A nearsighted man cannot clearly see objects more than 2.0 m away. The distance from the lens o

step 1 So we have a far point of two meters. That's the furthest away that this person can see comforta

A nearsighted man cannot clearly see objects more than 2.0...

A nearsighted man cannot clearly see objects more than $2.0 \mathrm{~m}$ away. The distance from the lens of the eye to the retina is $2.0 \mathrm{~cm}$, and the eye's power of accommodation is $4.0 \mathrm{D}$ (the focal length of the cornea-lens system increases by a maximum of $4.0 \mathrm{D}$ over its relaxed focal length when accommodating for nearby objects). (a) As an amateur optometrist, what corrective eyeglass lenses would you prescribe to enable him to clearly see distant objects? Assume the corrective lenses are $2.0 \mathrm{~cm}$ from the eyes. (b) Find the nearest object he can see clearly with and without his glasses.

Key Concepts

Accommodation in the Human Eye

Accommodation refers to the ability of the human eye to adjust its focal length for focusing on objects at various distances by changing the shape of the crystalline lens. The range of accommodation, expressed in diopters, is important for understanding the eye’s limitations in focusing on near versus distant objects and plays a role in determining how much additional correction might be required for clear vision across different distances.

Vertex Distance

Vertex distance is the space between the corrective lens and the eye. This distance can affect the effective power of the lens when used in spectacle prescriptions. In optical prescriptions, accounting for vertex distance is essential to ensure that the corrective lenses produce the desired optical effect when placed at a typical distance from the eyes, which may slightly modify the required lens power calculated via the thin lens formula.

Optical Power (Diopters)

Optical power is the measure of a lens's ability to converge or diverge light, defined as the inverse of the focal length (in meters). It is measured in diopters (D). This concept is central to prescribing lenses because it quantifies the correction needed; in myopic corrections, negative diopter values indicate diverging lenses that help extend the far point of the eye to infinity.

Myopia

Myopia, or nearsightedness, is an optical condition in which distant objects appear blurry while close-up objects can be seen clearly. This occurs because the eye’s optical system focuses images in front of the retina when the eye is in its relaxed state. Understanding myopia is fundamental to diagnosing and correcting vision problems as it directly impacts how image formation occurs in the eye.

Corrective Eyeglass Lenses

Corrective eyeglass lenses are used to compensate for the eye’s inability to focus images properly on the retina. In the case of myopia, diverging (negative) lenses are prescribed to spread out light rays so that the image is shifted back onto the retina, thus allowing distant objects to be seen clearly. The design and placement of these lenses, including factors like vertex distance from the eye, are crucial for ensuring proper visual correction.

Thin Lens Equation

The thin lens equation, given by 1/f = 1/d_o + 1/d_i, is a fundamental concept in geometric optics that relates the focal length of a lens (f) to the distance of the object (d_o) and the image (d_i). This equation is essential for understanding how corrective lenses form images, whether virtual or real, and is pivotal in the design and prescription of lenses that adjust the effective focal distance of the eye.

Transcript

00:01 So we have a far point of two meters.

00:08 That's the furthest away that this person can see comfortably.

00:13 And we want to find some prescription with refractive power p, which is equal to 1 over f.

00:20 We know that one over the focal length of the lens.

00:23 That will give them perfect far vision.

00:27 In perfect far vision, we have a p of infinity.

00:30 That's like the test for far vision.

00:33 You want to be able to see two infinite lengths.

00:37 So in order to achieve that, because this person's current far point is two meters, the lens will need to project the image two meters in front of the person's eye.

00:49 So we need a cue of negative two meters.

00:52 Now, because the lens itself for the eyeglass is two centimeters away from the eye, that's going to actually become negative 1 .98 meters.

01:07 Now we have the thin lens formula, 1 over f equals 1 over p plus 1 over q.

01:15 And we can just fill these values in there.

01:17 So as p goes to infinity, 1 over p approaches zero, because as the bottom of that fraction gets really large, the fraction overall gets very small.

01:28 So we're going to get that 1 over f is equal to 1 over negative 1 .98m.

01:38 And if you evaluate 1 over negative 1 .98 meters to give us the refractive power, we're going to get that this is negative 0 .51d.

01:56 It's actually negative 0 .505.

02:00 So i'm rounding up there.

02:04 Okay.

02:06 Now, knowing that, knowing, having this information about this person's eyesight, can we figure out their near point? so i'm going to keep that in there because we might need it.

02:21 We know that their minimum refractive power is at their far point, right? so the lens is refracting the least at the farthest point of the eye.

02:32 What is the refractive power of the eye at that time? well, we know that the distance there, the p, is 2 meters.

02:40 That's the furthest this person can see comfortably.

02:43 That's their far point.

02:44 And we know that the distance from the lens to the image in the eye at that point is to centimeters.

02:51 So it's 0 .02 meters.

02:56 Now we can use p is equal to 1 over p plus 1 over q.

03:06 Can use this formula to find the minimum refractive power of the eye.

03:10 So at the far point, what is the refractive power? and if you plug in those values, you will find that the refractive power is 50 .5 diopters.

03:24 Remember that diopters aren't always inverse meters...

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