A Geiger counter is used to detect charged particles emitted by radioactive nuclei. It consists of a thin, positively charged central wire of radius Rₐ surrounded by a concentric conducting cylinder of radius Rᵦ with an equal negative charge (Fig. 23–57). The charge per unit length on the inner wire is λ (units C/m) . The interior space between wire and cylinder is filled with low-pressure inert gas. Charged particles ionize some of these gas atoms; the resulting free electrons are attracted toward the positive central wire. If the radial electric field is strong enough, the freed electrons gain enough energy to ionize other atoms, causing an “avalanche” of electrons to strike the central wire, generating an electric “signal.”


(a) Find the expression for the electric field between the wire and the cylinder, and (b) show that the potential difference between Rₐ and Rᵦ is


Vₐ - Vᵦ = ( λ / 2π∊₀ ) ln( Rᵦ/Rₐ) .
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Verified step by step guidance

1. Identify the geometry of the problem and the relevant parameters. The system consists of a coaxial cable setup with a central wire of radius Rₐ and an outer cylindrical shell of radius Rᵦ. The charge per unit length on the wire is λ.
2. Use Gauss's Law to find the electric field between the wire and the cylinder. Consider a cylindrical Gaussian surface of radius r, where Rₐ < r < Rᵦ. The electric field E is radial due to symmetry, and the Gaussian surface will enclose the charge from the central wire.
3. Apply Gauss's Law, \( \oint \vec{E} \cdot d\vec{A} = \frac{Q_{\text{enc}}}{\epsilon_0} \), where \( Q_{\text{enc}} \) is the charge enclosed by the Gaussian surface. Since the charge per unit length is λ, the enclosed charge is λ times the height of the cylinder. The area of the Gaussian surface is \( 2\pi r L \), leading to \( E(2\pi r L) = \frac{\lambda L}{\epsilon_0} \).
4. Solve for E, the electric field, which simplifies to \( E = \frac{\lambda}{2\pi \epsilon_0 r} \). This expression shows that the electric field between the wire and the cylinder varies inversely with the radius r.
5. To find the potential difference Vₐ - Vᵦ, integrate the electric field from Rₐ to Rᵦ. The potential difference is given by \( V = -\int_{Rₐ}^{Rᵦ} E \, dr \), substituting the expression for E and integrating leads to \( Vₐ - Vᵦ = \frac{\lambda}{2\pi \epsilon_0} \ln\left(\frac{Rᵦ}{Rₐ}\right) \).

Key Concepts

Electric Field

The electric field is a vector field that represents the force exerted by an electric charge on other charges in its vicinity. It is defined as the force per unit charge and is directed away from positive charges and toward negative charges. In the context of the Geiger counter, the electric field between the charged wire and the cylinder is crucial for understanding how free electrons are accelerated towards the wire, leading to ionization and the generation of an electric signal.

Potential Difference

Potential difference, or voltage, is the work done per unit charge in moving a charge between two points in an electric field. It is a measure of the energy difference that drives the movement of charges. In the Geiger counter, the potential difference between the inner wire and the outer cylinder is essential for determining how much energy the freed electrons gain, which influences the ionization process and the resulting electric signal.

Ionization and Avalanche Effect

Ionization is the process by which neutral atoms are converted into charged ions through the removal of electrons. In the Geiger counter, when charged particles collide with gas atoms, they can ionize them, creating free electrons. The avalanche effect occurs when these free electrons gain sufficient energy from the electric field to ionize additional gas atoms, leading to a rapid multiplication of electrons that generates a detectable electric signal, crucial for the counter's operation.