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

 

[HalfLostAndSearching]

I would like to clarify that the first formula, N = N0eλt, is used to calculate the number of radioactive atoms remaining after a certain amount of time has passed. It is not directly related to distance.

To merge these formulas, we can use the concept of activity (A), which represents the rate at which the number of radioactive atoms decreases over time. This can be calculated by taking the derivative of N with respect to time (t):

A = -dN/dt = λN0eλt

We can then substitute this value for N in the second formula, C = k/x2, to get:

A = -dC/dx = k/x2

Rearranging this equation, we get:

dC = -k/x2 dx

Integrating both sides with respect to x, we get:

C = k/x + C0

Where C0 is a constant of integration. This equation takes into account both distance (x) and time (t) as C will change as x and t change.

Regarding the second part of the question, it is important to note that the formula C = k/x2 is a simplified model and does not account for all factors that may affect the readings of a Geiger counter. In reality, there are limitations to how close a Geiger counter can be placed to a radioactive source before it becomes saturated and gives inaccurate readings. Additionally, other factors such as shielding and environmental conditions can also affect the readings. Therefore, while this formula may provide a general understanding, it may not hold true in all situations and should be used with caution.