Calculating Photographic Exposure and Exposure Value

Hello. The previous lesson completed the story of how film records an image; this module turns to how a camera controls the image it delivers. We will begin with exposure: the amount of light reaching the recording surface, and the settings that determine whether that amount suits the film or sensor.

You have already seen how lens aperture affects image-plane irradiance. Here we add shutter duration and scene luminance, then distinguish the physical exposure from the exposure value used on cameras and light meters. By the end, you should be able to calculate both—and explain why changing ISO does not, by itself, change the light collected. Allow about 40 minutes, including the selected reading and short video segment.

From scene luminance to photographic exposure

A subject presents a luminance $L$ to the camera: luminous intensity per projected area in the camera’s direction, measured in $\mathrm{cd}{\mathrm{m}}^{-2}$. The image formed by the lens has an illuminance $E$, measured in lux, at the film or sensor. If that illuminance stays approximately constant while the shutter is open for $t$ seconds, the photographic exposure—often called light exposure—is

Its unit is lux-seconds. If the light varies during the exposure, the more general expression is $H=\int E(t)dt$; the simple product uses the time-averaged illuminance.

The distinction matters: luminance describes the scene; illuminance describes light arriving at the image plane; exposure includes how long it arrives. The following technical account connects them. Its PDF text extraction scrambles some equations, so consult the displayed equations on the pages rather than relying on copied text.

For a distant subject near the image center, the aperture admits rays over a cone whose small-angle solid angle scales as the entrance-pupil area divided by focal length squared. Since f-number is $N=f/D$, where $D$ is entrance-pupil diameter, the image-plane illuminance of an ideal, lossless lens is approximately

The factor $\pi /4$ comes from the geometry of the admitted cone; the inverse-square dependence on f-number is the essential result. If a fraction $T$ of light passes through the lens, a useful center-of-frame estimate is

Here $L$ is in $\mathrm{cd}{\mathrm{m}}^{-2}$, $t$ in seconds, and $H$ in lux-seconds. This is an approximation, not a universal lens specification: off-axis falloff, vignetting, close focusing, flare, and wavelength-dependent transmission can change the result. It is nevertheless a sound starting point for calculating delivered exposure.

Sensitivity specifies a requirement, not extra light

In the preceding module, a film’s characteristic curve related exposure to developed density. Film speed compresses part of that measured response into a usable rating. For a specified photographic criterion, a faster-rated material generally needs less $H$: doubling the ISO speed corresponds to roughly halving the required exposure, or one stop. It does not make a fixed lens and shutter deliver twice as many photons.

The manufacturer’s sensitometry workbook provides a concrete example of what lies behind a film-speed number. Read only its speed-determination passage; you do not need to work through the workbook’s questions.

For the black-and-white negative-film method illustrated there, the speed-point exposure, expressed in lux-seconds, is approximately

where $S$ is the numerical ISO speed. Thus an ISO 100 film has a speed-point exposure of about $0.008$ lux-seconds under the specified test conditions. That is a defined point near the foot of the film’s response curve, not a prescription that every scene area receive exactly $0.008$ lux-seconds. Other image regions receive different exposures, and film types and digital-camera ISO ratings do not all use this particular speed-point definition.

This separation between delivered and required exposure is especially useful when comparing film and digital capture. Changing film stock changes the recording material. Changing a digital camera’s ISO setting generally changes how its captured signal is amplified or rendered; at fixed scene, aperture, and shutter time, it does not increase the photons incident on the sensor. The sensor details come later in the course. For now, treat ISO as a sensitivity or output-setting input to the choice of camera settings, not as a factor in the optical equation for $H$.

Stops and exposure value

The exposure equation contains a product and a square. Photographic controls become easier to compare on a base-two logarithmic scale:

EV describes an aperture–shutter combination, with the time numerically referenced to one second. At fixed scene luminance and lens transmission, combinations with the same EV deliver approximately the same $H$. Increasing EV by one, while holding the scene fixed, halves delivered exposure: $N^2/t$ doubles, so $t/N^2$ halves. Opening the aperture by a factor of $\sqrt{2}$ in diameter doubles its area and adds one stop of light; doubling shutter time does the same.

This logarithmic approach was formalized in the mid-twentieth-century Additive System of Photographic Exposure Control, or APEX. Read the exposure equation and its logarithmic form, paying attention to the distinction between a camera setting and a meter’s recommendation.

A brief visual comparison shows what the EV number deliberately leaves out:

For example, $N=8$ at $1/125$ second gives

Opening to about $N=5.66$ and shortening the time to $1/250$ second keeps $N^2/t$ at approximately $8000$. It keeps the nominal light exposure unchanged but changes both depth of field and motion blur. In this context, reciprocity means trading aperture against time to preserve $H$; the photochemical reciprocity failure studied earlier is a limit to assuming that equal $H$ always produces an identical film response at extreme durations.

Calculating settings from a luminance reading

A reflected-light meter measures scene luminance and proposes camera settings for a chosen ISO rating. A conventional form of its calibration equation is

where $S$ is the numerical ISO speed and $K$ is the reflected-meter calibration constant. For a numerical example, take $K=12.5$, with luminance entered in $\mathrm{cd}{\mathrm{m}}^{-2}$ and time in seconds. This equation is a calibrated recommendation about how to place measured scene tones; it is not the optical derivation of $H$, and $K$ must not be substituted for lens transmission.

Suppose a subject region has luminance $L=1000\mathrm{cd}{\mathrm{m}}^{-2}$, and the selected speed is ISO 100. The meter’s target is

At $N=8$, this requires $t=64/8000=1/125$ second: the EV 13 combination above. If lens transmission is approximately $T=0.80$, our optical estimate independently predicts

Notice what each calculation answers. The meter equation recommends a setting for the specified luminance and ISO rating. The optical equation estimates the light exposure actually delivered at that image point, given an assumed lens transmission. The numbers are connected, but they represent different quantities.

Now keep the same scene but select ISO 200. The meter target becomes $N^2/t=16000$, an increase of one EV. Keeping $N=8$ calls for $1/250$ second and delivers about $0.0393$ lux-seconds—half the previous light exposure. If instead you merely change the ISO setting to 200 while retaining $N=8$ and $1/125$ second, $H$ remains $0.0785$ lux-seconds. ISO has changed the recommended exposure, not the light admitted by those unchanged settings.

A meter recommendation is not a guarantee of the desired rendering. A reflective meter responds to measured luminance; a bright subject and a dark subject under similar illumination can therefore produce different readings. Its calibration supplies a reference placement, while the photographer may choose another. A positive one-stop exposure compensation instructs the camera to admit twice the light relative to its metered recommendation; for a fixed scene, that means choosing a setting one EV lower, perhaps by doubling shutter time. We will examine how meters obtain their readings later in this module.

Conclusion

The core calculation is $H=Et$, with $E$ approximately proportional to $L/N^2$ for a given lens and imaging geometry. Shutter time scales light exposure linearly; f-number scales it inversely by its square. ISO specifies how much exposure a material or output setting requires for a chosen result, but does not alter the photons delivered at fixed optical settings.

EV packages aperture and shutter time into a base-two number: one EV higher means half the light exposure for the same scene. A luminance-meter equation then uses scene luminance, ISO speed, and calibration to recommend an EV and an aperture–time pair. In the next lesson, we will look inside the timing mechanism itself: how different shutters realize the requested exposure time, and what happens when their motion interacts with moving subjects and flash.