200-candle power lamp is hung 4 m above the centre of a circular area of 5 m diameter. The Illumination at the centre of the area is

The correct answer is option 1):(9.728 lux)

Concept:

The illumination on the circular area is,

\(E = \frac{{flux}}{{area}} = \frac{{CP \times \omega }}{A}\)

Where

A is circular area

ω = solid angle

CP = candle power

\(A = \frac{\pi }{4}{d^2}\)

ω = 2π (1 – cos θ) steradians

d = circular area diameter

\(\cos \theta = \frac{h}{{\sqrt {{h^2} + {{\left( {\frac{d}{2}} \right)}^2}} }}\)

Candle power of the lamp (CP) = 200 CP

Circular area diameter (d) = 5 m

Height (h) = 4 m

\(A = \frac{\pi }{4}{d^2} = \frac{\pi }{4} \times {\left( 5 \right)^2} = 6.25\pi \;{m^2}\)

\(\cos \theta = \frac{h}{{\sqrt {{h^2} + {{\left( {\frac{d}{2}} \right)}^2}} }} = \frac{4}{{\sqrt {{4^2} + {{2.5}^2}} }} = 0.848\)

ω = 2π (1 – cos θ) = 2π (1 – 0.848) steradians = 0.3π steradians

\(E = \frac{{CP \times \omega }}{A} = \frac{{200 \times 0.3\pi }}{{6.25\pi }} = 9.728\;lux\)