relationship between the size of an image formed on the retina and the corresponding object size and distance outside the eye.
In our exercise, the use of similar triangles allows us to find the real-world separation of two barely resolvable objects by linking it to their image separation on the fovea. We solved this by employing the formula:

• \(\frac{d_o}{d_f} = \frac{25\, \mathrm{cm}}{2\, \mathrm{cm}}\),

where \(d_o\) is the object separation, \(d_f\) is the fovea separation, 25 cm is the viewing distance, and 2 cm is the eye diameter. This equation shows that whatever happens on the fovea is a scaled-down version of what happens in the external world. By solving for \(d_o\), which is the separation perceived externally, we found it to be 50 micrometers.

Fovea

The fovea is the central focal point in the retina where the sharpest vision occurs. It is a small depression packed densely with cone cells that exclusively detect light. This high concentration of cones makes the fovea critical for activities requiring fine detail, such as reading and recognizing faces.
Unlike other parts of the retina where rod cells are also present, the fovea's cone-only setup is specialized for processing color and detailed vision in good lighting conditions. This specialization enhances the resolution and ability to distinguish fine details.
Understanding the fovea is key in optics and vision studies because it is central to how we perceive the world. The exercise showcases this by focusing on how two images need to be mapped onto the fovea with clear separation for us to see them distinctly. The arrangement and functioning of the fovea are integral to understanding how vision operates at its best resolution.