I am currently reading "Magnetic Monopoles" of Ya. Shnir. My problem is I can not retrieve a result the author provides in the first chapter of the first part. In this chapter, he studies the non-relativistic scattering of an electric charge on a magnetic one.
The author writes [p.5, near eq. (1.13)]:
... the appearance of an additional term in the definition of the angular momentum (1.11) originates from a non-trivial field contribution. Indeed, since a static monopole is placed at the origin, its magnetic field is given by (1.1). Then the classical angular momentum of the electric field of a point-like electric charge, whose position is defined by its radius vector r, and the magnetic field of a monopole is a volume integral involving the Poynting vector
˜Leg=14π∫r′×[E×B]d3r′=−g4π∫d3r′(∇′⋅E)ˆr′=−egˆr
where we perform the integration by parts, take into account that the fields vanish asymptotically and invoke the Maxwell equation
(∇′.E)=4πeδ(3)(r−r′)
...
The magnetic field is
B=gr3r
The generalised angular momentum is
L=r×mv−egˆr
The author gives how he got (L.2) from (L.1) but I do not know how to do? Have you any idea?
Answer
Answer expected by following author's hints. Leg=14π∫r′×[E×B]d3r′=14π∫[(B.r′)E−(E.r′)B]d3r′=14π∫[(gr′3r′.r′)E−(E.r′)gr′3r′]d3r′=g4π∫1r′[E−(E.ˆr′)ˆr′]d3r′=g4π∫[E.∇′]ˆr′d3r′.
Or, let U and v be arbitrary vectors: [U.∇]v=[U.∇vi]ei,
By integrating by parts we have : Leg=g4π∫[E.∇′]ˆr′d3r′=g4π∫E.∇′(ˆr′i)d3r′ei=g4π∫[∇′.(Eˆr′i)ei−(∇′.E)ˆr′]d3r′=g4π [∮ˆr′(E.da)−∫(∇′.E)ˆr′d3r′].
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