Question about a geiger counter's resolving time

[schattenjaeger]

Given that R=r/(1-rT) where R is the true counting rate and r is the observe counting rate

show that T=(R1+R2-R3)/(2R1*R2) where T is the resolving time

I've tried all kinds of plugging in and fiddling and what have you, but I can't quite figure it out. I understand in this equation R3 is the observed time of R1 + R2 together under the counter

The closest I got was solving that first equation for T=1/r-1/R and using R3 for r and R1+R2 for R, and doing that the numerator comes out correct but the denominator is R3(R1+R2)

 

[Jhero]

instead of 2R1*R2. Help!

First of all, great job on trying different approaches to solve this problem! It's always important to explore different methods and not give up easily.

Now, let's break down the first equation, R=r/(1-rT). This equation represents the relationship between the true counting rate (R) and the observed counting rate (r). The true counting rate is the actual number of counts that would be recorded if there were no errors or limitations in the observation process. The observed counting rate, on the other hand, takes into account any errors or limitations, such as the resolving time (T).

To solve for T, we need to isolate it on one side of the equation. We can do this by first multiplying both sides by (1-rT).

R(1-rT) = r

Next, we can distribute the R to the terms inside the parentheses.

R - rRT = r

Then, we can move all the terms with T to one side of the equation.

R - r = rRT

Now, we can divide both sides by rR to isolate T.

(R - r)/(rR) = T

Finally, we can substitute R1+R2 for r and R3 for R to get the desired equation.

T=(R1+R2-R3)/(2R1*R2)

We can also check if this equation is correct by plugging in different values for R1, R2, and R3 and seeing if it satisfies the original equation, R=r/(1-rT).

For example, let's say R1=10, R2=20, and R3=5. Plugging these values into the original equation, we get:

R=r/(1-rT)
(10+20-5) = (10+20-5)/(1-(10+20)T)
25 = 25/(1-30T)

Multiplying both sides by (1-30T), we get:

25-750T = 25

Subtracting 25 from both sides, we get:

-750T = 0

Dividing both sides by -750, we get:

T = 0

Now, plugging these values into the derived equation, we get:

T=(R1+R2-R3)/(2R1*R2)
(10+20-5)/(2*10*20)
=