How Can Radioactive Decay and Distance Formulas Be Combined in Physics?

[piisexactly3]

The formula for radioactive decay over time is N = N0eλt . The formula for how many number of atoms counted by the Geiger as distance (x) changes is C = k/x2. How can I merge these formulas to give one that accounts for distance and time? Secondly, how does the second formula work realistically given that C would tend to infinity as x decreases?

 

[tiny-tim]

hi piisexactly3!

The formula for radioactive decay over time is N = N0eλt . The formula for how many number of atoms counted by the Geiger as distance (x) changes is C = k/x2. How can I merge these formulas to give one that accounts for distance and time?

you simply multiply them …

the formula will be proportional to both e^λt and 1/x²

Secondly, how does the second formula work realistically given that C would tend to infinity as x decreases?

the geiger counter has a finite width w, which you assume is at a constant distance x from the source

when 2πx = w, the counter would need to be wrapped all the way round the source (360°)

and when 2πx = w/n the counter would need to be wrapped round it n times!

 

[piisexactly3]

Ok maybe I'm understanding this. You're saying that if the geiger counter was shaped like a sphere encasing the source and detecting every emmission, the k/x^2 formula would not hold true. But because it is a tube and cannot detect everything, the formula works and we don't get an infinite number of atoms detected?

When I times N0e^λt by k/x2, I get N² = N₀e^λtkx^-2 .
So what exactly is the constant k, does it involve use of the actual numbers of atoms there, as I don't see how we can derive how much the distance x is decreasing the detections by without knowing how many atoms there actually are, and I don't see how the k/x² formula uses that information... Unless k is N0e^λt? So how can I incorporate information about the width of the tube to ensure that this formula doesn't tend to infinity as x tends to 0?

I may have gone totally off track so sorry if I am fustrating you!

 

[tiny-tim]

hi piisexactly3!

Ok maybe I'm understanding this. You're saying that if the geiger counter was shaped like a sphere encasing the source and detecting every emmission, the k/x^2 formula would not hold true.

no, I'm saying it would hold true …

if its surface area was exactly enough to fit round the source once at that radius, it would count every emission

if we choose a radius n times smaller, so that the geiger counter's surface area is enough to wrap round the source n² times, it will count every emission n² times (once for every layer) …

so the k/x² formula does work! ​

…So how can I incorporate information about the width of the tube to ensure that this formula doesn't tend to infinity as x tends to 0?

i don't think the geiger counter is intended to be used that close to the source: if it is, the k/x² formula certainly doesn't work (the count doesn't go off to ∞), and you'd need to do some horrible integration