# Optical Distance Law

## Distance and geometric dilution of light

The illumination intensity on a surface is inversely proportional to the square of its distance from the light source and can be expressed as

*E = Φ / d^{2} (1)*

*where *

*E = light intensity, illumination (lux, lumen/m ^{2})*

*Φ *= *the quantity of light emitted by a lamp or a light source - luminous flux (lumen, lm)*

*d = distance from light source (m)*

Since

*E _{1} d_{1}^{2} = E_{2} d_{2}^{2} = constant ^{ }(2)*

*(2)* can be modified to

*E _{1} / E_{2} = d_{2}^{2} / d_{1}^{2 }(2b) *

### Example - Illumination Intensity from a Lamp at distance

The illumination intensity from lamp with luminance *10000 lumens* at distance *2 m* can be calculated with* (1)* as

*E _{1} = (10000 lumens) / (2 m)^{2}*

* = 2500 lux*

The illumination intensity at distance *5 m* can be calculated by modifying *(2b)* to

*E _{2} = E_{1} d_{1}^{2} / d_{2}^{2} *

* = (2500 lux) (2 m) ^{2} / (5 m)^{2}*

* = 400 lux*

### Cosine law of Illumination

*E = Φ / d^{2} cos θ (3)*

*where *

*θ* = angle between light source vertical line and sight line