Answer to Question #114552 in Optics for Jason

Question

As light passes into the eye, muscles will change the shape of the lens within the eye to ensure the rays are focused onto the retina (Fig A). The lens and cornea together can be approximated by one thin lens with an effective focal length (Fig B). For parallel rays coming from some distant object, the eye will not need to change the shape of the lens from its natural shape for the rays to be focused. The rays emitted from closer objects will not be parallel and the lens must change shapes. (a) When the eye is at rest looking at a distant object, its focal length is 2.50 cm. If the object is moved to 0.4 meters away from the eye, what must the new focal length be in order for the eye to see the object clearly. (b) Are the muscles within the eye flattening the lens or adding additional curvature to it to create the new focal length?

Answer 1

(a) The distance between the lens and the retina (lens-image distance) must be 2.5 cm. Apply thin lens formula:

$$\frac{1}{o}+\frac{1}{i}=\frac{1}{f},\\ \space\\ f=\frac{oi}{o+i}=\frac{40\cdot2.5}{40+2.5}=2.35\text{ cm}.$$

(b) The focal length decreases compared to the previous state. We know that the lens in the human eye can be represented as a convex lens with one flat surface. The focal length depends on the radius of one of its surfaces according to the following equation:

$$f=\frac{R}{n-1}.$$

Therefore, the muscles must decrease the radius making the lens rounder, increasing the curvature. The muscles "compress" the lens.