Physics · Optics · Grade 11

Luminous flux and illuminance

If we raise the lamp to twice the height, how many times dimmer is the desk lit?

The situation

A new classroom at a school is being fitted out. The electrician suggests replacing the old 100 W incandescent lamps with 40 W LED panels: “More light, a smaller electricity bill.” How much light is needed to read at a desk, what do we measure it with, and if we raise a lamp to twice the height, how many times dimmer is the desk lit?

Topic

Photometry: luminous flux and illuminance. Lighting a classroom: a lamp twice as far — four times dimmer

Key ideas

Luminous intensity
E = I cos α / r²
Photometer

Formulas

I = \frac{Φ}{Ω}, Φ = 4 π I

Luminous intensity $I$ (candela, cd) is the luminous flux $Φ$ (lumen, lm) per unit solid angle. For a lamp radiating equally in all directions, $Φ = 4 π I$ .

E = \frac{Φ}{S} = \frac{I \cos α}{r^{2}}

Illuminance (lux, lx = lm/m²): inversely proportional to the square of the distance and directly proportional to the cosine of the angle of incidence.

L = \frac{I}{S \cos θ}

Luminance (cd/m²): the luminous intensity per unit apparent area of the source surface.

\frac{I_{1}}{r_{1}^{2}} = \frac{I_{2}}{r_{2}^{2}}

Photometer balance: when the grease spot becomes invisible, the two sides of the screen are equally illuminated.

m_{1} - m_{2} = - 2.5 \lg \frac{E_{1}}{E_{2}}, m - M = 5 \lg \frac{d}{10 pk}

Pogson's formula (stellar magnitudes) and the distance modulus: $M$ is the magnitude at which the star would be seen from 10 parsecs.

Theory

Radiant flux and luminous flux

The power a lamp emits is the radiant flux (W). The eye, however, perceives different colours differently: it is most sensitive to yellow-green light (≈ 555 nm), less to red and violet, and not at all to infrared. The flux “weighted” by the eye's sensitivity $V (λ)$ is the luminous flux, measured in lumens; at 555 nm, 1 W of radiation equals 683 lm. Most of the radiation of an incandescent lamp is infrared — so it gives about 12–14 lm/W, while an LED gives about 100 lm/W. On the graph in “Classroom” mode compare both spectra with $V (λ)$ .

Luminous intensity, illuminance, luminance

Luminous intensity $I$ (cd) is the “strength” of a source in a given direction; illuminance $E$ (lx) is the flux falling on a unit area of a surface; luminance $L$ (cd/m²) tells how “dazzling” the surface of the source is: if a lamp of the same $I$ has a small surface (a clear bulb, a thin glowing filament), its luminance is very high. On a sunny day the illuminance reaches tens of thousands of lux, while on a full-moon night it is far below 1 lx.

Laws of illuminance

The flux from a point source is spread over a sphere of area $4 π r^{2}$ at distance $r$ — the inverse square law: if the distance doubles, $E$ falls by a factor of 4. If the rays strike a surface obliquely, the same flux is spread over a larger area — the cosine law. $E$ is directly proportional to the luminous intensity of the source. In the laboratory you measure $E$ with a lux meter for different $r$ , $α$ and lamps, and see that the dependence of $E$ on $1 / r^{2}$ , $\cos α$ and $I$ is a straight line. The model: the lamp is a point source radiating equally in all directions; light reflected from the walls is not taken into account.

Photometers

The Bunsen photometer: the grease spot on a paper screen reflects less light and transmits more. If one side of the screen is brighter, the spot looks dark from that side and bright from the other; when both sides are equally lit the spot disappears. Then $I_{2} = I_{1} {(r_{2} / r_{1})}^{2}$ — this gives the luminous intensity of an unknown (for example incandescent) lamp by comparison with a known standard lamp. A modern lux meter is a silicon photodiode with a $V (λ)$ filter: a photoelectric measurement.

Lighting a classroom

Lighting standards set the average illuminance on the working surface and its uniformity $U_{0} = E_{m i n} / E_{a v g}$ . For example, the 2011 edition of the European standard EN 12464-1 requires 300 lx on desks in an ordinary classroom, 500 lx for evening classes and adult-education rooms, and $U_{0} \geq 0.6$ ; allowing for lamps ageing and dimming, a design usually takes a value above the standard. The model counts only the light that falls directly from the lamps — in a real room light reflected from the white walls and ceiling adds to it. An LED panel directs light downwards, while an incandescent lamp scatters half of it onto the ceiling.

Stellar magnitudes and photoelectric photometry

Astronomers measure the brightness of stars by magnitude $m$ : the larger the magnitude, the fainter the star. In 1856 Pogson fixed the scale: a difference of 5 magnitudes is exactly a factor of 100 in illuminance. Apparent magnitude $m$ means how a star looks from Earth, and absolute magnitude $M$ how it would look at a distance of 10 pc (32.6 light-years). A photoelectric photometer on a telescope (a photomultiplier or CCD) measures the flux of starlight; in the UBV system, measurements through three filters (ultraviolet, blue, yellow) give the colour index $B - V$ , from which the star's temperature is estimated (the “Spectra” experiment).

Did you know?

In the SI system the candela is defined like this: the luminous efficacy of monochromatic radiation of frequency 540·10¹² Hz (green, about 555 nm) is exactly 683 lm/W. The eye is most sensitive to green light around this value.
In 1856 Norman Pogson fixed the scale of stellar magnitudes: a difference of 5 magnitudes means exactly a factor of 100 in illuminance, so 1 magnitude is about a factor of 2.512.

Inside the experiment

The experiment starts from a real-life situation: you predict the result, change the controls, run the experiment and draw a conclusion. Inside are missions, theory, a quiz and a guided tour in which the camera moves and explains the key points.