What does squaring a number mean, and why does it appear in physics?

Two lengths supply two factors

A panel two tiles wide and two tiles tall has two rows of two tiles: four altogether. A panel with three tiles in each direction contains nine. We multiply the number of tiles in a row by the number of rows. That is why the area of a square with side $a$a is $a^2$(a)^(2).

Distinguish multiplying by two from squaring. For the number 3, the first operation gives 6 and the second gives 9. The small raised two means two identical factors; it does not mean adding two or doubling. Factors can also be fractions: the square of one half is one quarter.

If the side is 3 metres long, the area is 9 square metres, or 9 m². That is nine squares measuring 1 m by 1 m. A metre measures length; a square metre measures area. The symbol m² does not mean “two metres”.

The geometric picture uses positive lengths. As an arithmetic operation, squaring is also defined for zero and negative numbers: for example, $(-3)^2=9$( − 3())^(2) = 9. This does not give our panel a negative side length.

Light crosses an ever larger surface

Now imagine a small source in empty space. It operates steadily and emits equally in every direction; this is called isotropic emission. We treat it as a point source: its size is negligible compared with the distance to the receiver. There is no material absorbing or scattering the light, and no obstruction.

Draw an imaginary spherical surface around the source. Every outward ray crosses it. Each second, the energy passing through the whole surface equals the energy emitted each second by the source. A larger sphere does not remove some of that power. It simply has a larger area for the light to cross.

When the sphere’s radius doubles, its surface expands in two directions and its area becomes four times as large. At three times the radius, the area is nine times as large. The shape is round, but the square appears for the same reason as with our tiles: area has two dimensions. University of Reading: deriving the relationship from spherical geometry and energy conservation.

In the figure, first compare the tile counts and then the areas of the imaginary spheres. We always measure distance $r$r from the source’s centre.

The same power gives each square metre a smaller share

A small receiver collects only part of the light. Face it towards the source, perpendicular to the rays, and keep its area unchanged. At distance $2r$2r, each square metre of the imaginary sphere receives one quarter as much as at distance $r$r. The same receiver therefore collects one quarter of the power. At $3r$3r, it collects one ninth.

This is the inverse square law for our propagation model: multiply the distance by a factor and the received power per unit area is divided by that factor squared. Three times the distance gives one ninth of the flux; half the distance gives four times the flux. Throughout, the source emits just as much power as before.

We mean received energy flux per unit area, rather than the subjective sensation of brightness. Geometric spreading also does not mean that each photon in this model gradually loses energy. The same receiving surface intercepts a smaller fraction of the total power at a greater distance.

A faint star can be powerful

A star’s own power, emitted across all wavelengths, is called its luminosity. The flux reaching us depends on luminosity, distance and losses along the way. A weak signal alone therefore does not prove that a star is distant: it could be nearby but emit little power. OpenStax: stellar luminosity and apparent brightness.

The comparison “twice as far gives one quarter of the flux” applies to the same unchanged source, or to sources with equal luminosities under otherwise equal conditions. It does not work for two arbitrary stars without further information.

Check the source and geometry first

A distant star often approximately fits the model of a small isotropic source. Close to a large luminous panel, however, its different parts lie at different distances and directions from the receiver. Replacing them with a single point may no longer work. We should also check before applying our rule to a receiver close to a star’s visible surface. NBS/NIST: limits for extended sources, pp. 30–31.

A laser beam does not distribute its total power uniformly over an entire sphere. We need to know how the beam’s cross section changes; $L/(4\pi r^2)$L/(4π(r)^(2)) does not describe every distance from its exit aperture. Dust, fog, shadows or changes in the source itself also change the received signal. Here, the inverse square law describes geometric spreading under the stated conditions.

Check your reasoning: a receiver moves from $2r$2r to $3r$3r. What fraction of its previous flux remains? Not one ninth: its previous flux was already one quarter of the flux at $r$r. The new distance is one and a half times the previous distance, so the area increases by a factor of 2.25. We obtain $4/9$4/9 of the previous flux, about 44%.

Squaring appears here because area changes in two directions. The inverse square adds one further step: conserved power is divided by that area. Looking at the night sky, you can now distinguish “little light reaches us” from “this star emits little light”. The next mathematical step in the course is angles and a star’s altitude above the horizon.