Question

A student is studying a book placed at 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 lamp so that maximum illuminance is produced at the position of the book?

• A. $R$
• B. $\frac{R}{2}$
• C. $\frac{R}{\sqrt{2}}$
• D. $\frac{R}{\sqrt{3}}$

Answer 1

The correct option is C $\frac{R}{\sqrt{2}}$

We have Intensity,
$I$ = $\frac{I_{o}cos\theta }{(x^2+R^2)}=\frac{I_{o}x}{(x^2+R^2{)}^{3/2}}$

If we consider $I$ to be maximum then
$\frac{dI}{dx}=0$

$⟹\frac{I_o}{(x^2+R^2{)}^{3/2}}$ $-\frac{x\left(3/2\right)\left(2x\right)}{(x^2+R^2{)}^{5/2}}$ = 0

$⟹R^2-2x^2=0$

$⟹x=\frac{R}{\sqrt{2}}$