If you are wearing contact lenses with a dioptric power of 3.4 and your eye has a diameter of 2.6 cm, what is the effective focal length of your eye-contact lens combination, in centimeters?
To find the effective focal length, we need to use the lens formula:
1/f = 1/v - 1/u
Where:
f = focal length of the lens
v = image distance
u = object distance
In this case, the contact lens is acting as a converging lens, so its focal length is positive. Let's assume the object distance (u) is the distance from the eye to the lens. Since the contact lens is directly on the eye, the object distance will be equal to the eye's diameter. Therefore,
u = 2 × 2.6 cm = 5.2 cm
The image distance (v) is the distance from the lens to the position where the image is formed. Since the contact lens is on the eye, the image distance is the distance between the lens and the retina, which is approximately the focal length (f) of the eye. Therefore,
v ≈ f
Substituting the values into the lens formula, we have:
1/f = 1/v - 1/u
1/f = 1/f - 1/5.2
Simplifying the equation by multiplying through by f,
1 = 1 - f/5.2
Rearranging and solving for f:
f/5.2 = 1 - 1
f/5.2 = 0
f = 0 × 5.2
f = 0 cm
Since the focal length is calculated to be 0 cm, this means the effective focal length of the eye-contact lens combination is practically the focal length of the eye, which can be considered to be infinite for practical purposes. This is because the contact lens adapts to the curvature of the eye, making its contribution to the focal length negligible.