Coaxial Conductor System with Uniform Space Charge: A static space-charge distribution fills the region between two long coaxial conductors as shown. The center conductor is held at a potential of Vo with respect to the outer conductor, which is at ground potential. The charge density in the region between the conductors is uniform, given by p(r) Po. Find the field and potential inside by a V direct integration of Poisson's equation with appropriate boundary conditions. Once the potential + is found, find the total induced charge on each conductor. At what critical voltage does the charge on the anode go to zero? Note: the r in this problem is interpreted as the radial distance from the z-axis as in cylindrical coordinates. ++ ++. XX + xx xx xx x xxxx xx ++ + +++ +
Coaxial Conductor System with Uniform Space Charge: A static space-charge distribution fills the region between two long coaxial conductors as shown. The center conductor is held at a potential of Vo with respect to the outer conductor, which is at ground potential. The charge density in the region between the conductors is uniform, given by p(r) Po. Find the field and potential inside by a V direct integration of Poisson's equation with appropriate boundary conditions. Once the potential + is found, find the total induced charge on each conductor. At what critical voltage does the charge on the anode go to zero? Note: the r in this problem is interpreted as the radial distance from the z-axis as in cylindrical coordinates. ++ ++. XX + xx xx xx x xxxx xx ++ + +++ +
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Transcribed Image Text:**Coaxial Conductor System with Uniform Space Charge**
A static space-charge distribution fills the region between two long coaxial conductors as shown. The center conductor is held at a potential of \( V_0 \) with respect to the outer conductor, which is at ground potential. The charge density in the region between the conductors is uniform, given by \( \rho(r) = \rho_0 \).
**Problem:**
1. Find the field and potential inside by a direct integration of Poisson’s equation with appropriate boundary conditions.
2. Once the potential is found, determine the total induced charge on each conductor.
3. Identify the critical voltage at which the charge on the anode goes to zero.
**Note:** The \( r \) in this problem is interpreted as the radial distance from the z-axis as in cylindrical coordinates.
**Diagram Explanation:**
The diagram displays two concentric cylindrical conductors. The inner conductor has a radius labeled as \( a \), while the outer conductor has a radius labeled as \( b \). The space between the conductors is marked with positive charges, indicating the presence of a uniform space charge. The inner conductor is connected to a voltage source labeled \( V_0 \).
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