Online RPS training course interactive resources - the inverse square law

It could be assumed by some that the inverse square law is somehow specific to work with ionising radiation (since it will appear at some point in just about any radiation safety training course at the RPS level). However, the law is far more universal than that, and can be applied to visible light (so any part of the electromagnetic spectrum which includes x-rays and gamma rays), gravity and sound (intensity). 

Quite often an expression of the following form is introduced early on.

\[ \dfrac{1}{r^2} \]

where \( r \) is the distance between a point source of energy (electromagnetic, gravity etc) and a measurement or calculation point. Whilst not a difficult expression for many, for others the simple fact of using something mathematical can be off putting (especially before the concept is understood conceptually). 

The Ionactive inverse square balloon demonstration

Way back Ionactive delivered monthly face to face radiation safety training courses at a venue in Buckinghamshire. We had a pack of white blow up balloons in our kit bag and a bright red thick marker pen. We would draw a small red square on the surface of the balloon and proceed to blow it up slowly in front of the delegates, with the red square facing them. We would then ask them to summarise what they had seen. 

After getting over the initial shock of this mad RPA messing around with balloons, they would begin to describe observations such as:

• The balloon got bigger (!). 
• The red square drawn on the balloon became bigger as the surface expanded.
• The red colour of the square become paler, pinker - fading away.

Importantly we did this BEFORE introducing:

\[ \dfrac{1}{r^2} \]

So the balloon is far from a perfect sphere - but that matters little for demo purposes. We then told the delegates to imagine that at the the centre of the balloon was a point source of ionising radiation (e.g. gamma ray source). The distance between the centre of the balloon and the outer surface was \( r \). As the balloon expands (i.e. the surface of the sphere moves away from the centre), so the value of \( r \) increases. The radiation from this point source is being radiated evenly outward from the centre of the sphere, and it can be shown that the surface area of the sphere (skin of the balloon) is proportion to the square of its radius (i.e. \( r^2 \) ). This means that as the emitted ionising radiation is further from the point source it gets spread out over this increasingly larger surface area. Therefore, the intensity of the radiation (e.g. the dose rate) is inversely proportional to the square of the distance ( \( r^2 \) ) - so gets less as the distance increases from the source. This is where the red coloured square comes in - it is increasing in size in proportion to \( \frac{1}{r^2} \), whilst the intensity of the colour is decreasing proportionally the same (i.e is becoming less red and more pink, and will eventually fade to be indistinguishable from the colour of the balloon - i.e. background). 

Don't worry if the above paragraph gives you more than a pause for thought, in our online training courses we concentrate on the visuals and the practicality of the inverse square law, Here we are just explaining how the use of the balloon translates to a more precise definition of the law.  

The inverse square law widget

Once upon a time… a balloon and a marker pen would suffice. Today… a fully interactive radiation protection widget. Same physics, modern learning. 

This widget was created for Ionactive by Dr Chris Robbins who has worked extensively with us over the last few years (and over decades if we go back to the very beginning). It is in our collection of interactive resources which can be explored here:  Radiation protection widgets. It is a virtual balloon inflation demonstration for modern times (with more precise physics and added bells and whistles). Take a look below and have a play. 

A few things to note. 

• The reference dose rate at 1 m (i.e. r = 1m) will change each time this page is reloaded so be aware any figures below may differ from what you observe when you have a play. 
• You can change the distance (from the point source) by moving the "Reference distance" bar (red circle). 
• Note how the sphere (balloon skin) expands with increasing distance (and contracts with decreasing distance).
• Note how the red square increases in size with distance (higher spread) and also fades in colour (i.e. radiation intensity or "dose rate" reduces).

The attenuation factor needs some explanation - but before we look at this, let's spell out what every RPS level course will state at some point:

• If you double the distance from a point source of electromagnetic radiation (i.e. x-rays / gamma rays) you will 1/4 the dose rate. 
• If you halve the distance from a point source of radiation you will realise a 4 fold increase in the dose rate. 

Try that now. Move the red circular slider between 0.5m, 1m , 2m and 4m and notice what happens. Each time you double the distance you will see a 1/4 reduction in intensity, and moving the other way (halving the distance) will yield a four fold increase in intensity. These reductions (or increases) are expressed by the attenuation factor. 

