Problem 75 - Fundamentals of Physics

Question

Figure \(23-61\) shows a Geiger counter, a device used to detect ionizing radiation, which causes ionization of atoms. A thin, positively charged central wire is surrounded by a concentric, circular, conducting cylindrical shell with an equal negative charge, creating a strong radial electric field. The shell contains a low-pressure inert gas. A particle of radiation entering the device through the shell wall ionizes a few of the gas atoms. The resulting free electrons (e) are drawn to the positive wire. However, the electric field is so intense that, between collisions with gas atoms, the free electrons gain energy sufficient to ionize these atoms also. More free electrons are thereby created, and the process is repeated until the electrons reach the wire. The resulting "avalanche" of electrons is collected by the wire, generating a signal that is used to record the passage of the original particle of radiation. Suppose that the radius of the central wire is \(25 \mu \mathrm{m},\) the inner radius of the shell \(1.4 \mathrm{~cm},\) and the length of the shell \(16 \mathrm{~cm} .\) If the electric field at the shell's inner wall is \(2.9 \times 10^{4} \mathrm{~N} / \mathrm{C},\) what is the total positive charge on the central wire?

Step-by-Step Solution

Answer

The total positive charge on the wire is approximately \(2.266 \times 10^{-9} \, \mathrm{C}\).

Step 1: Understand Gauss's Law

To solve this problem, we can use Gauss's Law which relates the electric field to the charge enclosed. Gauss's Law is given by the formula: \[ \Phi_E = \frac{Q}{\varepsilon_0} \] where \( \Phi_E \) is the electric flux through a closed surface, \( Q \) is the charge enclosed by that surface, and \( \varepsilon_0 \) is the vacuum permittivity \( (8.85 \times 10^{-12} \, \mathrm{C}^2/\mathrm{N} \cdot \mathrm{m}^2) \).

Step 2: Set Up for the Cylindrical Symmetry

Since the Geiger counter has cylindrical symmetry, we'll consider a cylindrical Gaussian surface co-axial with the wire, with radius \( r = 1.4\, \mathrm{cm} \) (the inner shell radius), and length \( L = 16\, \mathrm{cm} = 0.16\, \mathrm{m} \).

Step 3: Calculate Electric Flux

The electric field is radially outward and constant over our Gaussian surface, so the electric flux \( \Phi_E \) is given by: \[ \Phi_E = E \cdot 2\pi r L \] where \( E = 2.9 \times 10^4 \, \mathrm{N/C} \). Thus, \( \Phi_E = 2.9 \times 10^4 \, \mathrm{N/C} \times 2\pi \times 0.014\, \mathrm{m} \times 0.16\, \mathrm{m} \).

Step 4: Solve for \( Q \) Using Gauss's Law

Substitute the expression for electric flux into Gauss's Law and solve for \( Q \): \[ 2.9 \times 10^4 \times 2\pi \times 0.014 \times 0.16 = \frac{Q}{\varepsilon_0} \]\[ Q = (2.9 \times 10^4 \mathrm{N/C} \times 2\pi \times 0.014 \mathrm{m} \times 0.16 \mathrm{m}) \times 8.85 \times 10^{-12} \, \mathrm{C}^2/\mathrm{N} \cdot \mathrm{m}^2 \].

Step 5: Calculate \( Q \)

Carrying out the multiplication and simplification: \[ Q = (2.9 \times 10^4) \times (0.002816 \times \pi) \times 8.85 \times 10^{-12} \]\[ Q = (815.344) \times 8.85 \times 10^{-12} \times \pi \]\[ Q \approx 2.266 \times 10^{-9} \, \mathrm{C} \].

Key Concepts

Ionizing Radiation

Ionizing radiation refers to types of radiation that have enough energy to remove tightly bound electrons from atoms, thus ionizing them. This can include radiation from sources such as alpha particles, beta particles, gamma rays, and X-rays. When ionizing radiation enters a material, like the inert gas within a Geiger counter, it interacts with the atoms, causing them to lose electrons.
These free electrons then move towards a charged electrode due to the attractive electric field. In a Geiger counter, the ionizing radiation initiates a cascading effect, starting with the primary ionization that frees electrons. These electrons then gain energy from the electric field, collide with other gas atoms, and create additional ionizations, amplifying the original signal into what is termed an 'electron avalanche'.
This process highlights the critical role of ionizing radiation in detecting radioactive particles by substantially increasing the number of charge carriers that can be collected to generate a measurable signal.

Electric Field

An electric field is a region around a charged object where other charged objects experience a force. It is represented by the symbol \( E \) and measured in newtons per coulomb (N/C). In the context of a Geiger counter, the electric field is created by the positively charged wire and negatively charged cylindrical shell.
These charges generate a radial electric field directed from the wire outwards to the shell. The strength of this electric field impacts how quickly and efficiently electrons move through the gas. A stronger field means electrons accelerate more between collisions, enabling more effective ionization of gas molecules.
Understanding electric fields is crucial in analyzing the operation of a Geiger counter, as it governs the movement of electrons and the resultant detection process. The equation \( E = \frac{F}{q} \) describes the electric field \( E \) as the force \( F \) per unit charge \( q \). Essentially, it dictates how charged particles will move when subjected to an electric field.

Gauss's Law

Gauss's Law is a fundamental principle that connects electric fields to the charges producing them. It states that the total electric flux through a closed surface is proportional to the charge enclosed by that surface. The law can be expressed by the equation: \[ \Phi_E = \frac{Q}{\varepsilon_0} \]where \( \Phi_E \) is the electric flux, \( Q \) is the total enclosed charge, and \( \varepsilon_0 \) is the vacuum permittivity constant.
In a Geiger counter setup, Gauss's Law helps determine the total charge on the central wire by considering the cylindrical symmetry. By using a Gaussian surface that follows the shape of the device, we can easily relate the measurable electric field to the otherwise unknown charge.
The symmetry ensures that the field is consistently radial and simplifies calculations, giving us a practical tool to solve for charge using measured electrical parameters.

Electric Flux

Electric flux quantifies the flow of electric field lines through a given surface. It is a useful concept for understanding how electric fields interact with surfaces and can be mathematically described as:\[ \Phi_E = E \cdot A \cdot \cos(\theta) \]where \( E \) is the electric field strength, \( A \) is the area through which the field lines pass, and \( \theta \) is the angle between the field lines and the normal to the surface.
In the specific case of the Geiger counter, the electric flux across the cylindrical shell allows us to infer what's happening inside the detector without directly accessing the charges. Because of the radial nature of the field and the symmetry, \( \cos(\theta) = 1 \) for our surfaces, and flux simplifies to \( E \cdot A \).
This simplification leads to straightforward calculations where the flux through a cylindrical surface depends only on the cylindrical characteristics (radius and length) and the electric field present, vital for linking the external field measurements with the internal charge distribution as determined by Gauss's Law.