- a simple geiger counter is constructed out of a setup similar to what we just used in problem 4. a thin wire with a linear charge density ? is held in the center of a cylindrical tube, which is filled with an inert gas. when a high speed particle coming from a radioactive decay, cosmic ray, etc. enters the gas-filled tube, collisions with the gas knock electrons off of their atoms. the free electrons then accelerate towards the positively charged wire, and collide with other gas atoms causing them to lose electrons as well. this leads to a cascade of electrons accelerating towards the wire, which will be turned into a measurable current upon reaching the wire. for this problem, the tube
5. A simple Geiger counter is constructed out of a setup similar to what we just used in problem 4. A thin wire with a linear charge density \lambda is held in the center of a cylindrical tube, which is filled with an inert gas. When a high-speed particle coming from a radioactive decay, cosmic ray, etc., enters the gas-filled tube, collisions with the gas knock electrons off of their atoms. The free electrons then accelerate towards the positively charged wire and collide with other gas atoms, causing them to lose electrons as well. This leads to a cascade of electrons accelerating towards the wire, which will be turned into a measurable current upon reaching the wire. For this problem, the tube…
Answer:
To complete the analysis of the Geiger counter setup described, we need to address several key physical principles and calculations. Here’s a detailed breakdown:
1. Electric Field Inside the Cylindrical Tube
The electric field inside the cylindrical tube due to the charged wire can be derived using Gauss’s Law. For a cylindrical geometry with a linear charge density \lambda:
Gauss’s Law:
\oint \mathbf{E} \cdot d\mathbf{A} = \frac{Q_{\text{enc}}}{\epsilon_0}
For a cylindrical Gaussian surface of radius r and length L centered on the wire, the enclosed charge Q_{\text{enc}} is:
Q_{\text{enc}} = \lambda L
The electric field \mathbf{E} is radially symmetric and constant over the surface, so:
E \cdot (2\pi r L) = \frac{\lambda L}{\epsilon_0}
Solving for E:
E(r) = \frac{\lambda}{2\pi \epsilon_0 r}
2. Motion of Electrons in the Electric Field
The free electrons, once knocked off the gas atoms, will experience a force due to the electric field:
F = eE = e \cdot \frac{\lambda}{2\pi \epsilon_0 r}
Where e is the elementary charge.
The acceleration of the electrons is given by Newton’s second law:
a = \frac{F}{m_e} = \frac{e \lambda}{2\pi \epsilon_0 m_e r}
Where m_e is the mass of an electron.
3. Cascade Effect
As the electrons accelerate towards the wire, they ionize additional gas atoms, leading to a cascade or avalanche effect. This avalanche results in a large number of electrons reaching the wire, creating a measurable current pulse.
4. Measurable Current
The current I generated can be related to the number of electrons n_e reaching the wire per unit time t:
I = \frac{n_e e}{t}
The exact calculation of n_e depends on the initial ionization event and the multiplication factor due to the avalanche process.
5. Practical Considerations
- Tube Dimensions: The radius and length of the tube affect the electric field distribution and the sensitivity of the counter.
- Gas Pressure and Type: The type of inert gas (e.g., Argon) and its pressure influence the ionization energy and the efficiency of the electron cascade.
- Voltage Applied: The potential difference between the wire and the tube wall determines the strength of the electric field and the rate of electron acceleration.
Conclusion
A Geiger counter operates by detecting the ionization produced by high-energy particles in a gas-filled tube with a central charged wire. The electric field accelerates the freed electrons, causing further ionization and resulting in a detectable current pulse. This setup is crucial for measuring radiation levels in various applications, from scientific research to safety monitoring.
If you have specific parameters or additional details for the problem, such as tube dimensions, gas type, or applied voltage, we can further refine these calculations.