Problem 45 - 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 behind 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 \(25 \mathrm{~mm}\). (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) Radius of curvature needed is 0.875 cm. (b) It focuses text in front of retina. (c) Actual focus of mountain is in front, showing need for lens.

Step 1: Understanding the Problem

We need to determine the radius of curvature for the cornea that will focus parallel light rays from a distant object onto the retina, taking the index of refraction into account. For part (b), we will analyze if the same curvature can focus nearby objects like text on a screen. For part (c), we will consider if the actual physical curvature of the cornea properly focuses distant objects.

Step 2: Using Lensmaker's Equation (Part a)

We're assuming the cornea behaves like a lens. The lensmaker's equation for a lens with one side in air is: \( \frac{1}{f} = (n-1) \left( \frac{1}{R_1} - \frac{1}{R_2} \right) \). Here, for the cornea, we have \(R_2 = \infty\) since the back side is assumed flat, so \( \frac{1}{R_2} = 0 \). We need to find \(R_1 = R\) such that the focal length \(f = 25 \text{ mm} = 2.5 \text{ cm}\). We substitute these values in the equation: \[ \frac{1}{2.5} = (1.35 - 1) \left( \frac{1}{R} \right) \].

Step 3: Solving for Radius of Curvature (Part a)

Simplifying the equation: \( \frac{1}{2.5} = 0.35 \times \frac{1}{R} \), therefore, \(R = 2.5 \times 0.35 = 0.875 \text{ cm} \). Thus, the radius of curvature must be \(0.875 \text{ cm}\) to focus a distant mountain on the retina.

Step 4: Understanding Focal Capabilities (Part b)

We check if a screen 25 cm away would be focused by calculating the image distance using the thin lens equation: \( \frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i} \), substituting \(f = 2.5 \text{ cm}\) and \(d_o = 25 \text{ cm}\). Thus \( \frac{1}{2.5} = \frac{1}{25} + \frac{1}{d_i} \).

Step 5: Solving for Image Distance of Text (Part b)

Calculating from the equation: \( \frac{1}{d_i} = \frac{1}{2.5} - \frac{1}{25} \). Therefore, \( \frac{1}{d_i} = 0.4 - 0.04 = 0.36 \), resulting in \(d_i = \frac{1}{0.36} \approx 2.78 \text{ cm}\). The text focuses at 2.78 cm, in front of the retina.

Step 6: Analyzing the Actual Curvature (Part c)

Using the real radius of curvature \(R = 0.5 \text{ cm}\), we calculate where it focuses the mountain. Plugging into the lens equation: \( \frac{1}{f} = 0.35 \times \frac{1}{0.5} = 0.7 \), thus \(f = \frac{1}{0.7} \approx 1.43 \text{ cm}\). So, it focuses in front of the retina at 1.43 cm, suggesting additional focusing power (lens help) is needed.

Key Concepts

Lensmaker's Equation

The lensmaker's equation is a fundamental principle in optics that helps us understand how lenses form images. In the context of the human eye, it approximates the cornea as a lens, because the cornea bends light to help focus images on the retina.
The equation is presented as:

• \( \frac{1}{f} = (n-1) \left( \frac{1}{R_1} - \frac{1}{R_2} \right) \)

Here:

• \( f \) is the focal length,
• \( n \) is the refractive index of the lens material (for the eye, it's the index of the aqueous humor behind the cornea),
• \( R_1 \) and \( R_2 \) are the radii of curvature of the two surfaces of the lens.

When considering the cornea:

• The back surface is considered flat, making \( R_2 = \infty \), and thus \( \frac{1}{R_2} = 0 \).

By substituting these values, the equation simplifies, enabling us to solve for the radius of curvature \( R_1 \) that focuses images perfectly on the retina. Understanding and applying this equation helps us determine if a lens (or cornea) is appropriately shaped for clear vision.

Refraction

Refraction is the bending of light as it passes from one medium to another. This principle is critical in optics because it explains how lenses, including our eye’s cornea, can focus light.
In the human eye, when light enters through the cornea, it is bent or refracted so that it can be focused properly on the retina to form a clear image. The amount of bending is determined by the refractive index of the material.
The refractive index is a measure of how much light slows down in a medium, compared to in a vacuum. The refractive index of the aqueous humor in the eye is about 1.35, which bends light enough for the cornea to function effectively as a lens.
This bending is crucial in everyday vision:

• It affects how images are focused, which determines the clarity of what we see.
• Any problems in refraction can lead to common vision issues such as myopia or hyperopia.

Therefore, understanding refraction helps us appreciate how light behaves in our eyes and how corrective lenses work to address refractive errors.

Radius of Curvature

The radius of curvature is a measure of the curvature of a lens surface. For the cornea, this is a critical factor in determining how well it can focus light onto the retina.
In practical terms:

• A smaller radius means a more curved surface, which bends light rays more sharply.
• A larger radius indicates a flatter surface, bending light less.

For the eye's cornea, the ideal radius of curvature ensures that parallel rays from distant objects converge directly on the retina. This is crucial for clear vision of distant mountains, for example.
When the radius is not optimal:

• Light may focus in front of or behind the retina, causing blurry vision.

In our exercise, determining the right radius of curvature using the lensmaker's equation is key to understanding how the eye achieves focus and why sometimes additional optical correction is necessary.