Charged Insulating Rod vs. Conducting Shell
A long solid insulating cylinder of radius a = 2.0 cm carries a uniform volume charge density ρ = +3.0×10⁻⁶ C/m³. It is coaxial with a thin conducting cylindrical shell of radius b = 6.0 cm carrying a net charge per unit length of −2.0×10⁻⁷ C/m. A student claims that because the insulator is uniformly charged, the field outside the conductor can be found by treating the whole insulator as a line of charge at the axis. A second student argues that the conducting shell redistributes charge and that the field between the rod and shell must be computed with Gauss's law using only the enclosed charge. The system is in electrostatic equilibrium.
1. Determine the magnitude and direction of the electric field at r = 4.0 cm, and state which student's reasoning is correct for that region.
Read the worked answer
For a Gaussian cylinder of radius r with a < r < b, the enclosed charge per unit length is λ_ins = ρπa² = (3.0×10⁻⁶)(π)(0.020)² = 3.77×10⁻⁹ C/m. Gauss's law gives E(2πrL) = λ_ins L/ε₀, so E = λ_ins/(2πε₀r) = (2kλ_ins)/r = 2(8.99×10⁹)(3.77×10⁻⁹)/(0.040) ≈ 1.7×10³ N/C, directed radially outward. The second student is correct: only enclosed charge matters, and the shell's charge does not enter because it lies outside this Gaussian surface. The first student's shortcut happens to give the same result here only because the insulator is uniform; it is not a valid general argument.
2. Find the charge per unit length on the inner and outer surfaces of the conducting shell, and the field just outside the shell at r = 8.0 cm.
Read the worked answer
In equilibrium the field inside the conductor (a < r < b region is vacuum; inside the metal b ≤ r ≤ b+t) must be zero, so a Gaussian surface inside the metal encloses zero net charge: λ_ins + λ_inner = 0 → λ_inner = −3.77×10⁻⁹ C/m. The net shell charge is λ_shell = −2.0×10⁻⁷ C/m, so λ_outer = λ_shell − λ_inner = −2.0×10⁻⁷ + 3.77×10⁻⁹ ≈ −1.96×10⁻⁷ C/m. Outside at r = 8.0 cm, enclose everything: λ_enc = λ_ins + λ_shell = 3.77×10⁻⁹ − 2.0×10⁻⁷ ≈ −1.96×10⁻⁷ C/m. E = 2k|λ_enc|/r = 2(8.99×10⁹)(1.96×10⁻⁷)/(0.080) ≈ 4.4×10⁴ N/C, directed radially inward.