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, \[I_o L = k ............\left(\text{constant}\right)\]

\[ \Rightarrow I = \frac{k}{r^3}\]

\[ \Rightarrow I\alpha\frac{1}{r^3}\]