5 Marks Questions

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

Let the height of the source is ‘h’ and the luminous intensity in the normal direction is I₀.

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, $\frac{\text{dE}}{\text{dh}}=0$

$\Rightarrow\frac{\text{l}_0\Big[(\text{R}^2+\text{h}^2)^\frac{3}{2}-\frac{3}{2}\text{h}\times(\text{R}^2+\text{h}^2)\frac{1}{2}\times2\text{h}\Big]}{(\text{R}^2+\text{h}^2)^3}$

$\Rightarrow\big(\text{R}^2+\text{h}^2\big)^\frac{1}{2}\big[\text{R}^2+\text{h}^2-3\text{h}^2\big]=0$

$\Rightarrow\text{R}^2-2\text{h}^2=0\Rightarrow\text{h}=\frac{\text{R}}{\sqrt{2}}$

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Question 25 Marks

The illuminance of a small area changes from 900 lumen/m² to 400 lumen/m² when it is shifted along its normal by 10cm. Assuming that it is illuminated by a point source placed on the normal, find the distance between the source and the area in the original position.

Answer

Let, I = Luminous intensity of source

E_A= 900 lumen/m²

E_B = 400 lumen/m²

Now, $\text{E}_\text{a}=\frac{\text{l}\cos\theta}{\text{x}^2}$ and $\text{E}_\text{B}=\frac{\text{l}\cos\theta}{(\text{x+10)}^2}$

So, $\text{l}=\frac{\text{E}_\text{A}\text{x}^2}{\cos\theta}=\frac{\text{E}_\text{B}(\text{x+10})^2}{\cos\theta}$

$\Rightarrow900\text{x}^2=400(\text{x+40})^2$

$\Rightarrow\frac{\text{x}}{\text{x+10}}=\frac{2}{3}$

$\Rightarrow3\text{x}=2\text{x}+20$

$\Rightarrow \text{x}=20\text{cm}$

So, The distance between the source and the original position is 20cm.

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