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ID: past-exam-of-the-physics-b-course-of-the-university-of-cambridge/2017/paper-2/section-a/a4/solution

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Past exam of the physics B course of the University of Cambridge / 2017 / Paper 2 / Section A / A4 / Solution by

Codex  0 2026-10-18

The symmetry is electromagnetic reciprocity applied to mutual inductance. For thin closed circuits in vacuum, the magnetic vector potential generated by circuit B is

A_{B} (r) = \frac{μ _{0} I _{B}}{4 π} \oint_{B} \frac{d ℓ _{B}}{∣ r - r _{B} ∣} .

(1)

Stokes theorem converts the magnetic flux through circuit A into its magnetic vector potential circulation. Dividing by

I_{B}

gives Neumann's mutual-inductance formula,

M_{A B} = \frac{Φ _{A \leftarrow B}}{I _{B}} = \frac{μ _{0}}{4 π} \oint_{A} \oint_{B} \frac{d ℓ _{A} \cdot d ℓ _{B}}{∣ r _{A} - r _{B} ∣} .

(2)

The dot product and separation are unchanged by interchanging A and B, so

M_{A B} = M_{B A} .

(3)

Each has units

Wb A^{- 1} = H

. The winding orientations must be used consistently to determine the common sign. The same reciprocity holds in ordinary linear reciprocal magnetic media; it is not an unrestricted statement about biased nonreciprocal materials. Equivalently, a reversible magnetic energy function has equal mixed derivatives with respect to the two currents.

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