Question 15 Marks
A student is studying a book placed near the edge of a circular table of radius $R$. A point source of light is suspended directly above the centre of the table. What should be the height of the source above the table so as to produce maximum illuminance at the position of the book?
Answer
View full question & answer→Let the height of the source is $‘h\ ’$ and the luminous intensity in the normal direction is $I_0.$
So, illuminance at the book is given by,
$\text{E}=\frac{\text{l}_0\cos\theta}{\text{r}^2}=\frac{\text{l}_0\text{h}}{\text{r}^2}=\frac{\text{l}_0}{(\text{r}^2+\text{h}^2)^{ \frac{3}{2}}}$
For maximum $E, \frac{\text{dE}}{\text{dh}}=0$
$\Rightarrow\frac{\text{l}_0\Big[(\text{R}^2+\text{h}^2)^\frac{3}{2}-\frac{3}{2}\text{h}\times(\text{R}^2+\text{h}^2)\frac{1}{2}\times2\text{h}\Big]}{(\text{R}^2+\text{h}^2)^3}$
$\Rightarrow\big(\text{R}^2+\text{h}^2\big)^\frac{1}{2}\big[\text{R}^2+\text{h}^2-3\text{h}^2\big]=0$
$\Rightarrow\text{R}^2-2\text{h}^2=0$
$\Rightarrow\text{h}=\frac{\text{R}}{\sqrt{2}}$
So, illuminance at the book is given by,
$\text{E}=\frac{\text{l}_0\cos\theta}{\text{r}^2}=\frac{\text{l}_0\text{h}}{\text{r}^2}=\frac{\text{l}_0}{(\text{r}^2+\text{h}^2)^{ \frac{3}{2}}}$
For maximum $E, \frac{\text{dE}}{\text{dh}}=0$
$\Rightarrow\frac{\text{l}_0\Big[(\text{R}^2+\text{h}^2)^\frac{3}{2}-\frac{3}{2}\text{h}\times(\text{R}^2+\text{h}^2)\frac{1}{2}\times2\text{h}\Big]}{(\text{R}^2+\text{h}^2)^3}$
$\Rightarrow\big(\text{R}^2+\text{h}^2\big)^\frac{1}{2}\big[\text{R}^2+\text{h}^2-3\text{h}^2\big]=0$
$\Rightarrow\text{R}^2-2\text{h}^2=0$
$\Rightarrow\text{h}=\frac{\text{R}}{\sqrt{2}}$