• At a distance of 1 m, the attenuation factor is 1 [\( \dfrac{1}{1^2} \)]. 
• At a distance of 0.5 m, the attenuation factor is 4 [\( \dfrac{1}{0.5^2} \)] - meaning halving the distance yields a four fold increase in intensity (dose rate). 
• At a distance of 2 m, the attenuation factor is 0.25 [\( \dfrac{1}{2^2} \)] - meaning that doubling the distance between 1 m and 2 m yields a 1/4 reduction in intensity (dose rate). 
• At a distance of 4 m, the attenuation factor is 0.0625 [\( \dfrac{1}{4^2} \)] - meaning that increasing the distance between 1 m and 4 m yields a 1/16 reduction in intensity (dose rate). 

Note how the attenuation factor is calculated in real time as you move the slider (i.e. \( \dfrac{1}{r^2} \) is calculated). 

Also calculated and displayed is the dose rate (based on the reference dose rate at 1m). Note that in each case the dose rate at 1m is multiplied by the attenuation factor to give the new dose rate at a particular distance. 

More generally, we see that the dose rate can be calculated from knowing the dose rate at 1m (i.e r=1), and modifying by an attenuation factor based on \( \dfrac{1}{r^2} \).

This is all you need to appreciate the significance of the inverse square law by using the interactive widget. So no blowing up balloons, but use the widget and your imagination to explore how this would work face to face in a training room.  

Further thoughts on the inverse square law

What follows is additional information which moves beyond the purpose of the widget resource featured here, but which some readers (or our delegates) may appreciate.

• With respect to ionising radiation protection, the inverse square does not apply to alpha particles or beta particles. 
• The inverse square law strictly only applies to a point source of electromagnetic radiation (e.g. x-rays of gamma rays). It does not apply to the following source types (unless certain bounding conditions apply which you can read about further): a line radiation source, a planar radiation source or a volume radiation source. The supplied links will take you to the Ionactive glossary and from there you can dig as deep as you like. 
• It will not apply directly where the radiation source is collimated - think about blowing up the balloon where the sides are constrained by an object (a tube for example). 

Whilst not discussed in depth in this article, but certainly discussed in our online RPS training , is the practical use of the inverse square law in radiation protection. More generally, maximizing the distance from a source of radiation in many cases is a valid ALARP (as low as reasonably practicable) option. For readers interested in high activity sealed sources (HASS), the widget should adequately demonstrate that moving towards a potentially high dose rate source yields rapidly increasing exposure (following a power law, it is not an exponential increase). Conversely, moving away from such a source yields significant advantages early on (potentially between life threatening exposures and survivable exposures) , but meets the law of diminishing returns at greater distances from the source. This is why protection by distance is supplemented by the protection by time and shielding principles (discussed in a future article). 

The inverse square law widget shows that the concept goes beyond the somewhat simplistic (but valid) mantra of double the distance or half the distance (etc). With  \( \frac{1}{r^2} \) you can use the calculator to predict a dose rate at any distance (within the confines of the widget). More generally, \( \frac{1}{r^2} \) can be used to describe the dose rate or distance if you know three out of four variables (the four being dose rates at two distances and two specified distances).  This can be expressed as follows:

\( D_1 r_1^2 = D_2 r_2^2 \) , where

• D₁ is the dose rate at distance r₁
• D₂ is the dose rate at distance r₂

This can then be arranged as follows

\( \dfrac{D_1}{D_2} = \dfrac{r_2^2}{r_1^2} \) , or 

\( \dfrac{D_1}{D_2} = \left(\dfrac{r_2}{r_1}\right)^2 \)

From these expressions you can further rearrange to find the missing / unknown value. You could put this into a spreadsheet and make a little inverse square calculator for personal use. Of you could head over to our ALARP interactive resource - Radiation time, distance & shielding interactive ALARP widget. Here are some examples of the rearrangements possible.

\( D_2 = D_1 \left(\dfrac{r_1}{r_2}\right)^2 \) , where you need D₂

\( D_1 = D_2 \left(\dfrac{r_2}{r_1}\right)^2 \) , where you need D₁

\( r_2 = r_1 \sqrt{\dfrac{D_1}{D_2}} \) , where you need r₂

\( r_1 = r_2 \sqrt{\dfrac{D_2}{D_1}} \) , where you need r₁