# Relation among Illumination, Brightness, and Luminous Intensity

Power SystemsUtilisation of Electrical PowerUtilization of Electrical EnergyElectrical Engineering

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## Illumination

The luminous flux received by the surface per unit area is known as illumination. It is denoted by the letter 'E' and is measured in Lux or Lumen/m2.

Mathematically, the illumination is given by the expression,

$$\mathrm{Illumination,\mathit{E}\: =\: \frac{Luminous \: Flux\left ( \phi \right )}{Area \left (\mathit{A} \right )}\: =\:\frac{\mathit{C\, P}\times \omega }{\mathit{A}}}$$

## Luminous Intensity

Luminous intensity is defined as the amount of luminous flux emitted into a solid angle of a space in a specified direction. It is denoted by 'I' and is measured in Candela.

Mathematically,

$$\mathrm{Luminous\:Intensity,\mathit{I}\: =\: \frac{Luminous \: Flux}{Solid \: angle}}$$

## Brightness

The luminous intensity per unit surface area of the projected surface in the given direction is known as brightness of that surface. It is denoted by 'L' and is given by,

$$\mathrm{Brightness,\mathit{L}\: =\: \frac{Luminous \: Intensity\left ( \mathit{I} \right )}{Projected\: Area}\: =\:\frac{\mathit{I}}{\mathit{A}\, cos\, \theta }}$$

## Relation among Illumination, Luminous Intensity and Brightness

Consider a sphere with radius 'r' meters, having a source of 1 candle power and luminous intensity of 'I' candela at its center. Then, from the definitions, we have,

$$\mathrm{Brightness,\mathit{L}\: =\:\frac{\mathit{I}}{\pi \mathit{ r^{\mathrm{2}}}}\: \: \: \cdot \cdot \cdot \left ( 1 \right )}$$

And,

$$\mathrm{Illumination,\mathit{E}\: =\:\frac{\phi }{\mathit{A}}\: =\:\frac{C\, P\times \omega }{\mathit{A}}}$$

For a sphere,

$$\mathrm{\mathit{A}\: =\:4\pi \mathit{r}^{2}\: \: and\: \: \omega \: =\:4\pi }$$

$$\mathrm{\mathit{\therefore E}\: =\:\frac{\mathit{I}\times 4\pi }{4\pi \mathit{r}^{2}}\: =\:\frac{\mathit{I}}{\mathit{r}^{2}}\: \: \: \cdot \cdot \cdot \left ( 2 \right )}$$

From equations (1) & (2), we get,

$$\mathrm{\mathit{E}\: =\:\pi \mathit{L}\: =\:\frac{\mathit{I}}{\mathit{r}^{2}}\: \: \: \cdot \cdot \cdot \left ( 3 \right )}$$

Equation (3) gives the relation among luminous intensity (I), illumination (E) and brightness (L).