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

Question

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:

$\text{I}\propto\frac{1}{\text{r}}$
$\text{I}\propto\frac{1}{\text{r}^2}$
$\text{I}\propto \frac{1}{\text{r}^3}$
$\text{I}\propto\frac{1}{\text{r}^4}$

✓

Answer

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

Explanation:

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_oL $=\text{K} \big(\text{constant}\big)$

$\Rightarrow\text{I}=\frac{\text{k}}{\text{r}^3}$

$\Rightarrow\text{I}\alpha\frac{1}{\text{r}^3}$

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