The tube of a Geiger counter consists of a thin straight wire surrounded by a coaxial conducting shell. The diameter of the wire is 0.0025 cm and that of the shell is 2.5 cm. The length of the tube is 10 cm. What is the capacitance of a Geiger-counter tube ?
The problem will be solved under the assumption that the electric field generated is that of an infinitely long line of charge. A schematic side view of the tube is shown in Figure 1. The radius of the wire is rw, the radius of the cylinder is rc, the length of the counter is L, and the charge on the wire is +Q. The electric field in the region between the wire and the cylinder can be calculated using Gauss' law. The electric field in this region will have a radial direction and its magnitude will depend only on the radial distance r. Consider the cylinder with length L and radius r shown in Figure 1. The electric flux [Phi] through the surface of this cylinder is equal to
According to Gauss' law, the flux [Phi] is equal to the enclosed charge divided by [epsilon]0. Therefore
The electric field E(r) can be obtained using eq.(7):
The potential difference between the wire and the cylinder can be obtained by integrating the electric field E(r):
Using eq.(2) the capacitance of the Geiger tube can be calculated:
Substituting the values for rw, rc, and L into eq.(10) we obtain
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