question_answer

A lamp (point source) 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 centre?

A)  \[\frac{r}{2}\]                  

B)         \[\frac{r}{\sqrt{2}}\]

C)  \[\frac{r}{\sqrt{3}}\]                     

D)         \[\frac{r}{\sqrt{7}}\]

E)  \[\sqrt{3r}\]

Correct Answer: C

Solution :

\[{{E}_{1}}=\frac{1}{{{h}^{2}}}\]                               ...(1) and         \[{{E}_{2}}=\frac{I\cos \theta }{\left( \sqrt{{{h}^{2}}+{{r}^{2}}} \right)}\]                    ...(2)                                 \[=\frac{I\times h}{{{({{h}^{2}}+{{r}^{2}})}^{3/2}}}\] \[\therefore \]  \[\frac{{{E}_{2}}}{{{E}_{1}}}=\frac{1}{8}=\frac{{{h}^{3}}}{{{({{h}^{2}}+{{r}^{2}})}^{3/2}}}\] \[\Rightarrow \]\[{{({{h}^{2}}+{{r}^{2}})}^{1/2}}=2h\]\[\Rightarrow \]\[{{h}^{2}}=\frac{r}{\sqrt{3}}\]