A camera allows us to capture images of events that take place too quickly for our unaided eye to resolve. Another use of optical systems is to magnify—to see objects smaller or closer together than our eye can see.

The easiest way to magnify an object requires no extra optics at all; simply get closer! The closer you get, the bigger the object appears. Obviously the actual size of the object is unchanged as you approach it, so what exactly is getting “bigger”? Consider the green arrow in FIGURE 35.11a. We can determine the size of its image on the retina by tracing the rays that are undeviated as they pass through the center of the lens. (Here we’re modeling the eye’s optical system as one thin lens.) If we get closer to the arrow, now shown as red, we find the arrow makes a larger image on the retina. Our brain interprets the larger image as a larger-appearing object. The object’s actual size doesn’t change, but its apparent size gets larger as it gets closer.

FIGURE 35.11 Angular size.

Technically, we say that closer objects look larger because they subtend a larger angle $\theta$, called the angular size of the object. The red arrow has a larger angular size than the green arrow, ${\theta }_2>{\theta }_1$, so the red arrow looks larger and we can see more detail. But you can’t keep increasing an object’s angular size by moving closer because you can’t focus on the object if it’s closer than your near point, which we’ll take to be a normal 25 cm. FIGURE 35.11b defines the angular size ${\theta }_{\text{NP}}$ of an object at your near point. If the object’s height is $h$ and if we assume the small-angle approximation $\text{tan}\theta \approx \theta$, the maximum angular size viewable by your unaided eye is

Suppose we view the same object, of height $h$, through the single converging lens in FIGURE 35.12. If the object’s distance from the lens is less than the lens’s focal length, we’ll see an enlarged, upright image. Used in this way, the lens is called a magnifier or magnifying glass. The eye sees the virtual image subtending angle $\theta$, and it can focus on this virtual image as long as the image distance is more than 25 cm. Within the small-angle approximation, the image subtends angle $\theta =h/s$. In practice, we usually want the image to be at distance $s\prime \approx \text{∞}$ so that we can view it with a relaxed eye as a “distant object.” This will be true if the object is very near the focal point: $s\approx f$. In this case, the image subtends angle

FIGURE 35.12 The magnifier.

Let’s define the angular magnification $M$ as

Angular magnification is the increase in the apparent size of the object that you achieve by using a magnifying lens rather than simply holding the object at your near point. Substituting from Equations 35.6 and 35.7, we find the angular magnification of a magnifying glass is

The angular magnification depends on the focal length of the lens but not on the size of the object. Although it would appear we could increase angular magnification without limit by using lenses with shorter and shorter focal lengths, the inherent limitations of lenses we discuss later in the chapter limit the magnification of a simple lens to about $4\times$. More complex magnifiers with two or three lenses reach $20\times$, but beyond that one would use a microscope.