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Have you ever wondered why some people need glasses and others don’t? It’s because of the way light is focussed by the lens in your eye. This lens, like magnifying glasses, eyeglasses, and contact lenses, is considered a thin lens.

How lenses interact with light

Lenses bend light that passes through them. The direction and amount that the light bends depends on the curvature of the lens, the material the lens is made of, and the material in which the lens is immersed (for now, we’ll assume this is just air). If both sides of the lens curve outward, it is called a converging lens, and it will bend light from distant objects inwards toward a single point, called the focal point.
Figure of converging lens with light rays focussed on the opposite side
If both sides of the lens curve inward, it is called a diverging lens, and light from distant objects will bend outwards. Because the light is not being bent toward a single point, the focal point is not as obvious as it was in the case of the converging lens. We have to take the bent rays, and follow them back to the side of the lens that the light came from to make them come together and find the focal point. That means that the focal point is on the same side of the lens as the light rays were coming from.
Figure diverging lens with light rays diverging on the opposite side
In actuality, there are two focal points for every lens, the same distance from the lens, on opposite sides. The distance from the lens to the focal point is called the focal length. For converging lenses, the focal length is always positive, while diverging lenses always have negative focal lengths. However, these conventions are arbitrary, and physicists could just as easily have made the signs opposite.
Figure of converging lens with labeled negative and positive focal lengths

Thin lens rules and sign conventions

Now that we know how to find the focal point of a lens using a distant object, we can see what happens to light rays from objects that are closer to the lens. Let’s say we have a cat standing on one side of a converging lens. We know that there are two focal points, one on each side of the lens, and that if we take the focal length of the point across the lens from the cat as positive, then the one on the same side as the cat is negative.
In this case, the image of the cat will be across the lens from the actual cat, and it will be upside-down. Just as with the focal point, if the image is on the opposite side of the lens from the cat, the distance from the lens to the image will be positive.
Figure of cat with positive object distance, negative focal point, lens, positive image distance to inverted image of cat
What if the cat is closer to the lens than the focal point is? That is, what if the object distance is smaller than the focal length? The image will be on the same side of the lens from the object, and will be upright. The image will also be larger than the object. That means that the image distance will be negative.
Figure of large upright image of cat, negative focal point, smaller cat, lens, positive focal point
How about a diverging lens? This time, the image is not affected by whether the object is inside or outside the focal point. The image will always be on the same side of the lens as the object, upright, and smaller than the object. In this case the image distance for the diverging lens is negative.
Figure of focal point, cat, small image of cat, diverging lens, focal point
Magnification refers to a change in size of the object. If the magnification is greater than one, the image is larger than the object, but if the magnification is smaller than one the image is smaller than the object. For example, if the magnification is one half, then the image appears to be half the size of the object. The sign of the magnification tells us the orientation of the image. If the sign is positive, then the image is upright. If the sign is negative, then the image is upside-down. In the examples above, we can see that amount by which an object will be magnified changes depending on its distance from the focal point.

Consider the following

How might glasses help someone who can’t focus on distant objects? How might they help someone who can’t focus on close-up objects?
If you are using a magnifying glass to read fine print (object), and you move it too far from the page, the text looks like it has flipped over and gotten smaller (image). It turns out that this is because a magnifying glass is a converging lens and identifying the object distance where the image flips is one way of finding the focal length.
Ever wonder how a camera lens focuses? The lens has a fixed focal length, but the distance to the object changes as you point the camera at different things. The distance between the lens and the sensor or film strip has to change to make sure the sensor is sitting at the correct image distance.