# Physics of Geiger-Muller Counter

A Geiger-Muller tube is part of a Geiger counter, a device used to detect ionizing particles (radioactive products). It is a cylindrical capacitor with the outer metal cylinder at zero potential and the central wire at about 500 V. (a) Calculate the capacitance if its length is 15 cm, the outer radius is 2 cm, and central wire radius is 0.2 mm. When an ionizing particle enters the device, it creates free electrons, and ions, the gas breaks down, and the capacitor discharges. (b) How much energy is needed to recharge the Geiger-Muller tube? (c) Find the magnitude of the electric field at the surface of the wire. (d) Geiger-tubes are filled with an inert gas, such as argon. The dielectric strength of argon is 20% that of air, or about 1.2 MV/m. Will the electric field found in part (c) cause a continuous breakdown?

The capacitance of the cylindrical capacitor (per unit length L) is

$C/L=(2\pi\epsilon)/[ln⁡(r2/r1)]=(2\pi*8.85*10^{-12})/[ln⁡(0.02/0.0002)]=12.07 pF/m$

$C_{tot}=12.07*0.15=1.81 pF$

b) Energy is

$W=(CU^2)/2=(1.81*10^{-12}*500^2)/2=2.2625*10^{-7} J$

c) the charge on the capacitor is

$Q=CU=1.81*10^{-12}*500=9.05*10^{-10} C=0.905 nC$

Take a cylindrical closed surface having the radius R=r1 (wire radius) with the central wire inside it. Apply Gauss law (on cylinder lateral surface, because the electric flux sum on the 2 cylinder bases is zero):

$E(r_1)*S(r_1)=Q_{inside}/\epsilon$ with $S(r1)=2\pi r_1*L$

$E(r_1)=Q_{inside}/(2\pi \epsilon r_1*L)=(9.05*10^{-10})/(2π*8.85*10^{-12}*0.0002*0.15)=$

$=5.425*10^5 (V/m)=0.54 (MV/m)$

d) The field necessary for electric breakdown in Argon is 1.2 MV/m. This value is lower than the field at the inner armature of the capacitor, so that a continuous breakdown will not occur.