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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 +?.

1.Calculate the electric field in terms of ? and the distance r from the axis of the tube for r<a.

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

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

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

3.Calculate 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 constant ?0.

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

5.What is the charge per unit length on the outer surface of the tube?  Edit Comment  Solutions
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A) Region r < a,

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

Total electric flux through the curved surface is,

(B) Region a < r < b

Since the conducting tube is a charged tube all the charges will be residing on the outer surface. So, initially the charge per unit length outside the tube is +?.

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

So, by applying Gauss law for a coaxial cylindrical Gaussian Surface 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 L with radius r > b is,

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

By Gauss' law,

E3 2?rL = 2?L/?o

and points along radially outward direction.

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

(E)Â Â Charge per unit length on the outer surface ofthe cylinder = +2?  Edit Comment  Close Choose An Image
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