Question

Many physical quantities are connected by inverse square laws, that is, by power functions of the form\({\bf{f}}\left( {\bf{x}} \right){\bf{ = k}}{{\bf{x}}^{{\bf{ - 2}}}}\). In particular, the illumination of an object by a light source is inversely proportional to the square of the distance from the source. Suppose that after dark you are in a room with just one lamp and you are trying to read a book. The light is too dim and so you move halfway to the lamp. How much brighter is the light?

Short answer

The brightness is 4 times as much.

Step-by-step solution

Introduction

Inverse square law states that “the Intensity of the radiation is inversely proportional to the square of the distance”. “The intensity of the light to an observer from a source is inversely proportional to the square of the distance from the observer to the source”

Find the brightness of light

The light intensity\(f\left( x \right)\), that is the brightness of the light, varies as the inverse of

the square of the distance\(x\)to the light source, so we have\(f\left( x \right) = k{x^{ - 2}}\)

Suppose you are a fixed distance\(d\)from the lamp then the intensity there is

\(f\left( d \right) = k{d^{ - 2}}\)

You move in halfway closer so your new distance from the lamp is\(\frac{d}{2}\)

The light intensity now is given by

\(\begin{array}{l}f\left( {\frac{d}{2}} \right) = k{\left( {\frac{d}{2}} \right)^{ - 2}}\\\;\;\;\;\;\;\; = k{\left( {\frac{2}{d}} \right)^2}\\\;\;\;\;\;\;\; = {2^2}k \cdot \frac{1}{{{d^2}}}\\\;\;\;\;\;\;\; = 4 \cdot k{d^{ - 2}}\end{array}\)

Conclusion,

\(f\left( {\frac{d}{2}} \right) = 4f\left( d \right)\),

Hence, the brightness is 4 times as much.