Question

In a simplified model of the human eye, the aqueous and vitreous humors and the lens all have a refractive index of 1.40, and all the refraction occurs at the cornea, whose vertex is 2.60 cm from the retina. What should be the radius of curvature of the cornea such that the image of an object 40.0 cm from the cornea's vertex is focused on the retina?

Answer 1

To find the radius of curvature of the cornea, you can use the lens formula and plug in the given values.

Step-by-step explanation:

The radius of curvature of the cornea can be determined using the lens formula, 1/f = (n - 1) * (1/R1 - 1/R2), where f is the focal length, n is the refractive index, R1 is the radius of curvature of the object side surface, and R2 is the radius of curvature of the image side surface.

In this case, the object distance is 40.0 cm and the image distance is -2.60 cm (negative because the image is formed behind the mirror). Since the object is far from the cornea, it is approximately at infinity and the object distance can be considered as the focal length. The refractive index is 1.40.

Plugging in the values into the lens formula, we get: 1/40 = (1.40 - 1) * (1/R1 - 1/R2). Since the image distance is -2.60 cm, we have R2 = -2.60 cm.

Simplifying the equation gives: 1/40 = 0.40/R1 + 0.962/2.60. Solving for R1, we find: R1 = 9.34 cm.

Answer 2

To calculate the radius of curvature for the cornea, the thin lens equation is used with the given object distance and image distance to determine the focal length. Then, the lensmaker's formula relating the radius of curvature, refractive index, and focal length is used to find the radius required for the cornea to focus the image on the retina.

Step-by-step explanation:

To determine the radius of curvature of the cornea that focuses an object 40.0 cm from the cornea's vertex onto the retina, we will employ the thin lens equation:

1/f = 1/d

o

+ 1/d

i

where f is the focal length, do is the object distance, and di is the image distance.

Given that the image must be formed on the retina, di will be -2.60 cm (negative because the image is formed on the opposite side of the light source). The focal length f can be found using the object distance do = 40.0 cm.

So, the equation becomes:

1/f = 1/40.0 - 1/(-2.60)

Solving for f yields the focal length of the cornea. With the focal length known, we can use the relation between a lens's radius of curvature R, its refractive index n, and its focal length f, which is:

R = 2 * f * (n - 1)

Here, the refractive index of the cornea (and other parts) is 1.40. Substituting the known values, we can solve for the radius of curvature R.

The resulting radius will be reasonable for a human cornea, and it is essential for tasks such as fitting contact lenses, as the cornea's shape affects how light is refracted onto the retina. An accurate radius of curvature is also important for correcting visual impairments.