question_answer

A small lamp is hung at a height of 8 feet above the centre of a round table of diameter 16 feet. The ratio of intensities of illumination at the centre and at points on the circumference of the table will be               [CPMT 1984, 1996]

A)            1 : 1                                          

B)            2 : 1

C)            \[2\sqrt{2}:1\]                          

D)            3 : 2

Correct Answer: C

Solution :

                   Illuminance at A,                    \[{{I}_{A}}=\frac{L}{{{h}^{2}}}\]                    Illuminance at B,                      \[{{I}_{B}}=\frac{L}{\sqrt{{{({{h}^{2}}+{{r}^{2}})}^{2}}}}\cos \theta \]                    \[=\frac{Lh}{{{({{r}^{2}}+{{h}^{2}})}^{3/2}}}\]            \[\therefore \frac{{{I}_{A}}}{{{I}_{B}}}={{\left( 1+\frac{{{r}^{2}}}{{{h}^{2}}} \right)}^{3/2}}\]\[={{\left( 1+\frac{{{8}^{2}}}{{{8}^{2}}} \right)}^{3/2}}={{2}^{3/2}}=2\sqrt{2}:1\]