# Illuminance

Physical size
Surname Illuminance
Formula symbol ${\ displaystyle E _ {\ mathrm {v}}}$
Size and
unit system
unit dimension
SI Lux  (lx) L -2 · J

The illuminance E v ( English illuminance ) describes the area-related luminous flux that hits an illuminated object . Opposite it is the light intensity , which describes the luminous flux of a light source related to the spatial angle .

The SI unit of illuminance is lux (lx, from the Latin lux , light).

A related term is the luminous flux density, the surface density of the luminous flux through a surface element perpendicular to the beam direction.

## Definition and units of measure

A luminous flux of 1 lumen, which hits an area of ​​1 m 2 , illuminates this (averaged) with 1 lux

If the luminous flux falls on a uniformly illuminated surface , the illuminance on the surface is equal to the quotient of the incident luminous flux and the surface : ${\ displaystyle A}$${\ displaystyle \ Phi _ {\ mathrm {v}}}$${\ displaystyle E _ {\ mathrm {v}}}$${\ displaystyle \ Phi _ {\ mathrm {v}}}$${\ displaystyle A}$

${\ displaystyle E _ {\ mathrm {v}} = {\ frac {\ Phi _ {\ mathrm {v}}} {A}}}$

If the illuminance varies over the area, this mathematically simplified formula provides the illuminance averaged over the area. If the local variation of the illuminance is to be described in detail, the transition to the differential quotient gives:

${\ displaystyle E _ {\ mathrm {v}} = \ lim _ {A \ to 0} {\ frac {\ Phi _ {\ mathrm {v}}} {A}} = {\ frac {\ mathrm {d} \ Phi _ {\ mathrm {v}}} {\ mathrm {d} A}}}$

Illuminance is measured in the unit lux, which is defined as lumen per square meter (1 lx = 1 lm / m 2 ). A luminous flux of 1 lm, which is evenly distributed over an area of ​​1 m 2 , causes an illuminance of 1 lx there.

In the Anglo-American system of measurement , especially in North America, the unit foot-candle (fc) is also used, which means lumen per square foot . 1 fc corresponds to approximately 10.764 lux.

## Photometric law of distance

The light intensity of a light source assumed to be point-shaped is defined as the quotient of the emitted luminous flux and the solid angle into which the light is emitted. The solid angle element, in turn, is the quotient of a surface element in distance and the square of this distance. Thus: ${\ textstyle I _ {\ mathrm {v}} = {\ frac {\ mathrm {d} \ Phi _ {\ mathrm {v}}} {\ mathrm {d} \ Omega}}}$${\ textstyle \ mathrm {d} \ Omega \, = \, {\ frac {\ mathrm {d} A} {r ^ {2}}}}$${\ displaystyle \ mathrm {d} A}$${\ displaystyle r}$

${\ displaystyle E _ {\ mathrm {v}} = {\ frac {\ mathrm {d} \ Phi _ {\ mathrm {v}}} {\ mathrm {d} A}} = {\ frac {I _ {\ mathrm {v}}} {r ^ {2}}}}$.

If one also takes into account the possibility that the receiving surface can be inclined by the angle against the direction of radiation ( is the angle between the surface normal and the direction of radiation), one obtains the photometric distance law : ${\ displaystyle \ varepsilon}$${\ displaystyle \ varepsilon}$

${\ displaystyle E _ {\ mathrm {v}} \, = \, {\ frac {I _ {\ mathrm {v}}} {r ^ {2}}} \ cdot \ cos \ varepsilon}$.

The photometric law of distance thus states that the illuminance decreases with the square of the distance between the light source and the illuminated surface. When the lighting distance is doubled, four times as many luminaires are required to achieve the same illuminance.

The unit of luminosity, the candela, is defined as 1 cd = 1 lm / sr. If a light source emits light with a luminous intensity of 1 cd in the direction of a receiving surface that is 1 m away perpendicular to the beam direction, it generates an illuminance of 1 lx there.

