Light rays from an object 25 cm in front of an eye converge to a point on the retina at the back of the eye. The eye has a depth of 2.4 cm.
Calculate the optical power of the eye.
Using the thin lens formula, 1/s + 1/s' = 1/f, where s is the distance of the object from the lens, s' is the distance of the image from the lens, and f is the focal length of the lens (in this case, the eye).
We can rearrange the formula to solve for f: 1/f = 1/s + 1/s'
Since the object is 25 cm from the eye, we can substitute s = -25 cm (negative because the object is in front of the eye). We want to find s', the distance from the lens to the focal point on the retina.
Since the eye is a convex lens, we know that the focal point will be closer to the lens than the object. We can use similar triangles to find the image distance, s'.
Let x be the distance from the lens to the focal point, and 2.4 cm be the depth of the eye. Then, using similar triangles:
x/2.4 = (x+s')/s'
Expanding and solving for s', we get:
s' = (2.4x)/(x-2.4)
Now we can substitute into the equation for f:
1/f = 1/s + 1/s'
1/f = 1/-25 + 1/[(2.4x)/(x-2.4)]
1/f = -0.04 + (x-2.4)/(2.4x)
To find the value of x that makes this equation true, we can use algebra or a graphing calculator. The value of x that makes f = 60 diopters is approximately 0.0349 cm.
Therefore, the optical power of the eye is 60 diopters.
To calculate the optical power of the eye, we can use the formula:
Optical Power (P) = 1 / focal length (f)
First, let's find the focal length (f) of the eye.
The total distance from the object to the retina can be calculated as:
Distance = Eye Depth (2.4 cm) + Distance from Object to Eye (25 cm)
Distance = 2.4 cm + 25 cm = 27.4 cm
Since the light rays converge to a point on the retina, the focal length (f) is equal to the distance between the lens of the eye and the retina. Therefore, f = Distance / 2.
f = 27.4 cm / 2 = 13.7 cm
Now, we can calculate the optical power (P) using the formula:
P = 1 / f
P = 1 / 13.7 cm = 0.073 eye^-1
Therefore, the optical power of the eye is approximately 0.073 eye^-1.