35.2.1 Controlling the Exposure

The camera also must control the amount of light reaching the detector. Too little light results in photos that are underexposed; too much light gives overexposed pictures. Both the shutter and the lens diameter help control the exposure.

The shutter is “opened” for a selected amount of time as the image is recorded. Older cameras used a spring-loaded mechanical shutter that literally opened and closed; digital cameras electronically control the amount of time the detector is active. Either way, the exposure—the amount of light captured by the detector—is directly proportional to the time the shutter is open. Typical exposure times range from $1 / 1000 s$ or less for a sunny scene to $1 / 30 s$ or more for dimly lit or indoor scenes. The exposure time is generally referred to as the shutter speed.

The amount of light passing through the lens is controlled by an adjustable aperture, also called an iris because it functions much like the iris of your eye. The aperture sets the effective diameter $D$ of the lens. The full area of the lens is used when the aperture is fully open, but a stopped-down aperture allows light to pass through only the central portion of the lens.

The light intensity on the detector is directly proportional to the area of the aperture; a lens with twice as much area will collect and focus twice as many light rays from the object to make an image twice as bright. The aperture area is proportional to the square of its diameter, so the intensity $I$ is proportional to $D^{2}$ . The light intensity—power per square meter—is also inversely proportional to the area of the image. That is, the light reaching the detector is more intense if the rays collected from the object are focused into a small area than if they are spread out over a large area. The lateral size of the image is proportional to the focal length of the lens, as we saw in Equation 35.3, so the area of the image is proportional to $f^{2}$ and thus $I$ is proportional to $1 / f^{2}$ . Altogether, $I \propto D^{2} / f^{2}$ .

A photograph of a small aperture of a camera enclosed by the shutter. — An iris can change the effective diameter of a lens and thus the amount of light reaching the detector.

A photograph of a wide aperture of a camera enclosed by the shutter. — An iris can change the effective diameter of a lens and thus the amount of light reaching the detector.

By long tradition, the light-gathering ability of a lens is specified by its f-number, defined as

f-number = \frac{f}{D}

(35.4)

The f-number of a lens may be written either as $f / 4.0$ , to mean that the f-number is 4.0, or as F4.0. The instruction manuals with some digital cameras call this the aperture value rather than the f-number. A digital camera in fully automatic mode does not display shutter speed or f-number, but that information is displayed if you set your camera to any of the other modes. For example, the display $1/125 F5.6$ means that your camera is going to achieve the correct exposure by adjusting the diameter of the lens aperture to give $f / D = 5.6$ and by opening the shutter for $1 / 125 s$ . If your lens’s effective focal length is 10 mm, the diameter of the lens aperture will be

D = \frac{f}{f-number} = \frac{10 mm}{5.6} = 1.8 mm

NOTE The f in f-number is not the focal length $f;$ it’s just a name. And the $/$ in $f / 4$ does not mean division; it’s just a notation. These both derive from the long history of photography.

Because the aperture diameter is in the denominator of the f-number, a larger-diameter aperture, which gathers more light and makes a brighter image, has a smaller f-number. The light intensity on the detector is related to the lens’s f-number by

I \propto \frac{D^{2}}{f^{2}} = \frac{1}{(f-number)^{2}}

(35.5)

Historically, a lens’s f-numbers could be adjusted in the sequence 2.0, 2.8, 4.0, 5.6, 8.0, 11, 16. Each differs from its neighbor by a factor of $\sqrt{2}$ , so changing the lens by one “f stop” changed the light intensity by a factor of 2. A modern digital camera is able to adjust the f-number continuously.

A photograph of a camera with focal length and f-number information stamped on its lens. — Focal length and f-number information is stamped on a camera lens. This lens is labeled $5.8 - 23.2 mm 1 : 2.6 - 5.5$ . The first numbers are the range of focal lengths. They span a factor of 4, so this is a $4 \times$ zoom lens. The second numbers show that the minimum f-number ranges from F2.6 (for the 5.8 mm focal length) to F5.5 (for the 23.2 mm focal length).

The exposure, the total light reaching the detector while the shutter is open, depends on the product $I Δ t_{shutter}$ . A small f-number (large aperture diameter $D$ ) and short $Δ t_{shutter}$ can produce the same exposure as a larger f-number (smaller aperture) and a longer $Δ t_{shutter}$ . It might not make any difference for taking a picture of a distant mountain, but action photography needs very short shutter times to “freeze” the action. Thus action photography requires a large-diameter lens with a small f-number.

EXAMPLE 35.3 Capturing the action

Before a race, a photographer finds that she can make a perfectly exposed photo of the track while using a shutter speed of $1 / 250 s$ and a lens setting of F8.0. To freeze the sprinters as they go past, she plans to use a shutter speed of $1 / 1000 s$ . To what f-number must she set her lens?

MODEL The exposure depends on $I Δ t_{shutter}$ , and the light intensity depends inversely on the square of the f-number.

SOLVE Changing the shutter speed from $1 / 250 s$ to $1 / 1000 s$ will reduce the light reaching the detector by a factor of 4. To compensate, she needs to let 4 times as much light through the lens. Because $I is proportional to 1 / (f-number)^{2}$ , the intensity will increase by a factor of 4 if she decreases the f-number by a factor of 2. Thus the correct lens setting is $F 4.0$ .