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.” 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) .

Verified step by step guidance

1. Step 1: To find the electric field between the wire and the cylinder, we use Gauss's law. Gauss's law states that the electric flux through a closed surface is proportional to the charge enclosed by that surface. Mathematically, it is expressed as: ∮E·dA = Q_enclosed / ∊₀. Here, E is the electric field, dA is the infinitesimal area element, Q_enclosed is the charge enclosed by the Gaussian surface, and ∊₀ is the permittivity of free space.
2. Step 2: Choose a cylindrical Gaussian surface of radius r (where Rₐ < r < Rᵦ) and length L, coaxial with the central wire. The symmetry of the problem implies that the electric field E is radial and constant over the curved surface of the cylinder. The total flux through the Gaussian surface is then given by: ∮E·dA = E(2πrL), where 2πrL is the curved surface area of the cylinder.
3. Step 3: The charge enclosed by the Gaussian surface is the charge on the central wire within the length L. Since the charge per unit length on the wire is λ, the total charge enclosed is Q_enclosed = λL. Substituting this into Gauss's law, we get: E(2πrL) = λL / ∊₀. Simplify to find the electric field: E = λ / (2π∊₀r). This is the expression for the electric field between the wire and the cylinder.
4. Step 4: To find the potential difference between Rₐ and Rᵦ, recall that the potential difference is related to the electric field by the integral: Vₐ - Vᵦ = -∫(E·dr) from Rₐ to Rᵦ. Substituting the expression for E, we have: Vₐ - Vᵦ = -∫(λ / (2π∊₀r)) dr from Rₐ to Rᵦ.
5. Step 5: Evaluate the integral: Vₐ - Vᵦ = -(λ / (2π∊₀)) ∫(1/r) dr from Rₐ to Rᵦ. The integral of 1/r is ln(r), so: Vₐ - Vᵦ = -(λ / (2π∊₀)) [ln(Rᵦ) - ln(Rₐ)]. Simplify using the logarithmic property ln(a) - ln(b) = ln(a/b): Vₐ - Vᵦ = (λ / (2π∊₀)) ln(Rᵦ / Rₐ). This confirms the given expression for the potential difference.

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.

Gauss's Law

Gauss's Law relates the electric flux through a closed surface to the charge enclosed by that surface. It is mathematically expressed as the integral of the electric field over a closed surface being equal to the enclosed charge divided by the permittivity of free space. This law is essential for deriving the electric field between the wire and the cylinder in the Geiger counter, as it allows for the calculation of the electric field due to the linear charge distribution along the wire.

Potential Difference

The potential difference, or voltage, between two points in an electric field is defined as the work done per unit charge in moving a charge from one point to another. It is a measure of the energy available to move charges within the field. In the Geiger counter, calculating the potential difference between the inner wire and the outer cylinder is key to understanding how the electric field influences the behavior of the ionized gas and the resulting signal generation.