A very long conducting tube (hollow cylinder) has inner radius A and outer radius b. It carries charge per unit length +α, where α 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 +α. (b) What is the charge per unit length on (i) the inner surface of the tube and (ii) the outer surface of the tube?

Verified step by step guidance

1. Identify the system and the symmetry: The system consists of a long conducting tube (hollow cylinder) and a line of charge along its axis. Both have the same charge per unit length +α. The cylindrical symmetry of the problem suggests using Gauss's Law.
2. Choose a Gaussian surface: To find the charge per unit length on the inner surface, consider a cylindrical Gaussian surface inside the tube, just between the line of charge and the inner surface of the tube. The radius of this Gaussian surface should be less than A.
3. Apply Gauss's Law for the inner surface: Gauss's Law states that the electric flux through a closed surface is equal to the charge enclosed divided by the permittivity of free space, ε₀. Calculate the charge enclosed by the Gaussian surface, which is only due to the line of charge at the center.
4. Choose another Gaussian surface for the outer surface: This time, select a cylindrical Gaussian surface outside the tube, with a radius greater than b. This will help in determining the charge on the outer surface of the tube.
5. Apply Gauss's Law for the outer surface: Again, use Gauss's Law. The charge enclosed by this Gaussian surface includes the charge of the line and any net charge on the tube itself. Since the tube is neutral overall, the charge on the outer surface can be deduced by considering the total enclosed charge and the charge on the inner surface.

Key Concepts

Gauss's Law

Gauss's Law relates the electric flux through a closed surface to the charge enclosed by that surface. It states that the total electric flux is proportional to the enclosed charge, allowing us to calculate electric fields in symmetric charge distributions. This principle is crucial for analyzing the electric field around the conducting tube and the line of charge.

Charge Distribution in Conductors

In electrostatic equilibrium, charges in a conductor redistribute themselves such that the electric field inside the conductor is zero. This means that any excess charge resides on the surface of the conductor. Understanding how charge distributes on the inner and outer surfaces of the tube is essential for determining the charge per unit length on each surface.

Superposition Principle

The Superposition Principle states that the total electric field created by multiple charges is the vector sum of the electric fields produced by each charge individually. This principle is important for analyzing the contributions of both the line of charge and the charges on the tube's surfaces to the overall electric field in the region, which influences the charge distribution.