In lighting practice, extensive light sources are usually found. Here, more complex calculation methods based on the basic photometric law or working with visual factors must be used, which integrate the luminance distribution emanating from the luminous surface and the luminance distribution arriving on the receiving surface.

## Illuminance levels in practice

### Measurement

Luxmeter for measuring the illuminance

The illuminance is the photometric equivalent of the radiometric quantity irradiance (measured in watts per square meter, W / m 2 ). If electromagnetic radiation hits the receiving surface and generates the irradiance there, the illuminance caused by this radiation in lux (= lumens per square meter) can be determined by measurement or calculation by weighting the individual wavelengths of the radiation with the respective photometric radiation equivalent of the wavelength in question that describes the sensitivity of the eye. ${\ displaystyle E _ {\ mathrm {e}}}$${\ displaystyle E _ {\ mathrm {e}}}$

The illuminance is measured with a lux meter. At the Physikalisch-Technische Bundesanstalt  (PTB), illuminance levels between 0.001 lx and 100,000 lx can be achieved. This serves u. a. the calibration of illuminance meters.

### Illuminance levels required by standards

Target illuminance levels:

• Emergency lighting of escape routes : minimum illuminance at least 1 lux
• Workplaces depending on the work space, place and activity (indoor and outdoor) in accordance with Appendix 1 of ASR A3.4

### Examples of typical illuminance levels

 5 mW laser pointer, green (532 nm), 3 mm beam diameter 427,000 lx Modern operating room lighting, 3500  K 160,000 lx clear sky and sun in the zenith 130,000 lx 5 mW laser pointer, red (635 nm), 3 mm beam diameter 105,000 lx clear sky, sun height 60 ° (Central Europe midday in summer) Contributions: sun = 70,000 lx, sky light = 20,000 lx 90,000 lx clear sky, sun height 16 ° (Central Europe midday in winter) Contributions: Sun = 8,000 lx, sky light = 12,000 lx 20,000 lx overcast sky, sun height 60 ° (midday in summer) 19,000 lx Minimum requirement for dental treatment lights 15,000 lx In the shade in summer 10,000 lx overcast sky, sun height 16 ° (midday in winter) 6,000 lx Overcast winter day 3,500 lx Football stadium category 4 ( elite football stadium ) 1,400 lx Lighting TV studio 1,000 lx
 Dusk (sun just below the horizon) 750 lx Office / room lighting 500 lx Corridor lighting 100 lx living room 50 lx Street lighting 10 lx Twilight (sun 6 ° below the horizon) 3 lx Candle about 1 meter away 1 lx Full moon at its zenith, mean distance from the earth 0.27 lx Full moon night 0.05-0.36 lx Crescent at 45 ° height, mean distance from the earth 0.02 lx Starlight and airglow 0.002 lx Starry night sky (new moon) 0.001 lx Starlight 220 µlx Cloudy night sky without moon and extraneous lights 130 µlx Sirius 8 μlx

## Sample calculations

### Illuminance of a candle

The light intensity of a candle is about one candela (1 cd = 1 lm / sr ). It generates the illuminance at a distance of 2 m on a receiving surface perpendicular to the beam direction

${\ displaystyle E _ {\ mathrm {v}} = {\ frac {1 \ \ mathrm {cd}} {(2 \ \ mathrm {m}) ^ {2}}} = 0 {,} 25 \ {\ frac {\ mathrm {lm}} {\ mathrm {m} ^ {2}}} = 0 {,} 25 \ \ mathrm {lx}}$.

Objects that are vertically illuminated by a candle at a distance of approx. 2 m appear to be about as brightly illuminated as in the light of the full moon falling vertically.

### Luminous flux and luminous intensity of an isotropic light source

The illuminance generated by an isotropically radiating light source on a receiving surface at a distance of 3 m perpendicular to the direction of the beam is assumed to be ${\ displaystyle E _ {\ mathrm {v}}}$

${\ displaystyle E _ {\ mathrm {v}} = 20 \ \ mathrm {lx} \,}$.

