MCAT Review of Converging & Diverging Combinations of Lenses, Glasses for Nearsighted, Farsighted Human Eye

I wrote this for a student I tutor:

High-yield geometric optics for the MCAT:

This is a hopefully intuitive explanation to help pre-meds preparing to take the MCAT to understand converging and diverging lenses, the human eye, and combinations of lenses (i.e. people who wear glasses).

Most MCAT questions on geometric optics are likely to only require knowledge of the thin lens and lens strength equation (1/do + 1/di = 1/f = P), the magnification equation (M = hi/ho = -di/do), and the sign conventions.

That being said, the MCAT content outline does mention “Converging and diverging lenses“, “Combination of lenses” and “Optical instruments, including the human eye” (hint hint), and it’s clearly the most medically relevant application of geometric optics, so it’s safe to say that understanding the human eye in combination with convex and concave glasses is high yield for the MCAT.

Optical Instruments, Including the Human Eye

Convex, converging lens review:

A convex converging lens forms a real, inverted image on the other side of the lens if and only if the object is further from the lens than the focal length. In other words, if do > f, the image is real and inverted.

If the object is placed exactly at the focal length (do = f) of a converging lens, no image is formed.

If the object is placed between the focal length and twice the focal length (2f > do > f) of a converging lens, a real, inverted image will form on the opposite side of the lens that is larger and further away from the lens than the object.

Why it makes sense that the human eye is a convex, converging lens:

If the object distance is more than twice the focal length away (do > 2f) from the convex converging lens, it will form a real, inverted image on the opposite side of the lens that is smaller than the object and closer to the lens than the object.

This last case (do > 2f) is how the human eye works. The lens of the eye is a convex, converging lens whose focal length is about 1.7cm. The typical near point of the eye (the closest object distance the eye can focus on) is 25cm, which is more than twice the focal length of the eye’s lens (do > 2f).

In other words, the eye can only focus on objects that are much more than 2f away. This allows the lens of the eye to take a large, relatively far away object, refract it, and form a much smaller, inverted, real image on the retina, which is much closer to the lens than the object is.

Remember that the brain flips the inverted image formed on the retina so you have a conscious experience of seeing an upright image.

Combination of lenses

How does a magnifying glass work?

If the object is closer to the convex converging lens than the focal length (do < f), the image is virtual, upright, on the same side of the lens as the object, and larger than the object. This is how a magnifying glass works, and it’s why you have to hold the magnifying glass close to the object you’re looking at (do < f) or the image will become inverted.

How to correct farsightedness with convex, converging glasses:

This is also how a convex converging lens corrects farsightedness. Why? Farsighted people can’t focus on nearby objects. If the object is closer than the focal length of the convex converging glasses (do < f), the converging lens will form an upright, virtual image that is larger than the object behind the real object and therefore further away from the eye. That further away, virtual image serves as the object for the farsighted eye’s lens.

Keep in mind that a convex, converging lens will always have a positive focal point and therefore a positive power (since 1/f = P).

So to review, what does it mean if someone’s glasses have a focal length of 2cm or a power of 50 diopters and they’re looking at an object 1cm away? The focal length and power are positive, so it must be a convex converging lens, so the glasses must correct farsightedness. The object is closer to the lens than the focal length, so the image formed must be virtual, upright, behind the object, and larger than the object.

How to correct nearsightedness (myopia) with a concave diverging lens:

By contrast, a concave diverging lens always forms a virtual, upright image on the same side of the lens as the object. This is the kind of lens you would use to correct nearsightedness. Why?

Myopic people can’t focus on objects that are too far away. A pair of glasses with concave diverging lenses refracts the light from a far away object in such a way that a virtual, upright, smaller image is formed on the same side of the lens as the object. This virtual image formed by a concave diverging lens actually serves as the object for the nearsighted eye. This “virtual object” is closer to the eye than the real, far away object, so it’s easier for a nearsighted person to focus on. The image is also smaller than the object (reduced) with a diverging lens (-f), which gives the illusion that the image is as far away as the object is, even though it’s actually much closer and just a lot smaller.

Comparing and contrasting glasses for nearsightedness and farsightedness:

Both glasses for nearsightedness and farsightedness create virtual, upright images on the other side of the lens from the eye. This makes sense because glasses are worn much closer to the eye than the near point. Even the most nearsighted person has a near point further away than the distance from their eyes to their glasses.

So glasses for both conditions must create virtual image pseudo-objects for the eyes to focus on that are on the same side of the glasses as the object, on opposite side from the eye, and somewhat further away than the person’s near point. Since these virtual objects are on the same side of the lens as the real object, they are always virtual and upright.

The key difference is that concave diverging lenses do this for far away objects, creating smaller, closer virtual images, while convex converging lenses do this for nearby objects, making larger, further away virtual images.

Notice that in both cases, the image distance makes it easier for the person to see the image. For a nearsighted person, their concave diverging glasses make virtual images closer to their eyes than the object. For a farsighted person, their convex converging lenses make images that are further away than the object.

How do you remember whether the image formed is larger or smaller than the object? Also in both cases, the magnification of the image acts to “balance out” or compensate for the change in the image distance relative to the object distance. This is useful because it creates an optical illusion that the virtual image pseudo-object formed by the glasses is in the same place as the object, even though it isn’t.

So while a farsighted convex converging lens (+f, +P) creates an image further away, the image is bigger than the object, whereas a nearsighted concave diverging lens (-f, -P) creates an image that is closer but smaller than the object.

So what does it mean if the person’s glasses have a focal length of -2cm or a power of -50 diopters and the object is 2m away? The focal length and power are negative, so the glasses must have diverging, concave lenses for correcting nearsightedness. The image formed must be closer to the lens than the object, so it’s easier for the nearsighted person to see, and it must be smaller than the object (so it still looks like it’s as far away as the object is, even though it’s actually much closer).

I hope that was helpful.

Other resources to check out to deepen your understanding:

Converging Lenses – Object-Image Relations – The Physics Classroom

Thin lens sign conventions – Khan Academy article

Correcting nearsightedness/farsightedness – Student Doctor Network Forum

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