Chapter 16, Problem 70A

To determine

The height of streetlight position to have the same illumination on the ground under the lamp.

Answer to Problem 70A

  $2.3\text{ m}$ above the ground.

Explanation of Solution

Given:

The height of the streetlight bulbs above the ground is $d=3.3\text{ m}$

Streetlight contains two identical bulbs that means the luminous intensity $\left(I\right)$ of the bulbs are equal.

One of the two lamps is removed.

Formula used:

From the inverse square law of illumination, illuminance $\left(E\right)$ at any point on the surface is,

  $E=\frac{I}{d^2}$

  $\left(1\right)$

Where $I$ is the luminous intensity of the light source in the given direction, and

  $d$ is the distance between the surface and the light source

Calculation:

When one of the two bulbs is removed from the streetlight, luminous intensity is reduced to half of its initial intensity. That means $I$ is reduced to $\frac{I}{2}$

Substituting this in equation $\left(1\right)$ ,

  $E=\frac{\left(\raisebox{1ex}{$I$}\!\left/ \!\raisebox{-1ex}{$2$}\right.\right)}{d^2}=\frac{I}{2d^2}$

  $\left(2\right)$

This shows that the total illuminance on the ground is reduced to half of the initial illuminance (compared to equation $\left(1\right)$ ). Therefore, in order to get the initial illuminance on the ground even after the removal of one bulb, $d^2$ should be reduced by a factor of $2$ . That means $d$ should be reduced by a factor of $\sqrt{2}$ .

Hence, the height at which the streetlight should be positioned to have the same illumination on the ground under the lamp, if one of the two bulbs is removed is

  $\frac{3.3\text{ m}}{\sqrt{2}}=2.3\text{ m}$ above the ground.

Conclusion:

The height of streetlight position to have the same illumination on the ground under the lamp is $2.3\text{ m}$ above the ground.