A point source of light moves in a straight line parallel to a pl…

MCQ

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:

A
$\text{I}\propto\frac{1}{\text{r}}$
B
$\text{I}\propto\frac{1}{\text{r}^2}$
✓
$\text{I}\propto \frac{1}{\text{r}^3}$
D
$\text{I}\propto\frac{1}{\text{r}^4}$

✓

Answer

Correct option: C.

$\text{I}\propto \frac{1}{\text{r}^3}$

Let the distance between the parallel straight lines be $L$.
Angle with normal $=\theta$
We know,
$\text{I}=\frac{\text{I}_\text{o}\cos\theta}{\text{r}^2}$
From the above figure, we get
$\text{I}=\frac{\text{I}_\text{o}\sin(90^0-\alpha)}{\text{r}^2}$
$\Rightarrow\text{I}=\frac{\text{I}_\text{o}\sin\alpha}{\text{r}^2}$
$\Rightarrow\text{I}=\frac{\text{I}_\text{o}}{\text{r}^\text{2}}\Big(\frac{\text{L}_\text{o}}{\text{r}\text{}}\Big)$
$\Rightarrow\text{I}=\frac{I_\text{o}}{\text{r}^2}\Big(\frac{\text{L}}{\text{r}}\Big)$
$L =$ constant for parallel moving source
So $, I_o\ L =\text{K} \big($constant$\big)$
$\Rightarrow\text{I}=\frac{\text{k}}{\text{r}^3}$
$\Rightarrow\text{I}\alpha\frac{1}{\text{r}^3}$