Normalize a line density before integrating its field

Start with charge and units

For a line, dq = lambda(x)dx; lambda has units C/m. The assigned density is lambda 0(1+beta x/L), with L = 1 m. Integrating from 0 to L gives Q = lambda 0 L(1+beta/2). Holding lambda 0 fixed while changing beta would add charge, a different experiment from redistribution at fixed Q.

Keep source and observation coordinates separate

Call the source coordinate x and the observer z>L. Every source displacement is positive along the line, so dE = k lambda(x)dx/(z−x)². The source coordinate is the integration variable; z is held fixed during that integral. Crossing into the ideal line or approaching its endpoint is excluded from the controls.

Derive the endpoint expression

Using numerical metre coordinates with L = 1, write 1+beta x = (1+beta z)−beta(z−x). Integrating gives E = k lambda 0[(1+beta z)(1/(z−1)−1/z)−beta ln(z/(z−1))]. The logarithm is a ratio of distances, hence dimensionless. Restoring L requires keeping the factors beta/L, not treating a dimensional coordinate as an unlabelled pure number.

Audit approximation and limiting behavior

A midpoint sum approximates each small source cell by its center. Refinement reduces the displayed discrepancy in these supported cases, but neither cell count proves accuracy for arbitrary shapes or near-singular observers. Far from a bounded source, the leading contribution depends on total Q; changing beta affects smaller geometric corrections. Derive that limit before trusting a plotted flat-looking tail.

Worked original case

For total charge 2 nC on 0–1 m, beta = 1 gives lambda 0 = 4/3 nC/m. At observer z = 2 m the exact field is k lambda 0[3(1−1/2)−ln 2]≈9.66884 N/C. Eight midpoint cells give about 9.63396 N/C; sixty-four give 9.66829 N/C.

Investigation

At observer z = 2, compare beta = 0,1,2 while keeping total charge 2 nC fixed. Derive lambda 0 anew and compare exact,8-cell and 64-cell fields. Then increase observation distance and compare with the leading point-charge limit kQ/z². Explain why one small numerical discrepancy is not a general quadrature error bound.

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.