Image Formation: Lenses

There are two sorts of lenses: converging lens (Figure 1), and diverging lens (Figure 2). Notice how the converging lens brings the rays together, so the blue and green form another color, nice aqua (Figure 2a), while diverging lens separates rays (Figure 2b).

The converging lens is used as a magnifying glass, while the diverging lens makes objects look smaller and farther away (Figure 3a, b, c, and d).

Properties of Images Produced by Lenses

There are three properties of an image formed by a lens: location, orientation, and size. If the image is located behind the lens, it is real (the real rays form the image); otherwise, it is virtual (the image is created in our brain based on the assumption that the diverging rays were scattered on an existing object). If the image is in the same direction that the object, it is upright; otherwise, it is inverted. If the image is smaller in size than the object, it is reduced; otherwise, it is magnified.

You are invited to investigate the properties of images produced by lenses in the lab. However, before that, you can fill out the table based on ray diagrams

Ray Diagrams

Any object is a source of light (either emitted or reflected) that can be modeled with numerous rays. These rays are represented by lines, which is why that branch of physics is called geometric optics.

In order to find an image created by a lens, just two rays are needed. The diagrams below show three:

1. The parallel ray (red) is the ray that originates at the very top of the object and travels toward the lens parallel to the axis of symmetry. That ray will pass through the focal point of the lens (Figure 4)

2. The focus point ray (green) is the ray that originates at the very top of the object and travels toward the lens through the focal point. That ray is parallel to the axis of symmetry after passing through the lens (Figure 5)

3. The central ray (blue) is the ray that originates at the very top of the object and travels toward the center of the lens. That ray does not change its path (Figure 6)

These rays intersect at one point and form the image. Notice the symmetry of the red and green ray; the parallel incidence ray passes through the focal point, and the one that passes through the focal point is parallel on the other side of the lens.

Converging Lens Ray Diagram

Only two rays are needed to create the image on a lens. In the three diagrams below, the parallel and the central rays were used to find the image. Notice, that the three presented diagrams do not cover all possible cases of image formation.

Diverging Lens Ray Diagram

The rules of finding the image produced by a diverging lens are the same. Notice that the image produced by the diverging lens is always virtual, upright, and reduced in size.

Mathematics of Spherical Lenses

The math used to calculate distances from a lens is quite simple. What may cause confusion, are the signs of the distance of the image and the focal length. Like many other quantities in physics, it is an accepted convention that the focal length of a converging lens is positive and of a diverging lens is negative. All real distances are positive, while virtual ones are negative.

where d_o is a distance between the object and the mirror, d_i is a distance between the image and the mirror, and f is the focal length of the mirror.