Question

A lamp is hanging along the axis of a circular table of radius r. At what height should the lamp be placed above the table, so that the illuminance at the edge of the table is $\frac{1}{8}$ of that at its center

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

Answer 1

Answer: (D)

$\frac{r}{\sqrt{3}}$

Answer 2

$\frac{I_{center}}{I_{edge}} = \frac{(r^{2} + h^{2})^{3/2}}{h^{3}}$

$\Rightarrow 8 = \frac{(r^{2} + h^{2})^{3/2}}{h^{3}} \Rightarrow 2h = (r^{2} + h^{2})^{1/2}$

$\Rightarrow 4h^{2} = r^{2} + h^{2} \Rightarrow 3h^{2} = r^{2}$⇒ $h = \frac{r}{\sqrt{3}}$