Optics Question 501

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

[MP PET 2005]

Options:

A) $ \frac{r}{2} $

B) $ \frac{r}{\sqrt{2}} $

C) $ \frac{r}{3} $

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

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Answer:

Correct Answer: D

Solution:

$ \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} $

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



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