According to the photometric law of distance, this results in a light intensity for the light source

${\ displaystyle I _ {\ mathrm {v}} = 20 \ \ mathrm {lx} \ cdot (3 \ \ mathrm {m}) ^ {2} \, \ mathrm {sr} ^ {- 1} = 180 \ \ mathrm {cd} \ ,.}$

Integrated over the full solid angle of 4π sr, the luminous flux generated by the light source calculated to ${\ displaystyle \ Phi _ {\ mathrm {v}}}$

${\ displaystyle \ Phi _ {\ mathrm {v}} = 4 \ pi \ \ mathrm {sr} \ cdot 180 \ \ mathrm {cd} = 2260 \ \ mathrm {lm}}$.

### Dining room table

There is a small, practically point-shaped light source on the ceiling, which isotropically emits the luminous flux Φ v = 3000 lumens in a conical area with an opening angle α = 160 °. What illuminance levels does it generate on the r = 1.67 m lower table top

• at point A, which is perpendicular to the light source and
• in point B, which is also on the table top, but d = 1.15 m next to point A?

The opening angle of 160 ° corresponds to a solid angle of . Since the light source radiates isotropically, the light intensity is the same in all directions of the illuminated half-space and is: ${\ displaystyle \ Omega = \ left (1- \ cos \ left (\ alpha / 2 \ right) \ right) \ cdot 2 \ pi \, \ mathrm {sr} = 5 {,} 19 \, \ mathrm {sr }}$

${\ displaystyle I _ {\ mathrm {v}} \, = \, {\ frac {\ Phi _ {\ mathrm {v}}} {\ Omega}} \, = \, {\ frac {3000 \ \ mathrm { lm}} {5 {,} 19 \ \ mathrm {sr}}} \, = \, 578 \ \ mathrm {cd}}$.

Since the light source is assumed to be point-shaped, the photometric distance law can be used to calculate the illuminance. For point A the distance r = 1.67 m and the angle of incidence ε = 0 °, so

${\ displaystyle E _ {\ mathrm {v}} (A) \, = \, {\ frac {578} {1 {,} 67 ^ {2}}} \ cdot \, \ cos (0 ^ {\ circ} ) \ \ mathrm {lx} \, = \, 207 \ \ mathrm {lx}}$.

For point B the distance to the light source ( Pythagorean theorem ) is:

${\ displaystyle r '\, = \, {\ sqrt {r ^ {2} + d ^ {2}}} \, = \, {\ sqrt {1 {,} 67 ^ {2} +1 {,} 15 ^ {2}}} \ \ mathrm {m} \, = \, {\ sqrt {4 {,} 11}} \ \ mathrm {m} \, = \, 2 {,} 02 \ \ mathrm {m }}$

and the angle of incidence is:

${\ displaystyle \ varepsilon '\, = \, 90 ^ {\ circ} - \ arctan \ left ({\ frac {r} {d}} \ right) \, = \, 34 {,} 6 ^ {\ circ }}$

This results in:

${\ displaystyle E _ {\ mathrm {v}} (B) \, = \, {\ frac {578} {4 {,} 11}} \ cdot \ cos (34 {,} 6 ^ {\ circ}) \ \ mathrm {lx} \, = \, 116 \ \ mathrm {lx}}$.

