Chapter 16.1, Problem 10SSC

To determine

To Explain: Whether one bulb provides more or less illumination than two identical light bulbs at twice the distance.

Answer to Problem 10SSC

One bulb provides four times more illuminance than two identical bulbs

Explanation of Solution

Given:

Two identical light bulbs are given.

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:

From equation (1), it is found that illuminance $E$ is directly proportional to luminous flux $P$ and inversely proportional to square of distance between light bulb and screen i.e.

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

For a given luminous flux $P$ , the illuminance $E_1$ provided by one bulb at a distance $r_1$ from the screen can be calculated from equation (2)

  $E_1\propto \frac{P}{r_1{}^2}$

Or

  $E_1=\frac{P}{4\pi r_1{}^2}$  ...... (3)

For the same luminous flux $P$ , the illuminance $E_2$ provided by one bulb at a distance $r_2=2r_1$ from the screen can be calculated from equation (2)

  $E_1\propto \frac{P}{{\left(2r_1\right)}^2}$

  $E_1\propto \frac{P}{4r_1{}^2}$

Or

  $E_2=\frac{P}{4\pi \times \left(4r_2{}^2\right)}$  ...... (4)

Divide equation (3) by equation (4), we get

  $\begin{array}{l}\frac{E_1}{E_2}=\frac{P}{4\pi r_1{}^2}\times \frac{4\pi \left(4r_1{}^2\right)}{P}\\ \frac{E_1}{E_2}=\frac{4\cancel{r_1{}^2}}{\cancel{r_1{}^2}}\\ \frac{E_1}{E_2}=4\\ \boxed{E_1=4E_2}\end{array}$

Conclusion:

Hence, one bulb provides four times more illuminance than two identical bulbs.