Project loop elements and account for an arc’s return path

Use rotational symmetry carefully

For a circular loop of radius R in the xy plane carrying current counterclockwise viewed from +z, transverse Biot–Savart components cancel on the axis. The axial projection integrates to Bz = mu0 I R²/[2(R² + z²)^(3/2)]. This is an axis formula, not an off-axis field map. It is even in z for a fixed current orientation.

Recover the center and distant behavior

At z = 0, B = mu0 I/(2R). Far on the axis, B is approximately mu0 I R²/(2|z|³). Writing magnetic moment m = I pi R² gives the axial dipole form mu0 m/(2pi |z|³). For the fixed counterclockwise current viewed from +z, the signed axial component points along +z on both sides; distance in the far-axis denominator is nonnegative. A source’s far-field approximation does not determine its near-field structure uniquely.

Integrate only the arc contribution

At the center of a circular arc, source distance is R and each element gives dB = mu0 I dtheta/(4pi R) along one normal. Hence an angular span theta contributes mu0 I theta/(4pi R). A semicircle gives half the full-loop center field at the same I and R; a quarter-circle gives one quarter. These comparisons hold current fixed, not a total charge inventory.

Test the return geometry

Radial connecting segments have dl parallel or antiparallel to the displacement to the center, so their center contributions vanish. A remote or differently shaped return generally needs its own calculation. For an original two-arc packet with opposite orientation, radii R and 2R, and equal semicircle current I, net center field is mu0 I/(4R) − mu0 I/(8R) = mu0 I/(8R), provided the connecting radial portions contribute zero.

Worked original case

With I = 2 A, R = 1 m and z = 1 m, Bz ≈ 0.444288 microtesla. Full-loop center field is 1.25664 microtesla; the same-radius/current semicircle contribution is 0.628319 microtesla. Reversing current reverses each signed normal field.

Investigation

Derive the axial projection and compare center, near-axis and far-axis values. Then construct the opposite-semicircle packet and retain each signed contribution instead of summing magnitudes. State which return segments are omitted and why.

Changed-case practice

Try a new practice case

What fraction of full-loop center field comes from a quarter-circle at the same current and radius?

Numerical result in the requested units Check this practice result

This public, unsaved check is not a secure assessment. Explain the source integral, symmetry or charge boundary in the retained course evidence workspace.

Sources and model boundary

Original teaching checked 31 October 2026 against the current official framework, Unit 12. The displayed magnetic calculations use the stipulated introductory approximation mu0 = 4pi×10⁻⁷ SI, not a claim that this is exact in modern SI. All apparatus and records are virtual, supplied or generated. No real magnet handling, electrical current setup, particle apparatus or material operation is requested. Public feedback and retained reports are practice evidence, not independent mastery or performed laboratory time.

Supplemental primary teaching reference: The Biot–Savart law. The supplied cases, derivations and diagrams here are original; the source’s older exact-mu0 convention and classical spin language are not adopted literally.