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. Does that apply to Gamma, Beta and Alpha particles equally? The reason I ask is, being in SF and knowing 10 to 60% of mercury we receive here comes from China depending on the time of year. Lt. Gov. Newsom admitted they detected cesium and iodine last Saturday. You see, I’m concerned about the lack of data knowing that in the aggregate model in comparison to ton load (1,900) already spent 70% (IAEA)millions will die of exposure related diseases/cancer.

  2. It’s becoming apparent that the Japanese authorities are not handling any part of this disaster well.
    There are hundreds of thousands homeless, without clean drinking water, little proper food and haphazard electrical power.

    From MyWay : The nuclear crisis has compounded the challenges faced by a nation already saddled with a humanitarian disaster. Much of the frigid northeast remains a scene of despair and devastation, with Japan struggling to feed and house hundreds of thousands of homeless survivors, clear away debris and bury the dead.

    I’ve mentioned previously that Californian disaster experts have been exasperated with the equivalent Japanese agencies, their lack of co ordination, communications , and leadership style.

    The Fukushima Daiich incident is getting away from them and they refuse to acknowledge it. Yesterday two workers were injured by stepping in radioactive water. They were wearing ankle boots and little protective clothing.

    Finally it seems that some financial commentators are actually looking out the window (instead of staring at their computers) and seeing some real concern for a coming financial meltdown. There are already reports that Japan will possibly start to sell their U.S. bonds to raise funds for reparations.

  3. We’re assuming the data provided by the most dubious sources known to man, the government, is correct. A government proven time and time again to practice, “Don’t panic the herd at any and all costs.” So while the math refresher is appreciated, I can only imagine what the true radiation numbers really are. If your variables are wrong, the formula is moot.

    1. In the 50 s during the cold war, many peploe built fall-out shelters in/under their yards or under their homes. These usually consist of very thick concrete walls, contained ventilation systems, a separate well, chemical toilet, etc. Watch the movie Blast from the Past to get the general idea. It can be as elaborate as you can afford and want to make it.Seeing as I’ve actually seen footage of scientists observing nuclear tests in the desert at close range in similar shelters, I have to assume it is possible to survive the initial blast in such a shelter. The real key is surviving long-term how much food can you store, clean water, sewage problems, power, light, comfort items .uh, companionship? How about communication bring a short-wave radio perhaps?I’ve often pondered this very question. I’m a survivor and I like life. I wouldn’t cave to an apocalypse easily and I’d want to be as prepared as I could.

  4. Here’s some more math. 8 0f 18 monitors pulled off-line for abnormal readings equates to 45% of their equipment is either defective or mis-calibrated. Either the EPA is so incompetent they can’t maintain their own equipment or they’re hiding the true readings. And while we’re on it, the inverse square law applies to isotropic sources of radiation like the sun, an antenna or magnetism. The kind of radiation leaking from Japan acts more like a fluid and can be channeled, (Direct toward a particular end or object). Imagine smoke from a fire. It can be directed in a straight line with very little deviation nullifying the inverse square law of physics in this instance. The EPA is practicing the “Don’t panic the herd” law of government physics.

  5. This formula does not apply to the spread of radiation particles and is not the truth concerning Japan nuke plants.

  6. Just be aware this inverse square formula is meant for a stationary radiating object, not a moving mass of radiation. You could be 10 miles away from the power plant but directly down wind and receive lethal doses while people 2 miles up wind get nothing. No matter how far they’ve traveled, ingested alpha particles cause cancer, period.

  7. If you inhale any of these radioactive particles,what does that do to your inverse square law ?

    1. @Peter, The inverse-square law, and the way it was presented here (for the purpose of perspective), has nothing to do with inhaling radioactive particles (which is never a good thing).

  8. I think the distance rule applies to space between radioactive particles (fall out) and the person. For example, a radioactive rock (emitting 100 Rads/hr) lands three feet from you, then you will be exposed to X rems; if this rock lands 100 feet from you, then you will be exposed to Y rems. You can change Rad to Sv, and rems to Gy.

    The distance the blogger writes refers to physical distance the rock would travel. In this case, radioactive decay (half-life) is the determining factor. This the longer the rock remained in the upper atmosphere, the less radiation the rock would emit.

    If it took 1 hour for the rock to travel from Japan to SF, then it would emit a lot more radiation than if it took two weeks. And if you stayed far away from the rock itself, then you would receive mininal radiation.

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