A photographic plate is placed directly in front of a small diffused source in the shape of a circular disc. It takes 12s to get a good exposure. If the source is rotated by 60° about one of its diameter, the time needed to get the same exposure will be ___________ .
Options
6s
12s
24s
48s
Solution
24s
Here,
\[ t_1 = 12 s\]
\[ \theta_1 = 0^0 \]
\[ \theta_2 = {60}^0 \]
\[ t_2 = ?\]
Let the distance be r.
Let the incident luminosity be \[E_o.\]
We have,
\[ E_\theta1 = \frac{E_o {\cos\theta}_1}{r^2}\]
\[ t_1 \alpha \frac{1}{E_\theta1}\]
\[ \Rightarrow t_1 = \frac{r^2 k}{E_o {\cos\theta}_1}\]
\[ \Rightarrow 12 = \frac{r^2 k}{E_o \cos0}\]
\[ \Rightarrow \frac{r^2 k}{E_o} = 12\]
Similarly,
\[ t_2 = \frac{r^2 k}{E_o {\cos\theta}_2} = \frac{12}{\cos( {60}^0 )}\]
\[ \Rightarrow t_2 = 12 \times 2 = 24 s\]