## Relation to radiometric and other photometric quantities

 radiometric quantity Symbol a) SI unit description photometric equivalent b) symbol SI unit Radiant flux radiant power, radiant flux, radiant power ${\ displaystyle \ Phi _ {\ mathrm {e}}}$ W ( watt ) Radiant energy through time Luminous flux luminous flux, luminous power ${\ displaystyle \ Phi _ {\ mathrm {v}}}$ lm ( lumens ) Radiant intensity irradiance, radiant intensity ${\ displaystyle I _ {\ mathrm {e}}}$ W / sr Radiation flux through solid angles Luminous intensity luminous intensity ${\ displaystyle I _ {\ mathrm {v}}}$ cd = lm / sr ( candela ) Irradiance irradiance ${\ displaystyle E _ {\ mathrm {e}}}$ W / m 2 Radiation flux through the receiver surface Illuminance illuminance ${\ displaystyle E _ {\ mathrm {v}}}$ lx = lm / m 2 ( lux ) Specific radiation emission current density, radiant exitance ${\ displaystyle M _ {\ mathrm {e}}}$ W / m 2 Radiation flux through the transmitter surface Specific light emission luminous exitance ${\ displaystyle M _ {\ mathrm {v}}}$ lm / m 2 Radiance radiance, radiance, radiance ${\ displaystyle L _ {\ mathrm {e}}}$ W / m 2 sr Radiant intensity through effective transmitter area Luminance luminance ${\ displaystyle L _ {\ mathrm {v}}}$ cd / m 2 Radiant energy amount of radiation, radiant energy ${\ displaystyle Q _ {\ mathrm {e}}}$ J ( joules ) by radiation transmitted energy Amount of light luminous energy, quantity of light ${\ displaystyle Q _ {\ mathrm {v}}}$ lm · s Irradiation irradiation, radiant exposure ${\ displaystyle H _ {\ mathrm {e}}}$ J / m 2 Radiant energy through the receiver surface Exposure luminous exposure ${\ displaystyle H _ {\ mathrm {v}}}$ lx s Radiation yield radiant efficiency ${\ displaystyle \ eta _ {\ mathrm {e}}}$ 1 Radiation flux through absorbed (mostly electrical) power Luminous efficiency (overall) luminous efficacy ${\ displaystyle \ eta _ {\ mathrm {v}}}$ lm / W
a)The index "e" is used to distinguish it from the photometric quantities. It can be omitted.
b)The photometric quantities are the radiometric quantities, weighted with the photometric radiation equivalent K , which indicates the sensitivity of the human eye.

## literature

• Hans R. Ris: Lighting technology for practitioners. 2nd edition, VDE-Verlag GmbH, Berlin-Offenbach, 1997, ISBN 3-8007-2163-5 .
• Horst Stöcker: Pocket book of physics. 4th edition, Verlag Harry Deutsch, Frankfurt am Main, 2000, ISBN 3-8171-1628-4 .

## Individual evidence

1. Lexicon of Physics , Spectrum
2. a b c DIN 5031 radiation physics in the optical field and lighting technology. Part 3: Quantities, symbols and units in lighting technology. Beuth, Berlin 1982.
3. Measurement of light, photometry . Physikalisch-Technische Bundesanstalt, p. 15 .
4. Committee for workplaces: ASR A3.4 / 7 safety lighting, optical safety guidance systems. BAuA, accessed on February 18, 2019 .
5. publisher: BAuA - Technical Occupational Safety and Health (including technical rules) - ASR A3.4 Lighting - Federal Institute for Occupational Safety and Health. Retrieved February 18, 2019 .
6. P.K. Seidelmann (Ed.): Explanatory Supplement to the Astronomical Almanac. University Science Books, Mill Valley 1992, ISBN 0-935702-68-7 , p. 493.
7. a b c d DIN 5034 daylight indoors. Part 2: Basics. Beuth, Berlin 1985.
8. ISO 9680 Dentistry - Treatment lights
9. Alan Pears: Chapter 7: Appliance technologies and scope for emission reduction . In: Strategic Study of Household Energy and Greenhouse Issues (PDF), Australian Greenhouse Office, June 1998, p. 61. Archived from the original .
10. Christopher CM Kyba, Andrej Mohar, Thomas Posch: How bright is moonlight? . In: Astronomy & Geophysics . 58, No. 1, February 1, 2017, pp. 1.31–1.32. doi : 10.1093 / astrogeo / atx025 .
11. Brightness of Sirius of −1.46  mag inserted into the formula from: Jean Dufay: Introduction to Astrophysics: The Stars . Dover Publications, 1964, ISBN 978-0-486-60771-9 ( limited preview in Google Book Search [accessed November 4, 2019]). ; see also Apparent Brightness # Illuminance