Question

Light from a point source falls on a screen. If the separation between the source and the screen is increased by 1%, the illuminance will decrease (nearly) by
(a) 0.5%
(b) 1%
(c) 2%
(d) 4%

Answer 1

Correct option (c)
​Illuminance is given by:
$E=\frac{I_o\cos \theta }{r^2}$
$$\begin{gathered} \theta =0^0 \\ \frac{\Delta r}{r}=1\% \\ E=\frac{I_o}{r^2} \\ \mathrm{Differentiating}, \\ dE=-2\frac{I_o}{r^3}dr \\ \mathrm{As}\mathrm{approximation}\mathrm{differentials}\mathrm{are}\mathrm{replaced}\mathrm{by}\mathrm{\Delta }, \\ \mathrm{\Delta E}=-2\frac{{\mathrm{I}}_{\mathrm{o}}}{{\mathrm{r}}^3}\mathrm{\Delta r} \\ \Rightarrow \mathrm{\Delta E}=-2\frac{{\mathrm{I}}_{\mathrm{o}}}{{\mathrm{r}}^2}\left(\frac{\mathrm{\Delta r}}{\mathrm{r}}\right) \\ \Rightarrow \mathrm{\Delta E}=-2\mathrm{E}\left(\frac{\mathrm{\Delta r}}{\mathrm{r}}\right) \\ \Rightarrow \frac{\mathrm{\Delta E}}{\mathrm{E}}=-2\left(\frac{\mathrm{\Delta r}}{\mathrm{r}}\right) \\ \Rightarrow \frac{\mathrm{\Delta E}}{\mathrm{E}}=-2\times 1\%=-2\% \\ Since,n\mathrm{egative}\mathrm{sign}\mathrm{implies}\mathrm{decrease};hence,illuminancedecreasesby2\%. \end{gathered}$$