Negative Focal Length and Diverging Lenses

In Figure (1) we got a virtual image by moving the object inside the focal length. Another way to get a virtual image is to use a diverging lens as shown in Figure (2). Here we have drawn the three special rays, but the role of the focal point is reversed. The ray through the center of the lens goes through the center as before. The top ray parallel to the axis of the lens diverges outward as if it came from the focal point on the left side of the lens. The ray from the tip of the object headed for the right focal point, comes out parallel to the axis. Extending the diverging rays on the right, back to the left side, we find a virtual image on the left side.

Figure 1: When the object is located within the focal length, we get a virtual image behind the object.
You get diverging lenses by using concave surfaces as shown in Figure (34). In the lens maker’s equation,

.........(1)

you replace 1/r by – 1/r for any concave surface. If 1/f turns out negative, then you have a diverging lens. Using this negative value of f in the lens equation (with f = – |f |) we get

.........(2)

This always gives a negative image distance i, which means that diverging lenses only give virtual images.

Figure 2: A diverging lens always gives a virtual image.