A lot people in the NDE industry are intimidated by the inverse square law. In the Nondestructive Examination business it is most commonly used to establish boundaries for the use of Ionizing Radiation when performing Radiography (Taking X-Rays). These boundaries are required by Law and are there to protect the general public from overexposure to ionizing radiation. Once these boundaries are established the RT technician is responsible for controlling access in and out of the controlled area.
The Inverse square law may be used to determine the intensity of the radiation field at a given distance from a point source of radiation. In NDE/NDT we use both Gamma radiation sources (Radioactive Materials - RAM) and X-Rays from an X-Ray tube. Ideally we use the smallest size source that we can, physically (Focal spot size). The smaller the source is in physical size the sharper the image will be.
If you are getting ready to take the ASNT IRRSP examination you will need to know how to use this formula.
For those of you who are good at math this is a pretty simple formula to use. If I verbalize this (put it into words) it states that "the intensity of a source is inversely proportional to the square of the distance from that source".
Now, many people would be completely lost with this formula being written as it is. Those who do a lot of math or have had advanced math are currently yelling at the screen "Cross Multiply and divide!". I'm here to tell you that this is just multiplication and division. If you have a calculator and you can perform multiplication and division on that calculator, you can do this formula.
I've found that the issue isn't in doing the math, it is in getting the right information out of the word problem and plugging it in the right spot.
So, let's look at this in a different way. I'm going to rewrite this formula and I'm going to break down the steps that go into making it work.
Many people prefer to see this formula written as follows:
...for finding the new Intensity at a new distance, when we have been given an intensity at a starting distance.
...for finding the new Distance for a new intensity, when we have been given an intensity at a starting distance. This one has an extra step, we have to perform a square root calculation (this is a function on your calculator).
Intensity 2 = This is the new intensity we are looking for (I2)
Distance 2 squared = This is the new distance we have been given, multiplied by itself. ie; 1 foot squared = 1 x 1 = 1
Intensity 1 = The original intensity that we were given or we measured (I1)
Distance 1 squared = This was our original distance from the source where we measured Intensity 1, we multiply it by itself (Square it). ie; 25 feet squared = 25 x 25 = 625
I like to think of it this way I1 and D1 are where I am (or where I started), I2 and D2 are where I want to be (or where I will end up). One important thing to remember when applying this formula you will always start by multiplying [I1] x [D1 squared], you will always have a starting point. You will always have a place you are going to, if you know the new intensity you are looking for then that is the number you will divide by and it will give you a distance. If you know the new distance where you are going, you will divide by that value squared. We will demonstrate both of these situations.
Before I get neck deep in the math, the quick short hand for this is that if you Double your Distance you get 1/4 the intensity (Divide by 4), if you cut the distance in half you increase the intensity by 4 (multiply by 4). If you started at 25 feet and moved to 50 feet, you would only get 1/4 of the radiation you were previously getting, and if you moved to 12.5 feet you would be getting 4 times as much.
Now we'll dive in to this...
A very basic test question could go like this (It also happens to be a real world question). If I have a known intensity of 400 mR/hr at 10 feet, what will be my new intensity at 15 feet?
Let's get our values first;
Where am I starting from: 10 feet (D1), and 400 mR/hr (I1)
Where am I going: 15 feet (D2), our unknown here is the new intensity (I2)
Lets organize them this way:
I1 = 400 mR/hr
D1 = 10 feet
I2 = ?
D2 = 15 feet
Here are our steps; step 1, square the distances: 10ft^2 = 10 x 10 = 100, 15 ft^2 = 225
Step 2, make sure all units are the same, in this case the distances are all in feet, and our intensity is in mR/hr, our answer will be in mR/hr. so all units match.
Step 3, multiply I1 x D1^2, 400 x 100=40000 (remember we squared 10', or 10 x 10 to get 100)
Step 4, Divide that answer by D2^2, 40000/225=177.7 mR/hr (again we squared 15' to get 225)
4 simple steps that get you to 177.7 mR/hr for I2
Lets do a distance problem; I'm going to use the problem we just did, but I'm going to use the answer we just got for I2 and calculate for the distance, this can be used to check your answers on a test.
I1 = 400 mR/hr
D1 = 10 feet
I2 = 177.7 mR/hr
D2 = ?
Step 1, square the distance; D1^2, 10 x 10=100
Step 2, all units the same; I1 is in mR/hr and I2 is in mR/hr, all good.
Step 3, Multiply I1 x D1^2, 400 x 100=40000
Step 4, divide by I2, 40000/177.7 = 225.098
Step 5, Square root the answer, The square root key on the calculator should look like this:
Depending on your calculator you can input the 225 then press the square root key, or press the key then input the 225, if you get it wrong you may get an error message or an answer that makes no sense. the square root of 225 = 15. We have verified our answer.
One more example which may actually be more useful and will probably be on an IRRSP exam. You have 50 curies of IR-192 unshielded, where will the "Radiation Area" boundary be set per the NRC?
Let's get the known values out of this word problem.
I1 = the emissivity of the source x the activity of the source = Intensity
Ir-192 has an emissivity of 5.2 R/hr/ci @ 1 foot (per the NRC). The activity = 50 Ci
5.2 R/hr/Ci x 50 Ci = an intensity of 260 R/hr @ 1 ft, which is 260,000 mR/hr,
(260 R/hr x 1000 mR/R=260000 mR/hr, there are 1000 mR in 1 R, so 260 x
1000 gives you the mR/hr.)
D1 = 1 foot, all we were given was the emissivity at 1 foot, and we had to calculate the
actual intensity from that
I2 = 5 mR/hr, per the NRC the "radiation area" must be posted and is the area where a
person could receive greater than 5 mR in any one hour. Because we have no other
information we calculate for 5 mR/hr.
D2 = ?, this is our unknown. This will be a distance calculation.
Using our distance formula we write this:
Don't be intimidated, let's follow the steps.
Step 1; Square the distances: 1 squared is 1 x 1=1,
Step 2; all units the same, our answer should be in mR/hr, so we use the mR/hr value of our
starting intensity. We used 1 foot, so our answer will be in feet.
Step 3; multiply I1 x D1^2: we get 260000 x 1 = 260000, any number multiplied by 1 is the
number.
Step 4; divide by 5, 260000/5=52000
Step 5; square root of 52000 = 228.035 feet
There were some moving parts in that problem, things we had to know in order to figure out the right values to use in the formula.
I've been using English standard units or American units for this. The math works the same using international units. Emissivity is 0.48 R/hr/Ci @ 1 meter. If you are using Sv or mSv it will work the same, make sure your units match and then do the math, meters to meters, and mSv to mSv.
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