Problem 64 - University Physics with Modern Physics

Question

A Geiger counter detects radiation such as alpha particles by using the fact that the radiation ionizes the air along its path. A thin wire lies on the axis of a hollow metal cylinder and is insulated from it (Fig. P23.64). A large potential difference is established between the wire and the outer cylinder, with the wire at higher potential; this sets up a strong electric field directed radially out- ward. When ionizing radiation enters the device, it ionizes a few air molecules. The free electrons produced are accelerated by the electric field toward the wire and, on the way there, ionize many more air molecules. Thus a current pulse is produced that can be detected by appropriate electronic circuitry and converted to an audible "click." Suppose the radius of the central wire is 145\(\mu \mathrm{m}\) and the radius of the hollow cylinder is 1.80 \(\mathrm{cm} .\) What potential difference between the wire and the cylinder produces an electric field of \(2.00 \times 10^{4} \mathrm{V} / \mathrm{m}\) at a distance of 1.20 \(\mathrm{cm}\) from the axis of the wire? (The wire and cylinder are both very long in comparison to their radii, so the results of Problem 23.63 apply.)

Step-by-Step Solution

Answer

The required potential difference is approximately 2930 V.

Step 1: Understand the Problem

We need to find the potential difference required between a long, thin wire and a hollow cylindrical shell to produce a specified electric field at a certain distance from the wire. Given the potential difference, we can use the formula for the electric field near a long charged wire.

Step 2: Setup Formula for Electric Field

The electric field at a distance \( r \) from a long, straight, charged wire with linear charge density \( \lambda \), inside a cylindrical shell, is given by:\[ E = \frac{\lambda}{2\pi\varepsilon_0 r} \]Note that \( \varepsilon_0 \) is the permittivity of free space.

Step 3: Use Relationship Between Potential Difference and Electric Field

For a cylindrical geometry, the potential difference \( V \) between two points at distances \( r_1 \) and \( r_2 \) from the wire is:\[ V = \int_{r_1}^{r_2} E \, dr = \frac{\lambda}{2\pi\varepsilon_0} \int_{r_1}^{r_2} \frac{1}{r} \, dr \]This integral evaluates to:\[ V = \frac{\lambda}{2\pi\varepsilon_0} \ln\left(\frac{r_2}{r_1}\right) \]

Step 4: Calculate Linear Charge Density \( \lambda \) from Given Electric Field

We know the electric field at \( r = 1.20 \text{ cm} \) is \( 2.00 \times 10^{4} \text{ V/m} \). Rearrange the formula for the electric field to solve for \( \lambda \):\[ \lambda = 2\pi\varepsilon_0 rE \]\[ \lambda = 2\pi (8.85 \times 10^{-12} \text{ C}^2/\text{N} \cdot \text{m}^2)(0.012 \text{ m})(2.00 \times 10^{4} \text{ V/m}) \]Calculate \( \lambda \).

Step 5: Calculate Potential Difference \( V \)

Using the formula for \( V \) from Step 3 and the calculated \( \lambda \):\[ V = \frac{\lambda}{2\pi\varepsilon_0} \ln\left(\frac{r_2}{r_1}\right) \]Substitute \( r_1 = 145\mu m = 145 \times 10^{-6} \text{ m} \) and \( r_2 = 1.80 \text{ cm} = 0.018 \text{ m} \). Use the value of \( \lambda \) from Step 4.

Key Concepts

Electric Field

The electric field is a crucial concept when discussing how a Geiger counter operates. In simple terms, an electric field can be thought of as a force field that surrounds charged particles. It exerts forces on other charges within the field. This field is responsible for accelerating the electrons in a Geiger counter when ionizing radiation causes ionization of air molecules. When a high potential difference is applied between the wire and the cylindrical casing in a Geiger counter, a strong electric field is established. This field is radial, meaning it points outward from the wire towards the cylinder. It is this electric field strength that causes free electrons, produced when radiation ionizes air molecules, to accelerate towards the central wire. The required electric field for a Geiger counter like the one in question is notable at a distance of 1.20 cm from the wire. Understanding the electric field is essential for grasping how the device powers the detection of radiation.

Ionizing Radiation

Ionizing radiation is the type of radiation detected by a Geiger counter. It includes particles, such as alpha and beta particles, and various rays, like gamma rays, that have enough energy to remove electrons from atoms or molecules. This ionization process is what generates free electrons in the air surrounding the wire within the Geiger counter. Once these electrons are released, they are accelerated by the electric field present inside the counter. This acceleration allows them to collide with other air molecules, which generates more ions and free electrons. This cascading effect ultimately produces a measurable electric current or pulse. The detection of this pulse by electronic circuitry allows the Geiger counter to produce the familiar clicking sound, indicating the presence of ionizing radiation. Thus, understanding ionizing radiation is critical for interpreting how such devices effectively detect and measure radiation levels.

Cylindrical Geometry

The Geiger counter features a specific cylindrical geometry that is vital to its function. It consists of a thin wire positioned along the central axis of a hollow cylindrical metal shell. This arrangement provides a symmetrical field distribution, ensuring that the electric field is radial in nature. The wire has a very small radius compared to the outer cylinder, creating a sizable difference in geometry that enhances the counter's sensitivity to the electric field. Because this device is commonly elongated, the field and potential calculations assume a "long wire" approximation, simplifying the mathematical analyses. This cylindrical layout allows the effective capture and acceleration of ionized particles generated by ionizing radiation. The geometry thus ensures that any free electrons move efficiently toward the center wire, maximizing the signal generated from a radiation event.

Potential Difference

The potential difference between the wire and the cylindrical shell is what establishes the electric field necessary for the Geiger counter to function. This potential difference is achieved by applying a voltage, where the wire is kept at a higher voltage than the outer shell. Calculating this potential difference involves understanding the relationship between electric field strength and potential difference in a cylindrical configuration. Using the formula for potential difference in cylindrical coordinates, you integrate the electric field over the radial distance between the wire and the cylinder. This exercise particularly requires knowledge of the logarithmic relationship between the charge density, radii, and the potential difference. In the provided problem, the challenge is to compute the specific potential difference needed to generate an electric field of a given strength at a set radius, ensuring the effective operation of the Geiger counter. Hence, potential difference is fundamental in controlling the operation and sensitivity of the device.