Learning about the inverse square law and the direct square law

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Learning about the inverse square law and the direct square law can be quite confusing at first. Once you learn the formulas and dust the cobwebs off of your algebra skills to be able to solve for the variable using the formulas, knowing WHEN to use WHICH formula is sometimes the biggest challenge you will face. In order to determine this, we need to know what kind of information the question is asking you for. Let’s take a close look at each formula:
Inverse Square Law states: “The intensity is inversely proportional to the square of the distance.”

I1  / I2-    (D2)2/        (D1)2
Notice that the value for original intensity (I1) is in the numerator, and the value for the original distance (D1) is in the denominator, thus it is “inversely proportional to the square of the distance.”
Use this formula when the problem asks you to solve for a unit of radiation intensitydose, or exposure. Also, remember that radiation “intensity” is not measured in units of mAs, so if the question is asking you for a mAs value, this is not the formula for you. Units of radiation exposure or radiation dose are required for this formula (R – Roentgen, mR – milliRoentgen, rad, rem, Gy – gray, or Sv – Seivert).
Still confused? This simple tip could save you… you should now already have the fundamental knowledge that radiation intensity will decrease as the distance from its source increases. So look at your distance values: If the distance increases, then I2 should be a smaller number than I1. The opposite is true as well; if the distance decreases, the intensity will be stronger, and I2 will be a larger number than I1. This is important to remember when we discuss the direct square law:
Direct Square Law / Density Maintenance Formula:

mAs1/   mAs 2         =     (D1)2/(D2)2

Two main differences with this formula are: Instead of radiation intensity, we are using mAs values. Also, the original mAs and the original distance are both in the numerator – “direct” vs. “inverse.” We need to be using this formula when the question asks for a mAs value.
*** side note: You know that radiation exposure is directly proportional to mAs. In other words, if I double my mAs value, the radiation exposure value will double. Remember, these two units are distinct and separate, but related to one another.
After performing a few practice problems, you may notice that as the distance increases, the mAs value will increase. This is due to the “direct” relationship. As the distance from the radiation source to the image receptor increases, the mAs required to maintain density (density maintenance formula) will increase. So, if your D1 value is smaller than your D2 value, then your mAs1 value should be smaller than your mAs2 value. Conversely, if your D1 value is larger than your D1 value, then your mAs1 value needs to be larger than your mAs2 value.
What if the values presented in the question provide units of radiation exposure/intensity/dose AND mAs values? Don’t panic… just find out what the question is asking for, and apply what we have discussed.
Example: A radiographic exposure of the chest was taken at a distance of 72″ using 10 mAs and had an exposure of 50 mR. What would the exposure be at a distance of 80″?
The question is asking “What would the exposure be …?” Key word: exposure. This is your key term that determines we are looking for a unit of radiation intensity. First, fill in your variables:
I1 = 50mR
I2 = “x” or unknown
D1 = 72″
D2 = 80″
I trust your ability to solve once the equation is set up properly 😉 Remember that your distance is increasing, so your value for I2 should be smaller with this formula (this is how you can tell if you forgot to invert the distances – it will be a larger value if you forget).
Example: A radiograph of the knee produced 100mR of exposure when 70 kVp and 10 mAs was used at 40″. What new mAs would be required at a distance of 60 to maintain density”?
Key term: “What new mAs…to maintain density?” This one screams, “Density maintenance formula!!!”
Fill in your variables and solve:
mAs1 = 10
mAs2 = x
D1 = 40″
D2 = 60″
Your mAs2 value should be greater than mAs1 because your distance is increasing.

12 thoughts on “Learning about the inverse square law and the direct square law

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