A luminous done shaped as a hemisphere rests on a horizontal plane. Its luminosity is uniform. Determine the illuminance at the centre of that plane if its luminance equals `L ` and is independent of direction.

Explanation

From the definition of luminance, the energy emitted in the radial direction by an element `dS` o fthe surafce of the dome is
`d Phi = LS d Omega`
Here `L =` constant. The solid angle `d Omega` is given by
`d Omega = (dA cos theta)/(R^(2))`
where `dA` is the area of an element on the plane illuminated by the radial light. then
`d Phi = (Ld SdA)/(R^(2)) cos theta`
The illuminance at `0` is then
`E = int (d Phi)/(dA) = int_(0)^(pi//2) (L)/(R^(2)) 2pi R^(2) sin theta d theta cos theta = 2pi L int_(0)^(i) x dx = piL`