## Lens power

A colleague is out of town so I’m covering a couple class periods of general physics 2 these days. I’m tasked with going over thin lenses, a topic I really like. What I wanted to do was flesh out an approach I’ve used a couple of times here to get my thoughts together for Monday.

Consider the typical lens equation:

$\frac{1}{s_1}+\frac{1}{s_2}=\frac{1}{f}$

where $s_1$ is the object distance, $s_2$ is the image distance, and f is the focal length of the lens. There are lots of ways to go about presenting this to students:

1. Show them a lens and how it focuses far away light.
2. Just tell them it’s true
3. Have them do a lab where they measure the image and object distances to find a relationship between them.
1. I’ve tried that a few times but I never feel like they really figure out the functional form without some serious hints
4. Use the surface equation twice to derive the lensmakers formula

Once I’ve got the students willing to play around with that relationship, I like to have them start to consider using the concept of lens power instead of focal length. Power is just 1/f and has units of diopters (it’s what your eye doctor uses when writing prescriptions). I like power because I like to talk about lenses bending rays. The more power it has, the more it can bend them.

Consider again the lens equation, but this time set the right hand side to the power, and then multiply through by a random height that a ray might strike the lens at (above the optical axis).

$\frac{h}{s_1}+\frac{h}{s_2}=h \times \text{power}$

Written this way, it can be reinterpreted using some triangles (and the usual assumption that all angles are small so that both the sine and tangent of the angles are roughly equal to the angle in radians).

Setup showing the distances and angles for this approach

Now the left hand side is an expression of the total bend angle of the original ray. h/s1 is just $\alpha$, h/s2 is just $\beta$ and their sum, as seen in the image, is just $\Delta$, which, apparently, is just the height times the power.

For me, this really helps reinterpret the notion of lens power, as it works at any height. It also really helps try to figure out what happens to various rays as you think about changing their incoming angle a little. The total bend won’t change at all, for a given location on the lens.

The thing I really like about lens power is that it just adds for lenses close to each other. With this approach this makes sense because a ray would be bent twice, but both bends would happen at roughly the same height. The notion of powers adding for these situation is really great for talking about eyesight and eyesight correction.

“Normal” human eyes have a compound lens that can change in power from 60 to 64 (this assumes the depth of the eye is 17 mm). When you’re at 60, almost all the bending happens at the cornea surface. Light from infinity is bent to focus on the retina. So when you’re most relaxed (least power) you can see things infinitely far away. I like to say “you can do that all day.” When you’re at 64,  you are augmenting your cornea with 4 more diopters from your lens. I like to say you are tensed up at that point, because you get tired if you do it too long. This allows you to bend rays that are diverging to focus on your retina. When you’re maxed out at 64 (or 70 for some of my young students), you’re maxing out your bending capability, allowing yourself to see things at your near point.

If someone can’t focus on things far away, they’re eyes are too powerful. Adding in a (nearby) lens that has a negative power, helps out with that. If, on the other hand, you’re farsighted, your eye isn’t strong enough, and you have to tense up (just a little) just to see things infinitely far away. That’s not really the problem, though. Rather, society has decreed that you need to be able to focus on things 10 inches away, so your eyes are typically augmented with a positive lens to enable you to do that.

This also helps me explain magnifying glasses. They need to have a power greater than 4, because you use those 4 to allow yourself to relax, so that you can do it all day. Anything more than 4 allows you to pull things in closer than your normal near point so you can really see some great detail. Note that this assumes you’re holding the magnifying glass right up to your eye, which is how they’re designed to be used. Note also that eyepieces on microscopes and telescopes are just magnifying glasses, allowing you to stay relaxed (and giving up 4 of their power to allow you to do that).

So, what do you think of this way of using lens power? Useful? Boring? Hard to make use of? Other thoughts?

Professor of physics at Hamline University in St. Paul, MN
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### 5 Responses to Lens power

1. kuiperofsweden says:

I have used the (almost) constant deviation of the marginal rays; it is also in a PhET simulation. I would use your approach if I were to teach our opticien students. Thanks.

• Andy "SuperFly" Rundquist says: