A detector of radiation called a Geiger tube consists of a...

A detector of radiation called a Geiger tube consists of a closed, hollow, conducting cylinder with a fine wire along its axis. Suppose that the internal diameter of the cylinder is $2.50 \mathrm{cm}$ and that the wire along the axis has a diameter of $0.200 \mathrm{mm} .$ The dielectric strength of the gas between the central wire and the cylinder is $1.20 \times 10^{6} \mathrm{V} / \mathrm{m} .$ Calculate the maximum potential difference that can be applied between the wire and the cylinder before breakdown occurs in the gas.

Key Concepts

Dielectric Strength

Dielectric strength is the maximum electric field that an insulating medium (such as a gas, liquid, or solid) can withstand without undergoing electrical breakdown. It is a fundamental property that determines the limit at which a material can act as an insulator and is critical in designing devices that must operate at high voltages without failure.

Cylindrical Capacitor Geometry

A cylindrical capacitor consists of two coaxial conductors separated by a dielectric. The geometry, including the radii of the inner and outer conductors, significantly influences the electric field distribution between the conductors. This configuration is common in various devices and requires careful calculation to ensure that the applied potential does not exceed the breakdown limit of the intervening medium.

Electric Field in Coaxial Cylinders

The electric field in a coaxial cylindrical system varies inversely with the radial distance from the central conductor. It is strongest at the surface of the inner conductor and diminishes outward toward the outer conductor. Understanding this distribution is essential for predicting where breakdown is most likely to initiate and therefore determining the maximum allowable potential difference.

Integration of the Electric Field to Determine Potential Difference

Calculating the potential difference in a coaxial cylindrical geometry involves integrating the electric field over the radial distance between the conductors. This integration accounts for the non-uniform field distribution and directly links the device's geometry and the dielectric's properties to the maximum voltage that can be safely applied without causing breakdown.

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