Final answer:
The electric field magnitude for the region a < r < b is derived using Gauss's Law, which relates the electric field to the enclosed charge over a chosen Gaussian surface with symmetrical boundaries. The exact expression depends on the charge distribution, and it involves integrating the charge density over the enclosed volume.
Step-by-step explanation:
To derive the electric field magnitude in terms of the distance r from the center for the region a < r < b, we can use Gauss's Law. This law states that the electric field times the surface area of a closed surface (Gaussian surface) is equal to the enclosed charge divided by the permittivity of free space, ε0. To apply this to the scenario with cylindrical symmetry, you would choose a cylindrical Gaussian surface with radius r and length L that is coaxial with the charged cylinder.
For an area a < r < b, the charge enclosed by the Gaussian surface is proportional to the volume of the cylinder with radius r. The electric field E is uniform over the surface of the cylinder and directed radially outward. Therefore, Gauss's Law in this case simplifies to E(2πrL) = Q enclosed/ε0, where Q enclosed is the charge within the radius r. Solving for E gives us the electric field at a distance r from the center in the region a < r < b.
Note that the exact form of the electric field expression will depend on the charge distribution within the cylinders. If the charge distribution is uniform, we could express the charge density ρ as charge per unit volume and integrate it over the volume enclosed by the Gaussian surface to find Q enclosed. If the charge varies with radius, we integrate the charge density over the volume with respect to r to obtain Q enclosed. The integration limits would be from a to r