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
(a) $1\propto \frac{1}{r}$

(b) $1\propto \frac{1}{r^2}$

(c) $1\propto \frac{1}{r^3}$

(d) $1\propto \frac{1}{r^4}$

Answer 1

Correct option (c)

Let the distance between the parallel straight lines be L.
Angle with normal = ​θ
$$\begin{gathered} \mathrm{We}\mathrm{know}, \\ I=\frac{I_o\cos \theta }{r^2} \\ \mathrm{From}theabove\mathrm{fig}ure,weget \\ \mathrm{I}=\frac{{\mathrm{I}}_{\mathrm{o}}\cos \left({90}^0-\mathrm{\alpha }\right)}{{\mathrm{r}}^2} \\ \Rightarrow \mathrm{I}=\frac{{\mathrm{I}}_{\mathrm{o}}\mathrm{sin\alpha }}{{\mathrm{r}}^2} \\ \Rightarrow \mathrm{I}=\frac{{\mathrm{I}}_{\mathrm{o}}}{{\mathrm{r}}^2}\left(\frac{\mathrm{L}}{\mathrm{r}}\right) \\ \Rightarrow \mathrm{I}=\frac{{\mathrm{I}}_{\mathrm{o}}\mathrm{L}}{{\mathrm{r}}^3} \\ \mathrm{L}=\mathrm{constant}\mathrm{for}\mathrm{parallel}\mathrm{moving}\mathrm{source} \\ So,{\mathrm{I}}_{\mathrm{o}}\mathrm{L}=\mathrm{k}\left(\mathrm{constant}\right) \\ \Rightarrow \mathrm{I}=\frac{\mathrm{k}}{{\mathrm{r}}^3} \\ \Rightarrow \mathrm{I\alpha }\frac{1}{{\mathrm{r}}^3} \end{gathered}$$