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1. If C is count rate, then from a beta source with a very short half life I can write:

C=Coe^-ut, where u is decay constant and t is seconds

Rearranging; lnC = -ut + lnCo and plotting lnC on y axis and t on x axis

Now if I were to repeat the experiment with detector at double the distance how would the graph now look?

I'm assuming only the count rate will decrease - So y intercept moves down however gradient remains constant. Is this correct? And will the graph remain linear?
2. Also I know inverse square law doesn't apply since beta radiation is absorbed by air molecules. Any thoughts?
3. The decay constant only depends upon the radioactive source and isn't affected by where you put the detector - this is why the half life of a source is the same even if you put the detector in a different place. What is affected though is the initial rate, so I think you're right.
4. (Original post by Freerider101)
Also I know inverse square law doesn't apply since beta radiation is absorbed by air molecules. Any thoughts?
Strictly, inverse square doesn't apply to beta, so you are right. However, over smallish changes of distance typical of practical contamination monitoring or dose rate measurement, it isn't that far off.

Phil is correct about the effect of changing distance on the count rates- if you change the distance all your observed count rates will change proportionately

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