question_answer

A lamp rated at 100 cd hangs over the middle of a round table with diameter 3 m at a height of 2 m. It is replaced by a lamp of 25 cd and the distance to the table is changed so that the illumination at the centre of the table remains as before. The illumination at edge of the table becomes X times the original. Then X is                                [CPMT 1989]

A)            \[\frac{1}{3}\]                          

B)            \[\frac{16}{27}\]

C)            \[\frac{1}{4}\]                      

D)            \[\frac{1}{9}\]

Correct Answer: A

Solution :

                   Case I                     \[{{I}_{A}}=\frac{100}{{{2}^{2}}}=25\ cd\]                    and \[{{I}_{B}}=\frac{100}{{{(2.5)}^{2}}}\cos \theta \]                    \[=\frac{100}{{{2.5}^{2}}}\times \frac{2}{2.5}\]\[=\frac{200}{{{(2.5)}^{3}}}\]                    Case II,                     \[I{{'}_{B}}=X\,{{I}_{B}}=\frac{25}{{{(3.25)}^{3/2}}}\]                    so \[\frac{I{{'}_{B}}}{{{I}_{B}}}=\frac{25}{200}\times \frac{{{(2.5)}^{3}}}{{{(3.25)}^{3/2}}}\]            \[\Rightarrow X=1/3\]