Problem 24 - College Physics

Question

Focus of the eye. The cornea of the eye has a radius of curvature of approximately \(0.50 \mathrm{cm},\) and the aqueous humor bbehind it has an index of refraction of \(1.35 .\) The thickness of the cornea itself is small enough that we shall neglect it. The depth of a typical human eye is around 25 5 \(\mathrm{mm}\) . (a) What would have to be the radius of curvature of the cornea so that it alone would focus the image of a distant mountain on the retina, which is at the back of the eye opposite the cornea? (b) If the cornea focused the mountain correctly on the retina as described in part \((a),\) would it also focus the text from a computer screen on the retina if that screen were 25 \(\mathrm{cm}\) in front of the eye? If not, where would it focus that text, in front of or behind the retina? (c) Given that the cornea has a radius of curvature of about \(5.0 \mathrm{mm},\) where does it actually focus the mountain? Is this in front of or behind the retina? Does this help you see why the eye needs help from a lens to complete the task of focusing?

Step-by-Step Solution

Answer

(a) Required radius: 8.75 mm. (b) Text focuses slightly in front of retina. (c) Mountain focuses in front of retina; a lens is needed.

Step 1: Analyze the Problem

We're given a cornea with a radius of curvature of \(5.0\) mm (or \(0.50\) cm) and an index of refraction \(n=1.35\). We neglect cornea thickness. The human eye is approximately 25 mm deep. We need to use the refraction and thin lens formulas to solve parts (a), (b), and (c).

Step 2: Lensmaker’s Equation for Part (a)

We use the lensmaker's equation for a spherical surface: \[ \frac{n_2}{s} + \frac{n_1}{s'} = \frac{n_2 - n_1}{R} \]Where \(s\) is the object distance (approaching infinity for a distant mountain), \(s'\) is the image distance (25 mm), \(n_1=1\) is the refractive index of air, \(n_2=1.35\) is for the aqueous humor, and \(R\) is the radius of curvature we need to find.

Step 3: Solve for Radius of Curvature in Part (a)

When the mountain is a distant object, \(s \to \infty\) and this simplifies the lensmaker's equation to:\[ \frac{1}{s'} = \frac{n_2 - n_1}{R} = \frac{0.35}{R} \]Since \(s'=25\) mm, we solve for \(R\):\[ R = 25 \times 0.35 = 8.75 \text{ mm} \]

Step 4: Examine Focusing of Nearby Objects for Part (b)

We now investigate if an object at 25 cm (250 mm) can focus on the retina (at \(s'=25\) mm) given \(R=8.75\) mm. Using the lensmaker's equation, we have:\[ \frac{1.35}{250} + \frac{1}{25} = \frac{0.35}{8.75} \]Calculate to see if this equation balances or solve for \(s'\) to check the image distance.

Step 5: Calculate Image Distance for Part (b)

Solving the equation:\[ \frac{1.35}{250} + \frac{1}{s'} = \frac{0.35}{8.75} \]Rearrange and solve for \(s'\):\[ \frac{1}{s'} = \frac{0.35}{8.75} - 0.0054 \approx \frac{1}{24.39} \]So \(s' \approx 24.39\) mm.Since \(s'\) is slightly less than 25 mm, the focus is just in front of the retina.

Step 6: Analyze Current Structure for Part (c)

For \(R=5.0\) mm, solve where a distant object (\(s \to \infty\)) is focused. Use:\[ \frac{1}{s'} = \frac{0.35}{5.0} = 0.07 \to s' \approx 14.29 \text{ mm} \]Thus, the image of the distant mountain is focused at 14.29 mm, which is in front of the retina at 25 mm.

Step 7: Interpret Results

With a calculated \(R\) of \(8.75\) mm, neither a distant object nor a nearby one focuses precisely on the retina, indicating the need for an additional lens. The natural \(R=5.0\) mm places the distant focus in front of the retina, and nearby text also focuses incorrectly, underscoring the eye lens' necessity.

Key Concepts

Radius of Curvature

The radius of curvature is a fundamental concept in optics and relates to the bending of the lens surface. It is particularly important in the human eye where the cornea, acting as the eye's primary refractive surface, has a specific radius of curvature. This value determines how light is bent or focused by the cornea.
- In our example, the cornea's radius of curvature is given as approximately 5.0 mm.
- This curvature impacts how light from objects, whether distant like a mountain or near like a computer screen, meets the retina.
- If the curvature is not suitable, light may not focus correctly on the retina, causing blurred vision.
The radius of curvature directly influences how well the eye focuses light onto the retina. A larger radius means the cornea is flatter, and a smaller radius indicates a more curved cornea. Adjusting the cornea's curvature can help focus light properly on the retina.

Index of Refraction

The index of refraction is a measure of how much a material can bend, or refract, light. In the context of the human eye, it’s crucial for understanding how the different parts affect light as it enters the eye.
- The aqueous humor, the liquid behind the cornea, has an index of refraction of 1.35.
- Light travels slower through the aqueous humor compared to air, resulting in light bending as it enters the eye.
The difference in the index of refraction between air ( 1.0 ) and the aqueous humor affects how light is focused. This bending, or refraction, helps project a sharp image onto the retina. Without the correct index of refraction, the focused image can fall short or go beyond the retinal plane, affecting clarity of vision.

Lensmaker's Equation

The lensmaker's equation is a formula used to determine the focal properties of a lens based on its physical parameters like the radius of curvature and refractive indices.
- Mathematically, it’s expressed as:
\[ \frac{n_2}{s} + \frac{n_1}{s'} = \frac{n_2 - n_1}{R} \]
- Here, \(s\) is the object distance, \(s'\) is the image distance, \(n_1\) and \(n_2\) are refractive indices of air and the eye's internal fluid respectively, and \(R\) is the radius of curvature.
This equation is crucial for calculating how a lens, such as the eye’s cornea, will focus light. By knowing these parameters, one can determine how and where an image will be focused inside the eye. In our context, it shows that for different \(R\) values, the focus of light can change significantly.

Image Distance

Image distance in optics refers to the distance from the lens to where the image is formed, in this case, the retina.
- For clear vision, the cornea and other eye lenses together should focus the light directly onto the retina.
- In the human eye, this distance is consistently around 25 mm under normal conditions.
Using calculations from the lensmaker's equation, if light from distant or close objects does not converge accurately at this 25 mm, vision appears blurred.
- In the given exercise, differences in image distance for a nearby text compared to a distant scene highlight how precise the focus needs to be.
Understanding image distance helps comprehend vision problems like myopia or hyperopia. A precise image distance is paramount for sharp vision.

Refraction in the Eye

Refraction in the eye is the process through which light bends as it passes through different substances before hitting the retina. This bending is crucial because it allows the eye to focus images correctly.
- The cornea and lens in the eye work together to refract light in such a way that all rays converge exactly on the retina.
- Accurate refraction is essential for sharp images.
In this exercise, changes in the radius of curvature, and variations in index of refraction, revealed how small changes can have significant effects on image clarity. Without precise refraction, the image may focus before reaching the retina or continue beyond it, causing the need for corrective lenses.
This complex yet efficient process ensures we perceive the world in clear, focused detail.