Question

There are 2 concentric cylinders. These cylinders are very long with length L. The inner cylinder has a radius R1 and is a solid conductor. The outer one has a radius of R2 for its inner wall. It is a hollow cylinder with a conducting shell of thickness t.
The inner cylinder is charged and has a surface charge density of –σ. The outer cylinder is “grounded”, which means it has access to unlimited amounts of charge, charges are free to enter the object or leave the object.
a) Approximately, what is an expression for the total charge on the inner cylinder (as a function of R1, R2, L, or σ)? –the cylinder is very long, so the cap area will be very small compared to the sleeve area.
b) Approximately, what (if any) is the total charge that appears on the outer cylinder (as a function of R1, R2, L, or σ)?
c) Approximately, what (if any) is the surface charge density that appears on the inner wall of the outer cylinder (as a function of R1, R2, L, or σ)?
d) Approximately, what (if any) is surface charge density that appears on the outer wall of the outer cylinder (as a function of R1, R2, L, or σ)?
e) What is an expression for the electric field in the space between the inner cylinder and the inner wall of the outer cylinder (use r to denote a distance from the central axis of the cylinders) ?
f) What is the change of electric potential between the two cylinders?
g) Using all of the information from parts a,b,c, d, e, and f: calculate the capacitance of the entire system.

Answer 1

To get the charge along the inner cylinder, we use Gauss Law
E = d R1/2εo
For the outer cylinder the charge can be calculated using
E = d R2^2/2εoR1
where d is the charge density
Use these two equations to get the charge in between the cylinders and the capacitance between them.