34.5.1 Converging Lenses
These basic observations about lenses are enough to understand image formation by a thin lens. A thin lens is a lens whose thickness is very small in comparison to its focal length and in comparison to the object and image distances. We’ll make the approximation that the thickness of a thin lens is zero and that the lens lies in a plane called the lens plane. Within this approximation, all refraction occurs as the rays cross the lens plane, and all distances are measured from the lens plane. Fortunately, the thin-lens approximation is quite good for most practical applications of lenses.
NOTE We’ll draw lenses as if they have a thickness, because that is how we expect lenses to look, but our analysis will not depend on the shape or thickness of a lens.
FIGURE 34.29 shows three important situations of light rays passing through a thin converging lens. Part a is familiar from Figure 34.28. If the direction of each of the rays in Figure 34.29a is reversed, Snell’s law tells us that each ray will exactly retrace its path and emerge from the lens parallel to the optical axis. This leads to Figure 34.29b, which is the “mirror image” of part a. Notice that the lens actually has two focal points, located at distances on either side of the lens.
FIGURE 34.29 Three important sets of rays passing through a thin converging lens.
Figure 34.29c shows three rays passing through the center of the lens. At the center, the two sides of a lens are very nearly parallel to each other. Earlier, in Example 34.3, we found that a ray passing through a piece of glass with parallel sides is displaced but not bent and that the displacement becomes zero as the thickness approaches zero. Consequently, a ray through the center of a thin lens, with zero thickness, is neither bent nor displaced but travels in a straight line.
These three situations form the basis for ray tracing.
Many cars have automatic windshield wipers that adjust their speed in response to the intensity of the rain. The technology—simpler than it might seem—is based on total internal reflection. An infrared laser just below the rearview mirror shoots out a laser beam that strikes the front of the windshield glass from the inside at a 45° angle. This is larger than the critical angle of a glass-air boundary, so on a dry day the light rays undergo total internal reflection within the glass and return to a sensor. But 45° is smaller than the critical angle of a glass-water boundary, so water drops on the windshield cause some of the light to refract. The decrease in the laser-beam intensity returning to the sensor is detected by the car’s computer, which judges the amount of rain that strikes the windshield and adjusts the wiper speed accordingly.