A point source of light moves in a straight line parallel to a plane table. Consider a small portion of the table directly below the line of movement of the source. The illuminance at this portion varies with its distance r from the source as ___________ .
Options
\[1 \propto \frac{1}{r}\]
\[1 \propto \frac{1}{r^2}\]
\[1 \propto \frac{1}{r^3}\]
\[1 \propto \frac{1}{r^4}\]
Solution
\[1 \propto \frac{1}{r^3}\]
Let the distance between the parallel straight lines be L.
Angle with normal = θ
We know,
\[I = \frac{I_o \cos\theta}{r^2}\]
From the above figure, we get
\[I = \frac{I_o \cos\left( {90}^0 - \alpha \right)}{r^2}\]
\[ \Rightarrow I = \frac{I_o \sin\alpha}{r^2}\]
\[ \Rightarrow I = \frac{I_o}{r^2}\left( \frac{L}{r} \right)\]
\[ \Rightarrow I = \frac{I_o L}{r^3}\]
L = constant for parallel moving source
So, \[I_o L = k ............\left(\text{constant}\right)\]
\[ \Rightarrow I = \frac{k}{r^3}\]
\[ \Rightarrow I\alpha\frac{1}{r^3}\]
