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9. The picture to the right shows a "coaxial capacitor". Basically, the wire of radius a running down the middle serves as the positively charged "plate" and the outer hollow wire of inner radius b serves as the negatively charged "plate". The potential of the inner cylinder with respect to the outer, Vis given by Equation 1 below (note that it isn't the same as a parallel plate capacitor because this has a different geometry), where is the positive linear charge density accumulated on the inner wire (note that - accumulates on the outer wire). The electric field between the two wires (ie., a

9. The picture to the right shows a "coaxial capacitor". Basically, the wire of radius a running down the middle serves as the positively charged "plate" and the outer hollow wire of inner radius b serves as the negatively charged "plate". The potential of the inner cylinder with respect to the outer, Vis given by Equation 1 below (note that it isn't the same as a parallel plate capacitor because this has a different geometry), where is the positive linear charge density accumulated on the inner wire (note that - accumulates on the outer wire). The electric field between the two wires (ie., a

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9. The picture to the right shows a "coaxial capacitor". Basically, the wire of radius a running down the middle serves as the positively charged "plate" and the outer hollow wire of inner radius b serves as the negatively charged "plate". The potential of the inner cylinder with respect to the outer, V is given by Equation 1 below (note that it isn't the same as a parallel plate capacitor because this has a different geometry), where is the positive linear charge density accumulated on the inner wire (note that - accumulates on the outer wire). The electric field between the two wires (i.e., a <r< b) is given by Equation 2. These equations work well when the coaxial cable is much longer than a and b. Equation 1: V ab 2mc In In Equation 2: E(r) = - Inin+ It turns out this basic setup is used in a lot of applications, much like its cousin the parallel plate capacitor. Some examples include high frequency data transmission (i.e., cable TV), the axons of our neurons, electrostatic precipitators (which remove pollutants from smoke in power plants), and the one we're going to analyze here: Geiger counters. 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 (see figure below). 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 outward. When ionizing radiation enters the device, it ionizes a few air molecules. The fee electrons produced are accelerated 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". Counter Free electron Radiation a) Suppose the radius of the central wire is 145 µm and the radius of the hollow cylinder is 1.80 cm. What potential difference between the wire and the cylinder produces an electric field of 2.00×10* V/m at a distance of 1.20 cm from the axis of the central wire? (Assume the Geiger counter is much longer than 2.00 cm). b) Now suppose the radius of the central wire is 127 um and the radius of the hollow cylinder is 2.00 cm. If the tube uses an operating voltage of 850 V, what is the electric field strength at the outer surface of the central wire (r= a) and the inner surface of the hollow cylinder (>= 6)?
Transcribed Image Text:9. The picture to the right shows a "coaxial capacitor". Basically, the wire of radius a running down the middle serves as the positively charged "plate" and the outer hollow wire of inner radius b serves as the negatively charged "plate". The potential of the inner cylinder with respect to the outer, V is given by Equation 1 below (note that it isn't the same as a parallel plate capacitor because this has a different geometry), where is the positive linear charge density accumulated on the inner wire (note that - accumulates on the outer wire). The electric field between the two wires (i.e., a <r< b) is given by Equation 2. These equations work well when the coaxial cable is much longer than a and b. Equation 1: V ab 2mc In In Equation 2: E(r) = - Inin+ It turns out this basic setup is used in a lot of applications, much like its cousin the parallel plate capacitor. Some examples include high frequency data transmission (i.e., cable TV), the axons of our neurons, electrostatic precipitators (which remove pollutants from smoke in power plants), and the one we're going to analyze here: Geiger counters. 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 (see figure below). 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 outward. When ionizing radiation enters the device, it ionizes a few air molecules. The fee electrons produced are accelerated 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". Counter Free electron Radiation a) Suppose the radius of the central wire is 145 µm and the radius of the hollow cylinder is 1.80 cm. What potential difference between the wire and the cylinder produces an electric field of 2.00×10* V/m at a distance of 1.20 cm from the axis of the central wire? (Assume the Geiger counter is much longer than 2.00 cm). b) Now suppose the radius of the central wire is 127 um and the radius of the hollow cylinder is 2.00 cm. If the tube uses an operating voltage of 850 V, what is the electric field strength at the outer surface of the central wire (r= a) and the inner surface of the hollow cylinder (>= 6)?
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