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 $\frac{3}{8}$ of that at its center?

• A. $r\sqrt{\frac{3}{5}}$
• B. $\frac{r}{\sqrt{2}}$
• C. $\frac{r}{3}$
• D. $\frac{r}{\sqrt{3}}$

Answer 1

Answer: (A)

$r\sqrt{\frac{3}{5}}$

Illuminance at a point is inversely proportional to square of its distance from its lamp.

Let the lamp be at a height of h from the centre of table along the axis.

Let the illuminance at centre of table is L1.

Since illuminance is inversely proportional to square of distance.

L1 = $\frac{k}{h^2}$ where k is proportionality constant.

Now let illuminance at edge of table is L2.

Now distance between lamp and edge of table is $\sqrt{h^2+r^2}$.

Therefore L2 =$\frac{k}{h^2+r^2}$.

Since the given condition is L2 = $\frac{3}{8}$L1.

Substituting L1 and L2 in the above equation and solving for h we get

h= r$\sqrt{\frac{3}{5}}$