Radiation, Japan, and the Inverse-Square Law


When considering the potential for radiation spreading out from the severely damaged Fukushima Daiichi nuclear power plant, it is very important to understand the ‘inverse-square law‘, which helps to put in context the potential intensity of radioactive Fallout as it relates to distance.

What is the inverse-square law?
It is a physical law (Newton) stating that a specified physical quantity or strength is inversely proportional to the square of the distance from the source of that physical quantity.

For every doubling of distance away from the source that is emitting an ‘intensity’ (in this case, radiation), the radiation will be diluted to one-fourth the original quantity as it disperses into three-dimensional space to a point representing a doubling of distance.

If you double the distance between you and the source of radiation, you will theoretically be exposed to one-fourth the amount. There are variables of course, like the wind carrying concentrated amounts of radioactive particles in a particular direction, but the concept of the inverse-square law is helpful when putting distance versus dosage into perspective.

The Math – Inverse-Square Law

The Inverse-Square Law formula is as follows:

I1/I2 = (D2*D2)/(D1*D1)

I1 = Intensity at D1
I2 = Intensity at D2
D1 = Distance 1
D2 = Distance 2

To solve for the intensity at a location where an original set of measurements are known, we can solve for ‘I2’ by using the following version of the formula:

I2 = (I1*(D1*D1))/(D2*D2)

Average Radiation Dosage for Americans

First, lets put it in terms relating to the average dose per day that Americans get from natural and man-made radiation…

600 milliRem per year
6 milliSieverts per year

0.016438 mSv per day (milli Sieverts)
16.438 uSv per day (micro Sieverts)

0.000684932 mSv per hour (milli Sieverts)
0.684931507 uSv per hour (micro Sieverts)

Real world radiation examples relating to the Fukushima nuclear plant

Some recent reporting that appears valid, indicates that the radiation level at and around the immediate vicinity of the plant is around 500 uSv/h (micro Sieverts per hour). This is equivalent to 730 times the average radiation per hour that Americans receive!

The highest number I had read about earlier during the crisis was around 3,000 uSv/h. This is equivalent to 4,380 times the average radiation per hour that Americans receive!

The situation is obviously quite terrible nearby the plant and at least out to the current evacuation perimeter that has been recommended, 20 miles (the U.S. has recommended 50 miles).

Radiation making it to the U.S. and the Inverse-Square Law

I know that many in the U.S. have been highly concerned about radioactivity making its way here. And in fact as most of us have read reports that very small amounts have been detected. Having an engineering background, I know about the inverse-square law, and know that the levels here will remain very small in comparison to Japan. I also know that there are variables to this basic theory when we’re talking about radioactive Fallout.

Despite the inverse-square law, a great deal depends on how many particles make it on the wind currents. Also, the inverse-square law is somewhat challenged here because the particles will concentrate within fairly narrow wind patterns at first, while later on dispersing more.

Regardless of the variables though, here is the math while using the radiation numbers in Japan at Fukushima, 500 uSv/h (micro Sieverts per hour ).

You can run the numbers yourself with the formula above, but here are my results based on the following assumptions.

I1 = 500 uSv/h
D1 = 0.1 miles (about 500 feet around the plant)
D2 = 4,500 miles (distance to California)

Solve for I2 = (500*(0.1*0.1))/(4,500*4,500)

I2 = 0.000000246914 uSv/h (micro Sieverts per hour)
I2 = 0.000000000247 mSv/h (milli Sieverts per hour)

This is equivalent to 0.000036% of the average per hour that we normally get.

Now before some of you jump all over me, bear in mind that there are variables here that will offset these numbers…

Wind patterns, both low level and high level


The current situation today reveals that there is a reactor breach at No. 3

The risk is still there that a reactor may explode (or more than one)

The radiation levels there could shoot substantially higher

There is an added problem with Reactor No. 3 using MOX fuel (with Plutonium)

Accuracy of the numbers coming from TEPCO and the Japanese government

There will be long term effects from particles with long ‘half-life’s’ such as Cesium-137

We will have a ‘generations’ risk of consuming foods from the contaminated regions

The point here is, consider the perspective between what is currently happening there and what is or might happen here. We should remain vigilant, and continue to remain prepared for variables, changes in the current situation, and to realize that we have many risks around us – many of them out-of-sight and out-of-mind.

Here is a very informative video that simply explains some of the basics with regards to radioactive particles and measurements. Arnie Gundersen, Chief Nuclear Engineer

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  1. Mothra March 25, 2011
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