# Electric Potential, Capacitance and charge of a Geiger Counter

• inevitable08
In summary, the problem at hand is to find the potential difference, capacitance, and charge on the anode of a Geiger Counter. The Geiger Counter contains low-pressure neon and works by ionizing neon atoms with the collision of alpha particles or electrons. This creates a cascade of free electrons that travel towards the anode, which has a high potential compared to the cathode. We are given the diameter of the cathode and anode, length of the Geiger Counter, radius of neon atoms, energy required to ionize neon, and pressure and density of the gas in the Geiger Counter. Using Gauss Law and the Mean-Free Path equation, we can find the electric field inside the Geiger Counter. By using the energy
inevitable08

## Homework Statement

Find the potential difference, capacitance of the Geiger Counter and the charge on the anode.
So, we have a Geiger Counter with low-pressurized neon inside. What we want is to have an alpha particle or electron to travel into the Geiger Counter with enough energy to collide with a neon atom and ionize it. then we will cause that free electron to accelerate through the potential difference in the G.C. to collide with another neon atom and again ionize that atom. Basically we are going to have a cascade of free electrons traveling towards the anode. The inner wire, anode, is at a high potential while the outer cylinder (cathode) is at a lower potential.

Variables we know.
diameter of cathode: 15mm = 15 x10^-3 m
diameter of anode: 1mm = 1x10^-3 m
length of Geiger Counter: 40mm = 40x10^-3 m
radius of neon atom = 154 pm = 154 x10^-10m
energy to 1st ionize neon = 2080.7 kJ/mol
pressure in G.C. = 75 torr which equals 8.1x10^-3 kg/m^3 density

## Homework Equations

Gauss Law : E.A = q/εo
-∫E dr = ΔV
Mean-Free Path = 1/(pi*r^2*n_v)

## The Attempt at a Solution

Ok so we need to find how much joules it takes to ionize neon.
2080.7 kJ/mol = 1.04x10^-2 eV
1 eV = 1.602x10^-19 J
2080.7 kJ/mol * 1.04x10^-2 eV * 1.602x10^-19 J = 3.466 x 10^-18 J

next we find the atom/volume density of neon in the G.C.

8.1x10^-3 kg/m^3 * 1000g/kg * 1 mole/20.1797g * 6.02x10^23 atoms/mole = 2.416x10^23 atoms/m^3

this will be out number density n_v.

next we will find the distance the electron travels before it becomes in contact with a neon atom:

Mean-Free PAth = 1/(pi*r^2*n_v) r is the radius of the neon atom

1/(pi* (154 x10^-10m)^2 * 2.416x10^23 atoms/m^3) = 5.55x10^-9 m
so an electron will on average travel that distance before it becomes in contact with another neon atom.

Now let's find the electric field inside the G.C.
Well the cathode is a thin cylindrical shell so it produces no electric field inside the G.C. The wire produces a E though...
E.A = q/εo
Surface area of the wire is the derivative of the volume of a cylinder... 2*pi*r*L

E (2*pi*r*L) = q/εo
E = Q/(2*pi*r*L*εo)

now let's find the potential from the anode to the cathode (r --> R)

-∫Q/(2*pi*r*L*εo) dr
-Q/(2*pi*L*εo)∫ 1/r dr
-Q/(2*pi*L*εo)( ln(R) - ln(r) )

-Q/(2*pi*L*εo)*ln(R/r)

ok so we pretty much have everything. Distance, the potential equation, electric field equation, energy required to ionize now how do we put it all together? This is where I am lost. E is not uniform so I'm not sure if I can solve for the electric force then solve for acceleration and use Vf^2 = Vi^2 + 2*a*Δx or use the energy to find the velocity right before the electron collides with a neon atom. Then what do I do with the velocity. I am pretty sure in some way I am suppose to find the acceleration or force and then i can solve for Q ( the charge on the anode) then I can find everything else...right?

Ok i think i got it. We want to create a cascade of electrons from anywhere in the tube. This means i can find the acceleration of the electron that's needed to accelerate it to the velocity needed to ionize a neon atom. And the least likely place of that to happen is at the cathode (furthest from anode) so we have to make sure the acceleration atleast agrees with the total charge (Q) and electric field at hat point. So I find the acceleration needed then use F=ma and find the Force acted on it by the anode. Then using E = Fe/q I can find the total charge on the anode. Then plug the Q into my potential equation, find the potential and now I can find the capacitance which is C = q/V.

So this is what i did.

E = 1/2 m v^2
3.466x10^-18J = 1/2* 9.11x10^-31kg * v^2
v = 2.76x10^6 m/s
Vf^2 = Vi^2 + 2*a*x (assume initial v = 0 m/s)
2.76x10^6 ^2 = 2 *a * 5.55x10^-9m
a = 6.857x10^16 m/s^2

E = F/q
E = ma/q E = Q/(2*pi*R*L*εo) R = radius of cathode

Q = (2*pi*R*L*εo)*(ma/q)
Q = (2*pi*7.5x10^-3m*40x10^-3 * εo) * (9.11x10^-31kg*6.857x10^16 m/s^2)/1.602x10^-19C
Q = +6.506x10^-9C

Now plug Q into the ΔV integration equation from r --> R
you get 7919.68V...

C = Q/V
C = 6.506x10^-9C/7919.68V = 8.21x10^-13 C.

Finished? I hope so, I think that's right... lol

Last edited:

## 1. What is electric potential?

Electric potential is a measure of the electric potential energy per unit charge at a point in an electric field. It is typically measured in volts (V).

## 2. How is capacitance calculated?

Capacitance is calculated by dividing the charge stored on one of the conductors by the potential difference between the two conductors. It is measured in farads (F).

## 3. What is the charge of a Geiger Counter?

The charge of a Geiger Counter refers to the amount of electric charge that is produced by the ionization of gas molecules within the counter. This charge is typically measured in coulombs (C).

## 4. How does a Geiger Counter detect radiation?

A Geiger Counter detects radiation by using a gas-filled tube with a high voltage applied across it. When radiation passes through the tube, it ionizes the gas molecules, creating a pulse of electric current that can be measured.

## 5. Can a Geiger Counter measure different types of radiation?

Yes, a Geiger Counter can measure different types of radiation, including alpha, beta, and gamma particles. However, the sensitivity and detection capabilities may vary depending on the type of radiation.

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