A simple Geiger counter is constructed out of a setup similar to what we just
used in problem 5. 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
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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 has a radius of 1cm and is
10cm long; the wire has a radius of 10_m and _ = 10��8C=m.
(a) What is the force on a free electron located at the outer wall of the tube?
(b) What is the force on an electron immediately next to the thin wire?
(c) What is the work done on an electron pulled from the outer wall of the tube
to the wire?
(d) Calculate the electric flux through the walls of the tube perpendicular to the
wire (i.e. the walls the wire is attached to).
(e) Calculate the electric flux through the surface of the tube parallel to the
wire. Would this value change if the Geiger counter were box shaped rather
than cylindrical?
To answer these questions, we need to use the concepts of electric fields, Coulomb's law, and Gauss's law. Let's go through each question step by step.
(a) To find the force on a free electron located at the outer wall of the tube, we need to calculate the electric field at that point. The electric field at a distance r from an infinitely long wire with charge density λ is given by the equation:
E = λ / (2πε₀r)
where ε₀ is the permittivity of free space.
In our case, the charge density λ is given as _ = 10⁻⁸ C/m, and the distance r is equal to the radius of the tube, which is 1 cm (or 0.01 m).
So the force on the electron can be calculated using the equation:
F = qE
where q is the charge of the electron. The value of q is -1.6 x 10⁻¹⁹ C (since the electron has a negative charge).
(b) To find the force on an electron immediately next to the thin wire, we need to calculate the electric field at that point. The electric field of an infinitely long wire at a distance r from its center is given by the equation:
E = λ / (2πε₀r)
In this case, the distance r is equal to the radius of the wire, which is given as 10 μm (or 10⁻⁵ m).
(c) To find the work done on an electron pulled from the outer wall of the tube to the wire, we need to calculate the electric potential difference between those two points. The electric potential difference is given by:
ΔV = V₂ - V₁
where V₁ is the electric potential at the outer wall and V₂ is the electric potential at the wire. The electric potential at a distance r from an infinitely long wire with charge density λ is given by the equation:
V = λ / (2πε₀) * ln(r / r₀)
where r₀ is a reference distance. In this case, we can choose r₀ as the radius of the wire.
(d) To calculate the electric flux through the walls of the tube perpendicular to the wire, we can use Gauss's law. According to Gauss's law, the electric flux through a closed surface is given by the equation:
Φ = Q / ε₀
where Q is the net charge enclosed by the surface.
(e) To calculate the electric flux through the surface of the tube parallel to the wire, we can again use Gauss's law. The value of the electric flux will not change if the Geiger counter were box-shaped rather than cylindrical, as long as the charge distribution inside the box remains the same.
Please note that the specific calculations for each question require some additional information, such as the magnitude of the charge or the charge distribution within the tube, which are not provided in the question.