# Geiger Detector - What happens to the counting rate when we increase the distance?

In an experiment with a Geiger detector 164 counts/s are recorded when a point source is placed 8.3 cm from the detector. Knowing that the background count rate is 20 counts/s determine the rate of counts recorded by the Geiger when the source is shifted to a distance of 20.5 cm.

So far what I know is: The total ratio is the difference between the ratio with the radioactive source and the ratio of background counts (without the radioactive source):

$total \,ratio = r = g - b$

where g - ratio with the radioactive source (unit:counts/s) and b - ratio of background counts (unit:counts/s)

The uncertainty associated with the total counting rate is given by: $$\sigma_r=\sqrt{\sigma_g^2+\sigma_b^2}=\sqrt{\frac{G}{t_G^2}+\frac{B}{t_B^2}}=\sqrt{\frac{g}{t_G}+\frac{b}{t_B}}$$

where G - number of counts of radioactive source (unit:counts); B - number of counts of background (without the radioactive source) (unit:counts); $t_G$ is the time of counts of radioactive source; $t_B$ is the time of counts of background.

What is the mathematical expression that allows me to relate the distance to the counts ratios?

• As the Geiger tube moves further from the source it detects a smaller fraction of the particles emitted by the source. It is an example of the inverse square law. – Farcher Nov 4 '17 at 6:01
• @Farcher Is my resolution completely correct? – Maria Barroso Nov 4 '17 at 13:00

My resolution:

Rate of geiger = 164 counts/s

Rate of background = 20 counts/s

Total rate=Rate of geiger-Rate of background = 164 counts/s - 20 counts/s = 144 counts/s = $\big(\frac{N}{t}\big)_1$

$$\frac{\big(\frac{N}{t}\big)}{A}=\frac{1}{D^2}$$ $$A=\bigg(\frac{N}{t}\bigg)_1\times D_1^2=\bigg(\frac{N}{t}\bigg)_2\times D_2^2$$

$$\bigg(\frac{N}{t}\bigg)_2=\frac{\big(\frac{N}{t}\big)_1\times D_1^2}{D_2^2}=\frac{144 \,counts/s\,\times 8,3^2 \,cm^2}{(20,5)^2\,cm^2}=23,60537775 \,counts/s$$

As in the problem ask the rate of the Geiger counter, then to the result obtained we have to add the background rate, so the final result is given by:

rate of Geiger = 23,60537775 counts/s + 20 counts/s = 43,60537775 counts/s