Question

The illuminance of  a small area changes from 900 lumen m⁻² to 400 lumen m⁻² when it is shifted along its normal by 10 cm. Assuming that it is illuminated by a point source placed on the normal, find the distance between the source and the area in the original position.

Answer 1

Let the luminous intensity of the source be l and the distance between the source and the area, in the initial position, be x.

Given,

Initial illuminance (E_A)​ = 900 lumen/m²

Final illuminance (E_B)​​ = 400 lumen/m²

Illuminance on the initial position is given by,

`E_A=l costheta/x^2............(1)`

Illuminance at final position is given by

`E_B=(lcostheta)/(x+10)^2............(2)`

Equating luminous intensity from (1) and (2), we get

`l=(E_Ax^2)/costheta=(E_B(x+10)^2)/costheta`

`rArr900x^2=400(x+10)^2`

`rArrx/(x+10)=2/3`

⇒ 3x = 2x + 20

⇒ x = 20 cm

The distance between the source and the area at the initial phase was 20 cm.