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 E

Calculate the magnitude the electric field in terms of ? and the distance r from the axis of the tube for r>b.

Express your answer in terms of the variables ?, r, and constants ? and ?0.

Find the direction of the electric field in terms of ? and the distance r from the axis of the tube

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?

Answer 1

(A) Region r < a,

Consider a Gaussial cylinder with r as radius and with r<aand coaxial with the other cylinder. By applying Gauss' law forthis Gaussian cylinder,

Total electric flux through the curved surface is,

pointing radially outwards.

(B) Region a < r < b

Since the conducting tube is a charged tube all the chareswillbe residing on the outer surface. So, initially thechargecper unit length outside the tube is +?.

When another line charge distribution +? C per unit lenthis along the axis, a charge distribution -? Coulomb per unitlength will be induced on the inner surface of the conductingcylinder. So, the total charges on the outer surface of thecylinder is + 2? coulomb per unit length.

So, by applying Gauss law for a coaxial cylindrical Gaussiansurface with radius r between a and b gives,

Hence E₂ = 0 in the region a < r <b.

(C) Region r > b:

Here total charge enclosed by a Gaussian surface of length Lwith radius r > b is,

Q_encl = (+2? - ? + ?)L =2?L

By Gauss' law,

E₃ 2?rL = 2?L/?_o

and points along radially outwarddirection.

(D) As discussed earlier Charge per unit length on the innersurface of the cylinder = -?

(E)  Charge per unit length on the outer surface ofthe cylinder = +2?