Inverse square law radiation
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The Inverse Square Law
Inverse square law radiation: Isotropic radiation from a point source follows the inverse square law, which embodies the fact that the flux from a radiation source (e.g., a light emitter) diminishes by a factor that is the square of the distance between source and observer. This is because in three-dimensional space, the photon number flux, energy flux, or power (energy per unit time) that passes through an imaginary sphere enclosing the source cannot depend on the size of the sphere because photons and energy are conserved. (Of course, there must be no interaction between the radiation and anything inside that sphere- it goes without saying that the inverse square law does not apply if there is matter blocking the path between source and observer!) The vital conditions of isotropy and a point-source emitter are required for the simple formulas discussed below to apply. Photon numbers and energy are still conserved of course if these conditions don’t apply, it’s just that the formulas to account for the departures will no longer be simple.
In the simplest case, if the radiation source is isotropic and has a total luminosity of $L$ (in say, $\rm erg \ s^{-1}$), and the flux measured at a distance $r$ from the source is $F$ (in say, $\rm erg \ cm^{-2} \ s^{-1}$), then the surface area of the sphere times the flux must be equal to the total luminosity (power) at any distance (all $r$) from the source:
\[
4\pi r^{2} F \ = \ L \ .
\]
Then
\[
F \ = \ \frac{L}{4\pi r^{2}}
\]
and the latter is the inverse square law. The same basic relation applies if instead of $L$ we have the total number of photons $s^{-1}$ emitted isotropically in all directions, say $N_{0}$, and instead of $F$ we have the number of photons $s^{-1}$ intercepted per unit area at some distance $r$ from the source, say $n(r)$. That would give the following formula (using concrete units to make it even more clear):
\[
n(r) \ ({\rm photons \ cm^{-2} \ s^{-1}}) = \frac{N_{0} ({\rm photons \ s^{-1}})}{4\pi r^{2}}
\]
Note that if the radiation source is anisotropic, the inverse square law still applies but you have to take account of the angular distribution of the radiation. The basic formula will give only the average flux, and inferring luminosity from measured flux will incur an error whose size and sense depends on the angular distribution of the radiation emission, and the relative orientation of the source to the observer’s line of sight.
In general, for anything that is conserved and spreads out isotropically in three-dimensional space and having originated from a point, the inverse square law is followed. For example if you were in zero-gravity conditions and threw a bunch of tennis balls out in all directions, the number of tennis balls passing through unit area would fall off as the inverse square of the distance from the point of projection of the balls.
Ratios
A common type of question involves calculating ratios when you are given a factor change in either the distance between emitter and receiver or the received flux (or other relevant equivalent quantity), and you have to calculate the corresponding change in the other quantity. In such cases you do not need to know any of the quantities in the particular inverse square law. For example, if the distance between emitter and receiver is increased by a factor of 3, the received flux decreases by a factor of $3^{2}$. (Equivalently, the received flux is one-ninth of that before the change.) As another example, you could be asked to calculate the change in the distance between emitter and receiver if the measured flux increased by a factor of 5. In this case the distance must have decreased by a factor of $\sqrt{5}$. In other words, if $F_{1}$, $r_{1}$, and $F_{2}$, $r_{2}$ are corresponding pairs of fluxes and emitter-receiver separations,
\[
\frac{F_{1}}{F_{2}} \ = \ \left(\frac{r_{2}}{r_{1}}\right)^{2} \ .
\]
Inverse square law for gravity
In the case of Newtonian gravity, the entity that is conserved in the spatial regime is not tangible as such, but it can be thought of as something related to the number of “lines of force” although this is a hard thing to define. Details of the equation for gravitational force are given in the appropriate section but here we just give the inverse square law formula for gravitational force without explanation of the terms:
\[
F \ = \ \frac{GM_{1} M_{2}}{r^{2}}
\]
Inverse square law for electrostatic force
As with gravity, the entity that is conserved is not tangible but is related to the number of “lines of force.” The inverse square law formula for electrostatic force is
\[
F \ = \frac{Q_{1} Q_{2} }{4\pi\epsilon_{0} r^{2}}.
\]
Again, details are provided in the appropriate section.
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