Reconcile coaxial volume charge with piecewise field

Declare infinite cylindrical symmetry

The core and shell extend indefinitely along a shared axis, and their volume densities are uniform within each region. Rotational and translational symmetry makeE radial and constant on the curved wall of a coaxial Gaussian cylinder. End-cap normals are axial, so their flux is zero. A finite object would have edge fields absent from this model.

Use charge per length consistently

The core contains 2 nC per metre across area pi×1², and the shell occupies area pi(2²−1²). For r≤1, enclosed line charge is 2r² nC/m. For 1≤r≤2 it is 2+lambda_shell(r²−1)/3. Outside 2 it is 2+lambda_shell. The coefficient formulas use numerical metre coordinates with the fixed stated radii.

Derive the radial field

For Gaussian lengthell, flux isE(2pi r ell) and enclosed charge islambda_enclosed ell. Cancellingell givesE = lambda_enclosed/(2pi epsilon 0 r) = 2k lambda_enclosed/r. The signed line charge determines inward or outward direction. The core center limit is zero rather than a division-by-zero evaluation.

Interpret cancellation locally

Equal opposite total line charges produce zero exterior field, but an interior Gaussian surface can still enclose positive charge. Thus external shielding in this ideal distribution does not imply every region is field-free. The materials here are insulating; if they were conductors, charge redistribution and zero field inside conductor material would impose different constraints.

Worked original case

The core carries+2 nC/m within radius 1 m and the shell carries−2 nC/m between radii 1 and 2 m. At r = 1.5 m, enclosed charge per length is 2−2(2.25−1)/3 = 7/6 nC/m, giving Er≈13.9806 N/C. Beyond radius 2 m the net enclosed charge per length is zero, so the ideal exterior field vanishes.

Investigation

Compare shell charges−2,−1 and+2 nC/m. Derive each enclosed-charge branch before plottingE. Check continuity at radii 1 and 2, then compare a point inside the core, inside the shell and outside both. State why this infinite insulating model is not a finite conducting cable calculation.

Public numerical transfer practice

Sources and evidence boundary

Original cases and teaching checked 30 October 2026 against the current official calculus-based electricity and magnetism framework and OpenStax electric fields. All displayed records are generated or supplied, not learner measurements. No charging, mains connection, high-voltage equipment, physical circuit or material handling is requested. Public feedback and retained self-reports are practice evidence, not independent mastery, performed laboratory time or AP authorization.