Question

A point source emitting uniformly in all directions is placed 0.5 m above a table top. The luminous flux of the source is $500\pi $ lumen. Find the illuminance at a small surface of the table top 0.8 m from the source
A. $125lux$
B. $122lux$
C. $118lux$
D. $115lux$

Answer 1

Hint: Look at the values given in the problem. Here, luminous flux($\Delta F$) and luminous intensity(I)are given. So, first find the luminous intensity using the solid angle concept. Then find the illuminance using the angle made between the point source and the table top.

Complete step by step answer:
 Luminous flux is defined as the amount of light passing through a unit area in one second. In other ways it is a measure of the total amount of visible light emitted by a point source. On the other hand luminous intensity is defined as the amount of visible light emitted by a point source in unit time per unit solid angle.

Intensity of luminous flux is given by $I = \dfrac{{\Delta F}}{{\Delta w}}$
Where, $\Delta F$ is the luminous flux and $\Delta w$ is the solid angle.
As the point source is emitting uniformly in all directions, hence there are $4\pi $ steradians about a point enclosed by a sphere.
$ \Delta w = 4\pi $
Also it is given that luminous flux, $\Delta F = 500\pi $
\[I = \dfrac{{500\pi }}{{4\pi }} = 125\]

Illuminance (E) is defined as the total luminous flux of light incident per unit area.
\[ E = \dfrac{{\Delta F}}{{\Delta A}}\\
\Rightarrow E = \dfrac{{I\cos \theta }}{{{r^2}}}\]
Here, \[\cos \theta = \dfrac{{0.5}}{{0.8}} = \dfrac{5}{8}\] and r = 0.8 m
Putting these values in \[E = \dfrac{{I\cos \theta }}{{{r^2}}}\] we get
\[E = 125 \times \dfrac{5}{8} \times \dfrac{1}{{{{0.8}^2}}} \\
\therefore E= 122\]lux

Hence the correct option is B.

Note:The SI unit of luminous flux is lumen. The unit of luminous intensity is 1 lumen per steradians. Steradian is the unit of solid angle. The unit of luminous intensity is also called candela. On the other hand, the SI unit of illuminance is lux. It should be noted that the luminous flux is the amount of light emitted by the source, while illuminance is the total amount of light received by an object.