A very long conducting tube (hollow cylinder) has inner radius a and outer radiu

Question

A very long conducting tube (hollow cylinder) has inner radius a and outer radius b. It carries charge per unit length ?? where ? is a positive constant with units of C/m. A line of charge lies along the axis of the tube. The line of charge has charge per unit length +?.

Part B

Find the direction of the electric field in terms of ?and the distance r from the axis of the tube forr<a.

Find the direction of the electric field in terms of  and the distance  from the axis of the tube for .

Part D

Find the direction of the electric field in terms of ?and the distance r from the axis of the tube for a<r<b.

Find the direction of the electric field in terms of  and the distance  from the axis of the tube for .

Part G

What is the charge per unit length on the inner surface of the tube?

Express your answer in terms of the given quantities and appropriate constants.

Part H

What is the charge per unit length on the outer surface of the tube?

Express your answer in terms of the given quantities and appropriate constants.

PLEASE SHOW ALL WORK! Thanks!

Explanation / Answer

let's choose a cylinder as the Gaussian surface that we will use to find out the electric field in each area.
Imagine a cylinder concentric with the line of charge of radius r< a. The charge enclosed by the cylinder is going to be aL (where L is the length of the imaginary cylinder, much shorter than l). The electric flux on the ends of the cylinder will be zero (the electric field from the left and right sides will essential cancel out because the length l is much greater than the radius). The electric flux will be only through the curved surface of the cylinder which has an area of 2*pi*r*L.
Plug these values into the formula: E*2*pi*r*L = a*L/e
E=2ka/r (where k = 1/(4*pi*e))
Use this same method by choosing a Gaussian surface (cylinder) with radii of a<r<b, and r>b.
For a<r<b, the electric field will be zero (the net charge enclosed by the Gaussian surface will be zero, electric filed within any conductor is zero).
For r>b, E=4ka/r

B

the direction of the electric field radially outward

D

the direction of the electric field radially inward

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