Question

The cornea of the eye has a radius of curvature of approximately 0.50 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 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 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 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?

Short answer

The cornea needs a 14 mm curvature to focus distant objects. It focuses text 16 mm in front of the retina and distant objects 14.3 mm in front of it, needing a lens to adjust focus.

Step-by-step solution

Understand the Problem

We are asked to determine the radius of curvature of the cornea so that it alone can focus a distant mountain on the retina. We are given the radius of curvature of the cornea, the index of refraction, and the depth of the eye. We will use refraction and lens formula concepts. Additionally, we need to determine if the cornea can focus text from a screen and where it focuses a distant mountain based on its actual radius of curvature.

Use the Lensmaker's Formula

We will use the lensmaker's formula for a converging lens: \( \frac{1}{f} = (n - 1)\left( \frac{1}{R_1} - \frac{1}{R_2} \right) \), where \( f \) is the focal length, \( n \) is the index of refraction, and \( R_1 \) and \( R_2 \) are the radii of curvature. Since the cornea is a single surface, \( R_1 = R \) and \( R_2 \rightarrow \infty \), simplifying to \( \frac{1}{f} = \frac{n-1}{R} \).

Calculate Focal Length for Distant Mountain

For (a), the image is formed on the retina, 25 mm (0.025 m) from the cornea. Since the mountain is far away, the object distance is infinity, and the lens formula becomes \( \frac{1}{f} = \frac{1}{d_i} \). Thus, \( f = d_i = 0.025 \) m.

Find the Required Radius of Curvature

Using the lensmaker's formula and the calculated focal length \( f = 0.025 \) m, solve for \( R \):\( 0.025 = \frac{1.35-1}{R} \).Thus, \( R = \frac{0.35}{0.025} \approx 0.014 \) m or \( 14 \) mm.

Check Ability to Focus on Nearby Object

For (b), use the lens formula \( \frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i} \) for a computer screen 0.25 m away, and solve:\( \frac{1}{0.014} = \frac{1}{0.25} + \frac{1}{d_i} \).This gives \( d_i \approx 0.016 \) m or 16 mm, in front of the retina.

Determine Focus with Actual Cornea

For (c), we need to find where the actual cornea with \( R = 0.005 \) m focuses a distant object. Using \( f = \frac{R}{n-1} \), we find \( f = \frac{0.005}{0.35} \approx 0.0143 \) m. Therefore, the image is formed 14.3 mm from the cornea, which is in front of the retina.

Conclusion

The eye on its own does not focus distant objects onto the retina with the given radius, explaining the need for the eye's lens, which flexibly adjusts focal length, to focus light onto the retina for clear vision.

Key concepts

Cornea Curvature

The curvature of the cornea plays a crucial role in the eye's ability to focus light correctly. To understand this, think of the cornea as the eye's primary lens. It is transparent and is the first surface that light hits when entering the eye. The radius of curvature of the cornea determines how much the incoming light rays can bend. The point where these rays converge is essential for focusing visual images on the retina.

Typically, the human cornea has a radius of curvature around 5 mm. However, in some problems, like the one described above, you might be asked to find a different radius that would allow the cornea to focus light from distant objects, like a mountain, onto the retina without assistance from the eye's lens. The desired curvature ensures that the focal point is precisely on the retina, delivering a sharp image. Changes in the curvature, even if minor, can significantly affect eyesight, demonstrating the importance of the cornea's shape for effective vision.

Lensmaker's Formula

Lensmaker's formula is a vital tool in optics that helps us calculate the focal length of lenses, with particular relevance to the cornea in eye calculations. The formula is given by:

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

where:

• \(f\) is the focal length of the lens,
• \(n\) is the index of refraction of the lens material relative to the surrounding medium,
• \(R_1\) and \(R_2\) are the radii of curvature of the lens's two surfaces.

For the cornea, which generally acts like a single refractive surface, we can simplify this formula because the second surface radius, \(R_2\), is considered to be infinite. Thus, the formula reduces to:

• \[\frac{1}{f} = \frac{n-1}{R} \]

This equation allows us to relate the cornea's radius of curvature with its ability to focus light, directly affecting how clearly we see objects both near and far. By adjusting the radius \(R\), opticians can determine whether the cornea can focus light directly onto the retina or if additional corrective lenses are needed.

Focal Length

The focal length of a lens or curved surface, like the cornea, is the distance at which light rays converge to a point after passing through it. A shorter focal length means that the lens is stronger and can bend light rays more sharply. This is critical for focusing light precisely on the retina.

When we apply this to the human eye, particularly the cornea, the focal length determines where the image of a distant object, like a mountain, will be formed. In an ideally focused eye, the image should land directly on the retina. However, if the focal length is too short or too long due to the cornea's curvature, the image will appear either in front of or behind the retina.

To find the appropriate focal length of the cornea for a given radius of curvature, we use the simplified lensmaker's equation \(f = \frac{R}{n-1}\), where \(R\) is the radius of curvature and \(n\) is the refractive index. The task is to match this focal length with the distance to the retina, normally about 25 mm. Variations demand compensations like glasses or contact lenses, which focus light correctly for clear vision.