Relation Cross-Scans

Toy Coupling Factor Cross-Scan

A fixed current and flux expose scaling, halving, and sign reversal in the toy energy budget. Exact arithmetic here means exact results for the stated model inputs; measured inputs still carry uncertainty and significant-figure limits.

highlighted = computed this step

Coupling row 1 gives energy 6

With current 3 and flux 2, coupling factor row 1 gives energy 6.

gIΦ=1⋅3⋅2=6g I\Phi=1\cdot 3\cdot 2=6
Coupling row 1The toy check keeps current and flux fixed while the factor changes.current=3 Aflux=2 WbcouplingFactor=1couplingEnergy=6 J

Coupling row 2 gives energy 3

With current 3 and flux 2, coupling factor row 2 gives energy 3.

gIΦ=12⋅3⋅2=3g I\Phi=\frac{1}{2}\cdot 3\cdot 2=3
Coupling row 2The toy check keeps current and flux fixed while the factor changes.current=3 Aflux=2 WbcouplingFactor=1/2couplingEnergy=3 J

Coupling row 3 gives energy -6

With current 3 and flux 2, coupling factor row 3 gives energy -6.

gIΦ=−1⋅3⋅2=−6g I\Phi=-1\cdot 3\cdot 2=-6
Coupling row 3The toy check keeps current and flux fixed while the factor changes.current=3 Aflux=2 WbcouplingFactor=-1couplingEnergy=-6 J

The toy factor scales and flips the energy budget

The same current and flux make six units of energy at factor one. Half factor halves the budget, and a negative factor reverses the sign.

gIΦE132612323−132−6\begin{array}{c|c|c|c}g&I&\Phi&E\\1&3&2&6\\\frac{1}{2}&3&2&3\\-1&3&2&-6\\\end{array}
Toy coupling cross-scanThe displayed row is the sign-reversal case.negative factor flips the energy sign