## 20130114

### Presentation: magnifiers

Look at this magnifying glass. Just look at it. Um, through it.

In this presentation we will look through, um, at how magnifiers magnify. (In the next presentation we'll see how these magnifying lenses are used as eyepieces in telescopes and microscopes.)

First, defining what magnifiers do, and to what.

The angular size Θ is not the actual size, it is a measure of how large an angle it subtends with your eye at the origin, and is a measure of how big something "seems" from your viewpoint.

The angular magnification M (upper-case M, to distinguish it from linear magnification lower-case m) is a numerical factor denoting how much larger the angular size of something appears as seen through a magnifier, compared to with just the unaided eye.

When a converging lens with a focal length f is used a magnifier, the angular magnification is the ratio of the angular size as seen through the magnifier, compared to the angular size as seen with an unaided eye. This is also the ratio of the near point (the nominal closest distance an unaided eye can focus on, 25 cm) to the the focal length of the magnifier.

Second, the process of magnification using a magnifier.

Let's start with a rather provocative statement: a magnifying lens doesn't really magnify. Here we see a close-up (but unfocused) view of a pliers, and the same pliers at the same distance using a magnifying lens. The angular size of the pliers is relatively unchanged (after accounting for extreme defocusing circle of confusion blurring)!

Consider an Ames room, which is constructed that a person moving along the back of the room (which is actually greatly skewed) will be farther away or closer to an observer's eye, causing the person's angular size to change unexpectedly. This is the main idea behind angular magnification, which is merely caused by bringing an object closer to an eye.

Bringing an object closer to an eye increases its angular size, but there is a practical limit to how close an object can be brought such that the eye still can focus on it--the near point, with the nominal value being 25 cm. Any closer would still increase the angular size of the object, but the eye would no longer be able to focus clearly on it.

Now compare these two views of the pliers, without and with a magnifying lens. Note that without the magnifying lens, the view is focused at ∞, where objects on the horizon are sharply in focus, and the pliers is out of focus. With the magnifying lens, the view is still focused at ∞, but now the pliers is in focus, meaning that the virtual image produced by the magnifying lens is at ∞. (How do you know that this is a virtual image? Which ray tracing best matches this?) This is because the pliers is on the focal point of the magnifying lens, which produces a virtual image (of the same angular size) out at infinity.

So to refine our understanding of what magnifiers actually do:
• The maximum angular size of an object as seen by an unaided (nominal) eye is when the object is placed 25 cm away.
• For a magnifying lens with a focal length f of less than 25 cm, the object can then be brought closer (up to the focal point of the magnifying lens), such that the magnifying lens allows the eye to be able to focus (at ∞) on an object held closer than 25 cm.
• Closer is bigger.
One could also think of the magnifying glass as increasing the accommodation ability of the eye to focus on objects nearer than 25 cm, such that objects can be brought closer in order to increase their angular size.