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# Relation among Illumination, Brightness, and Luminous Intensity

## 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/m ^{2}**.

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).

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