Reconstruct current sum and difference from signed field components

Write both source contributions

Two long z-directed wires lie at x = −1 and +1 m. At the midpoint line (0,y), the left current Il and right current Ir give Bx = −mu0 y(Il + Ir)/[2pi(1 + y²)] and By = mu0(Il − Ir)/[2pi(1 + y²)], using numerical metre coordinates. These signs follow the azimuthal right-hand field around each source, not a scalar sum of field magnitudes.

Invert independent component information

Define f = mu0/[2pi(1 + y²)]. Then current sum S = −Bx/(fy), difference D = By/f, and Il = (S + D)/2, Ir = (S − D)/2. The supported y range excludes zero so this chosen inverse does not divide by zero. At y = 0 the x component would not constrain the sum, illustrating a geometric identifiability limit.

Show why a magnitude can be ambiguous

At y = 1 m and Il = 2 A, Ir = +2 A gives B = (−0.4,0) microtesla, while Ir = −2 A gives B = (0,+0.4) microtesla. Both magnitudes are 0.4 microtesla. A magnitude-only probe cannot distinguish those two signed source hypotheses; an oriented component or a new observation location can.

Retain calibration and dynamics alternatives

The algebra assumes independently known wire positions, observer coordinates and component scale/sign calibration. A rotated sensor, gain error or omitted return current could mimic a source discrepancy. Additional same-calibration samples may improve a fit without resolving every shared bias. The static-wire inference also cannot exclude a magnetic contribution from changing electric fields without an appropriate time-dependent source account.

Worked original case

With Il = 2 A, Ir = −2 A and y = 1 m, Bx = 0 and By = +0.4 microtesla. The inferred sum is zero and difference 4 A, recovering the two currents. Changing only Ir to +2 A rotates the resulting field while preserving this particular magnitude.

Investigation

Predict both component signs for negative, zero and positive right current. Reconstruct the two currents from the component ledger without substituting their known inputs. Compare a magnitude-only alternative and propose a calibrated observation that genuinely distinguishes it.

Changed-case practice

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: Ampère’s 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.