How does the area of illumination change when the distance from the light source is doubled?

When the distance from a light source is doubled, the area of illumination increases four times due to the inverse square law of light. This law states that the intensity of light or illumination from a point source is inversely proportional to the square of the distance from that source.

To understand this, consider the formula for illuminance (E), which describes how light intensity decreases with distance. The illuminance is calculated as follows:

[ E = \frac{I}{d^2} ]

Where ( E ) is the illuminance, ( I ) is the luminous intensity, and ( d ) is the distance from the light source.

If the distance is doubled (let's say from ( d ) to ( 2d )), substituting this into the formula gives:

[ E = \frac{I}{(2d)^2} = \frac{I}{4d^2} ]

This shows that the illuminance at double the distance is one-fourth the original intensity. However, since the area of illumination (which is proportional to how much surface area is covered by light) increases as the inverse square of the distance from the source, the area that is illuminated becomes four times larger.

Hence