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Focus of the Eye. The comea 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 comes 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 correcly 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

01

Understanding the Lens Formula

We will use the lens maker's formula to find the focal length. For a spherical surface, the formula is given by:\[\frac{1}{f} = \frac{(n_2 - n_1)}{R}\]where \(f\) is the focal length, \(n_2\) is the refractive index of the aqueous humor behind the cornea, \(n_1\) is the refractive index of air (approximately equal to 1), and \(R\) is the radius of curvature, which we need to find. First, we calculate \(f\) using the condition that for a distant object, \(f\) is approximately equal to the depth of the eye (25 mm or 2.5 cm).

02

Calculating the Radius of Curvature

Rearrange the lens formula to solve for \(R\): \[R = \frac{(n_2 - n_1)}{1/f}\]. Substitute \(f = 2.5\, \text{cm}\), \(n_2 = 1.35\), and \(n_1 = 1\):\[R = \frac{1.35 - 1}{1/2.5} = 0.35 \times 2.5 = 0.875 \ cm\]Thus, the radius of curvature needed is approximately \(0.875 \ cm\).

03

Evaluating Close Object Focus

For an object at 25 cm from the eye (such as a screen), using the lens formula:\[\frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i}\]where \(d_o = 25 \ cm\) and \(f = 2.5 \ cm\). Solving for \(d_i\): \[\frac{1}{d_i} = \frac{1}{2.5} - \frac{1}{25} = 0.4 - 0.04 = 0.36\]\[d_i = \frac{1}{0.36} \approx 2.78 \ cm\]Thus, the cornea focuses the screen text at approximately 2.78 cm behind the cornea, which is not on the retina (2.5 cm), meaning the text is focused behind the retina.

04

Actual Cornea Focus of Distant Objects

Using the given radius of curvature, 0.5 cm, for distant objects like a mountain, calculate the focal point using \(R = 0.5\ cm\):\[f = \frac{R}{n_2 - n_1} = \frac{0.5}{0.35} \approx 1.43 \ cm\]The cornea focuses the mountain image at about 1.43 cm, which is significantly in front of the retina (2.5 cm), indicating the need for additional lens support to focus light precisely on the retina.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Lens Formula

The lens formula is an essential concept in optics, particularly when dealing with lenses and refracting surfaces. It helps us understand the relationship between the object distance from the lens, the image distance from the lens, and the focal length of the lens. In mathematical terms, the lens formula is expressed as:\[ \frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i} \]where:

• \( \frac{1}{f} \) is the focal length of the lens.
• \( \frac{1}{d_o} \) is the distance of the object from the lens.
• \( \frac{1}{d_i} \) is the distance of the image from the lens.

The focal length is a crucial factor in determining where the image will form when light passes through the lens. Understanding this relationship allows us to predict whether an image will appear on the retina or at some other point. Applying this formula helps illuminate how lenses, like the cornea of the eye, contribute to focusing images.

Focal Length

In optics, the focal length of a lens is a measure of how strongly the system converges or diverges light. It is the distance over which initially collimated (parallel) rays are brought to a focus. For a spherical lens, this crucial property tells us where the lens will bring light rays coming from infinity to a focus. For example, in the human eye, the focal length must match the depth of the eye to properly focus distant objects on the retina. If the focal length is shorter than the distance to the retina, the image will be formed in front of the retina, resulting in a blurred vision of distant objects. Conversely, if the focal length is longer, the image will form "behind" the retina, making the image appear out of focus. Thus, the precise calculation of focal length is vital for ensuring sharp vision.

Refractive Index

The refractive index, represented usually by \(n\), is an indicator of how much a medium changes the speed of light. It is defined by the ratio between the speed of light in a vacuum and the speed of light in the material medium. The refractive index is key to understanding how light bends, or refracts, when passing from one medium to another.In our exercise, the aqueous humor, a clear fluid filling the space between the cornea and the lens, has a refractive index of 1.35. This value indicates how much light will bend as it travels through it. The degree of bending is crucial for focusing light accurately on the retina. Differences in the refractive index between the cornea and the air (approximately 1) allow the cornea to focus light effectively, acting similarly but not identically to a lens. Knowing the refractive index helps us apply the lens formula to achieve accurate focus.

Radius of Curvature

The radius of curvature is a fundamental geometric attribute representing the radius of the sphere that aligns with the curvature of an optical surface, such as a lens or the cornea of the eye. A strong curvature corresponds to a smaller radius, leading to a greater bending (or refraction) of light rays. In optical terms, the radius of curvature directly impacts the focal length of the lens—a smaller radius results in a shorter focal length and vice versa. In our exercise context, adjusting the radius of curvature of the cornea impacts where the light is focused within the eye. If the radius is not appropriately configured based on the refractive index and eye dimensions, it can lead to visual defects like myopia or hyperopia, demonstrating why correct radius curvature is essential for optimal eye health and functionality.