Normalize a line potential and recover its field

Normalize before integrating

A line on 0 ≤ x ≤ L has lambda(x) = lambda0(1 + beta x/L). With L = 1 m and Q = 2 nC, lambda0 = Q/[L(1 + beta/2)]. Potential beyond the endpoint is V(z) = k integral lambda(x)/(z − x) dx. The localized source permits V(infinity) = 0; the observer must remain outside the singular ideal line.

Use the observer as the differentiation variable

For the declared one-metre source, integration gives V = k lambda0[(1 + beta z) ln(z/(z − 1)) − beta], with numerical coordinates expressed in metres. Differentiating with respect to z gives E = k lambda0[(1 + beta z)(1/(z − 1) − 1/z) − beta ln(z/(z − 1))]. The normalization is held fixed during this derivative. Changing source inventory while differentiating would describe a different experiment.

Transfer to the perpendicular bisector

For a uniform line from −a to a with fixed lambda, V(y) = k lambda integral dx/sqrt(x² + y²) = 2k lambda asinh(a/y), for y > 0. Differentiating yields Ey = 2k lambda a/[y sqrt(a² + y²)]. With a = y = 1 m and Q = 2 nC, V = 2k(1 nC/m) asinh(1) ≈ 15.8428 V. Doubling y gives V = 2k(1 nC/m) asinh(1/2), not half the exact finite-line value. The point-source approximation requires distance much larger than the half-length.

Worked original case

For beta = 0 and z = 2 m, lambda0 = 2 nC/m and V = k(2 nC/m) ln 2 ≈ 12.4594 V. The field is k(2 nC/m)(1 − 1/2) = 8.98755 N/C. A centered finite difference of V approximates this derivative but has truncation error.

Investigation

Compare beta = 0, 1 and 2 at the same total charge. Predict how moving more charge toward the near endpoint changes potential. Derive the field separately, compare the centered derivative discrepancy and test a changed perpendicular-bisector observer using the written expression.

Changed-case practice

Sources and evidence boundary

Original teaching and cases checked 30 October 2026 against the current official framework, Unit 9 and OpenStax electric potential. All records are supplied or generated, not measured by a learner. No physical charging, battery connection, high-voltage equipment or particle experiment is requested. Retain predictions, derivations and critique in the course evidence workspace; public feedback is not independent mastery or performed laboratory time.