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B.2 Magnetic vector potential

The definition of the canonical toroidal angular momentum Pϕ involves the magnetic vector potential. Hence we need to calculate the vecotor potential. Given a current source J(r,t), the vector potential can be calculated using

μ0 ∫ J (r′,t′) 3 ′ A(r,t) = 4π- Ω-|r-− r′|d r,
(627)

where

|r − r′| t′ = t−--c---,
(628)

is called the retarded time. For a steady-state source, J(r′,t′) = J(r). Then Eq. (627) is simplified as

μ ∫ J(r′) A(r) =--0 -----′d3r′, 4π Ω |r − r|
(629)

For a current flowing in a zero-thinkness wire, the above equation is written as

μ ∫ J(r′) A (r) =-0- -----′dS (r′)dl(r′), 4π |r− r |
(630)

where dS is a surface element perpendicular to the wire and dl is a line element along the wire. Using J(r′)dS(r′) = I(r′), the above eqaution is written as

μ0 ∫ I(r′) ′ A (r) = 4π- |r−-r′| dl(r) ∫ ′ = μ0- -I(r)′dl(r′). (631) 4π |r− r|

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