Understanding Capacitive Behavior in Charged and Grounded Concentric Cylinders
Capacitive Behavior in Concentric Cylinders
There are 2 concentric cylinders with length L. The inner cylinder, a solid conductor, has a radius of R1 and carries a surface charge density of -σ. The outer cylinder, with an inner wall radius R2, is a hollow conductor with a conducting shell of thickness t. The outer cylinder is grounded, allowing for free movement of charges.
Questions:
a) Approximately, what is an expression for the total charge on the inner cylinder?The total charge on the inner cylinder is given by Q = -σ∙(2πR1L).
b) What (if any) is the total charge on the outer cylinder?The total charge on the outer cylinder is equal and opposite to the charge on the inner cylinder, hence Q' = +σ∙(2πR1L).
c) What is the surface charge density on the inner wall of the outer cylinder?The surface charge density on the inner wall of the outer cylinder is -σ to neutralize the field within the conductor.
d) What is the surface charge density on the outer wall of the outer cylinder?The surface charge density on the outer wall of the outer cylinder would be zero if it remains connected to the ground.
e) What is the expression for the electric field in the space between the inner cylinder and the inner wall of the outer cylinder?The electric field E can be calculated using Gauss's law as E = σ/(2πεr), where ε is the permittivity of free space and r is the radial distance from the central axis of the cylinders.
f) What is the change of electric potential between the two cylinders?The change of electric potential ΔV between the two cylinders can be determined by V = -σ∙Ln(R2/R1)/(2πε).
g) Calculate the capacitance of the entire system using the provided information.The capacitance (C) of the system is determined by the stored charge and potential difference, C = Q/ΔV. The charged and grounded concentric cylinders exhibit capacitive behavior defined by charge distribution and Gauss's law.