Past exam of the mathematics course of the University of Cambridge / 2021 / ib / Paper 1 / 15D / a

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- Solution a

Solution

Since the magnetic field is

B = \nabla \times A

, Stokes theorem gives, for any oriented surface

S

bounded by

C

,

Φ = \int_{S} B \cdot d S = \int_{S} (\nabla \times A) \cdot d S = \oint_{C} A \cdot d x .

Under a gauge transformation

A \mapsto A + \nabla χ

, the integral changes by

\oint_{C} \nabla χ \cdot d x = 0

because

C

is closed. The flux expression is therefore gauge independent.

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