Question

A renowned hospital in the metropolitan area of Puerto Rico is interested in training volunteers as part of a visual health campaign that will be carried out in the island municipality of Vieques. Since you are a university student and you are finishing the second part of the General Physics course, you have been selected to be trained to become an assistant to an optical technician during the visual health campaign. One day he is learning how to put a contact lens in a patient's eye. Take a measurement with a "keratometer," which is used to measure the curvature of the front surface of the eye, the cornea. This instrument places an illuminated object of known size at a known distance "p" from the cornea. The cornea reflects some light from the object, forming an image of the object. The magnification (increase or magnification) "M" of the image is measured using a small viewing telescope that allows comparison of the image formed by the cornea with a second calibrated image projected onto the field of view by a prism arrangement. As part of your training, the technician asked you not to use the automatic calculator associated with the machine, but to perform the calculations yourself. Determine the radius of curvature R of the cornea for the measurements you make for the patient and the power of the lens: p = 30.0 cm and M = 0.0130.

Answer 1

So let's break this down step by step, as if we are solving a puzzle.

First, let's go over what we are dealing with:

1. We have an instrument called a keratometer, which helps in measuring the curvature of the cornea, the front surface of the eye.

2. This keratometer has an illuminated object at a known distance, ‘p’ from the cornea.

3. The cornea reflects some light and forms an image.

4. We have a telescope to compare the image size with the object size.

5. We are given p = 30.0 cm and the magnification M = 0.0130.

6. We need to find the radius of curvature R of the cornea.

Now, let's put on our physics hat!

In optics, when you deal with mirrors, the mirror equation is often used. For the cornea, we can treat it like a mirror (it reflects light, right?), so the same principle applies here.

The mirror equation is:

1/f = 2/R = 1/p + 1/q

Where:

- f is the focal length,

- R is the radius of curvature,

- p is the object distance (in this case, it's 30.0 cm from the cornea),

- q is the image distance (which we need to find).

Also, magnification (M) is defined as the ratio of the image distance to the object distance, or

M = -q/p

We can plug in the values and solve for q:

0.0130 = -q/30.0

q = -30.0 * 0.0130 ≈ -0.39 cm

Now that we have the image distance, let’s go back to the mirror equation to find the radius of curvature R:

1/f = 1/p + 1/q

= 1/30.0 + 1/(-0.39)

≈ 0.0333 - 2.5641

≈ -2.5308 cm⁻¹

Now, f = 1 / (-2.5308) ≈ -0.395 cm.

Remember that 1/f = 2/R, so we can find R:

2/R = -2.5308

R ≈ -2 / 2.5308 ≈ -0.79 cm.

Finally, the power of a lens or mirror is defined as the inverse of its focal length (in meters). Power (P) is measured in diopters (D).

P = 1/f (in meters)

= 1/(-0.395/100)

≈ -2.53 D (in diopters).

And voila! The radius of curvature of the cornea is approximately -0.79 cm, and the power of the lens is approximately -2.53 diopters.

Remember that the negative sign indicates that the curvature is concave. The cornea acts like a concave mirror, reflecting light inwards.

Hope this helps! Let me know if there's anything that needs further clarification.