The idea behind a magnifying glass is to make something look bigger, but how does this work? Well, the principle is so simple you probably already know it. The farther away something is, the smaller it looks, right? So to make something look bigger, just move it closer to your eye.
Seems straightforward, right? Unfortunately, it’s not. The problem is that your eye can only focus on something that is far enough for your eye’s lens to “accommodate.” Bringing something closer makes it bigger, but fuzzy, and that doesn’t help. Typically, this distance is about 10″ or 25 cm. Children can often focus closer than 25 cm, and people over 50 can’t normally focus even this close, so take the 10″/25cm as a rough approximation. If you’re as nearsighted as I am, you can focus 4″/10cm from your eye. It’s great for inspecting things, but not so good for driving.
The question is now, “how can I bring something closer to my eye, but still focus on it?” And the answer is, “a magnifying glass.”
A magnifying glass has two essential properties: its focal length and its diameter. The diameter, or how big the lens is doesn’t really matter that much. If you want to hold the lens far from your eye, then bigger is better, but it doesn’t change the magnifying power. The magnifying power is determined by the focal length.
Focal length is basically the distance from the magnifying glass to where it forms an image of a distant object. For example, you can make a small image of the moon by holding a magnifying glass a few inches (6-12 cm) from a piece of paper.
The magnifying power of a magnifying glass is 10″ (25 cm) divided by its focal length. For example, if you have a lens with a focal length of 4″ (10 cm) it will have a magnifying power of 10 divided by 4 (or 25 divided by 10), which is 2.5X.
The optical design of a magnifying glass is simple. Both sides must have the same curvature (so it doesn’t matter which way you hold the lens), and the glass is almost always simple soda-lime glass, so the focal length determines the design. The radius is given by
\(R = 2(n-1)F\)
where R is the radius of the surfaces, n is the refractive index of the glass (roughly 1.52) and F is the desired focal length. So for a 2.5X magnifying glass (focal length = 10 cm), R = 2*(1.52 – 1) * 10 = 10.4 cm.