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,

\[ t_2 = \frac{r^2 k}{E_o {\cos\theta}_2} = \frac{12}{\cos( {60}^0 )}\]

\[ \Rightarrow t_2 = 12 \times 2 = 24 s\]