Chapter 16.1, Problem 12SSC

To determine

The distance at which light bulb will provide the correct illumination.

Answer to Problem 12SSC

The light bulb should be at a distance of 0.71 m to provide correct illuminance.

Explanation of Solution

Given:

The distance is $r=1.0\text{m}$ .

Formula used:

The expression for illumination is given by

  $E=\frac{P}{4\pi r^2}$  ...... (1)

Here, $P$ and $r$ are luminous flux and the distance of the object from the light source respectively.

Calculation:

Since the illuminance is directly proportional to luminous flux $P$ and inversely proportional to the square of distance between light bulb and the screen i.e.

  $E\propto \frac{P}{r^2}$  ...... (2)

If the initial illuminance of the bulb is $E_i$ at a distance $r_i$ and final illuminance of the bulb is $E_f$ at a distance $r_f$ , then the initial illuminance of the bulb is half of the final illuminance of the bulb i.e.

  $E_i=\frac{E_f}{2}$

  $\frac{E_i}{E_f}=\frac{1}{2}$  ...... (3)

Since illuminance is inversely proportional to the square of distance between light bulb and the screen i.e.

  $E\propto \frac{1}{r^2}$  ...... (4)

Therefore, for initial illuminance, equation (4) becomes,

  $E_i\propto \frac{1}{r_i{}^2}$  ...... (5)

For final illuminance, equation (4) becomes,

  $E_i\propto \frac{1}{r_i{}^2}$  ...... (6)

Divide equation (5) by equation (6) both side,

  $\frac{E_i}{E_f}=\frac{r_f{}^2}{r_i{}^2}$  ...... (7)

Substitute $\frac{1}{2}$ for $\frac{E_i}{E_f}$ in equation (7)

  $\frac{1}{2}=\frac{r_f{}^2}{r_i{}^2}$

  $r_f=\sqrt{\frac{r_i{}^2}{2}}$  ...... (8)

Substitute 1.0 m for $r_i$ in equation (8)

  $\begin{array}{l}{r}_f=\sqrt{\frac{{\left(1.0\text{m}\right)}^2}{2}}\\ \boxed{r_f=0.71\text{m}}\end{array}$

Conclusion:

Hence, the light bulb should be at a distance of 0.71 m to provide correct illuminance.