Luminous intensity

Prof. Dr. Thorsten Jungmann (last updated 2025-09-03)

Luminous intensity (German: Lichtstärke) describes the visible radiation emitted in a particular direction in space per solid angle, weighted with the spectral luminous efficiency of the human eye.

Do not confuse

Illuminance $E_{\mathrm{v}}$ in lux (Lx) describes the light incident on an area;
luminance $L_{\mathrm{v}}$ in cd/m² describes how bright an area appears.
Luminous intensity $I_{\mathrm{v}}$ in cd depends on direction and refers to the solid angle.

1 Quantity symbol and unit

The quantity symbol of luminous intensity is $I_{\mathrm{v}}$
Its unit (SI base unit) is the candela (Latin for candle) with the unit symbol $\mathrm{cd}$.

$$[I_{\mathrm{v}}]=\mathrm{cd}$$

Definition via the photometric relationship:

$$I_{\mathrm{v}}=\frac{\mathrm{d}{\Phi }_{\mathrm{v}}}{\mathrm{d}\Omega }$$

with the luminous flux ${\Phi }_{\mathrm{v}}$ in lumens (lm) and the solid angle $\Omega$ in steradians (sr).

SI definition of the candela (at $540\mathrm{THz}$, monochromatic):

$$1\mathrm{cd}=\frac{1}{683}\frac{\mathrm{W}}{\mathrm{sr}}$$

2 Practical and worked examples

In the fundamentals of electrical engineering, no calculations are carried out with luminous intensity. It appears here primarily because I wanted to include the base quantities in full. Nevertheless, a few worked examples follow for those who are interested.

2.1 Isotropic point source

equal luminous intensity in all directions

$${\Phi }_{\mathrm{v}}=4\pi I_{\mathrm{v}}$$

Example: $I_{\mathrm{v}}=2.0\mathrm{cd}\Rightarrow {\Phi }_{\mathrm{v}}\approx 25.1\mathrm{lm}$.

2.2 Illuminance of a point source at distance $r$

$$E_{\mathrm{v}}=\frac{I_{\mathrm{v}}}{r^2}$$

Example: $I_{\mathrm{v}}=10\mathrm{cd}$, $r=2.0\mathrm{m}\Rightarrow E_{\mathrm{v}}=2.5\mathrm{lx}$.

2.3 From luminous flux and beam angle

Simplified estimate:
Solid angle of a rotationally symmetrical cone with half-angle $\theta$:

$$\Omega \approx 2\pi (1-\cos \theta )$$

Then:
$I_{\mathrm{v}}\approx \frac{{\Phi }_{\mathrm{v}}}{\Omega }$

Example: ${\Phi }_{\mathrm{v}}=300\mathrm{lm}$, half-angle $\theta =60^{\circ }\Rightarrow \Omega \approx \pi \mathrm{sr}$, hence $I_{\mathrm{v}}\approx 95.5\mathrm{cd}$.

3 Typical orders of magnitude

• Candle: $\approx 1\mathrm{cd}$
• Indicator light emitting diode (directional): $\approx 1\dots 20\mathrm{cd}$
• Torch (peak): $\approx 10^2\dots 10^3\mathrm{cd}$
• Vehicle headlamp (peak): $\approx 10^4\dots 10^5\mathrm{cd}$

4 Example phrasings

• ‘The light emitting diode has a luminous intensity of $8\mathrm{cd}$ in the main direction of emission.’
• ‘At $r=3\mathrm{m}$, the result is $E_{\mathrm{v}}=I_{\mathrm{v}}/r^2=1.1\mathrm{lx}$.’
• ‘The isotropic source with $I_{\mathrm{v}}=5\mathrm{cd}$ delivers ${\Phi }_{\mathrm{v}}\approx 63\mathrm{lm}$.’

5 Related quantities and relationships

• Luminous flux:
  ${\Phi }_{\mathrm{v}}=\int I_{\mathrm{v}}\mathrm{d}\Omega ,I_{\mathrm{v}}=\frac{\mathrm{d}{\Phi }_{\mathrm{v}}}{\mathrm{d}\Omega }$
• Illuminance (point-source approximation):
  $E_{\mathrm{v}}=\frac{I_{\mathrm{v}}}{r^2}$

6 Related components

• Light emitting diode