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In most classical or quantum optics texts an angular momentum is considered for the EM radiation as the following:

[itex] J = ε_0 ∫_V r × [E(r, t) × B(r, t)] d^3 r [/itex]

Then it is claimed that:

"Using the usual formula for a double vector product and integrating by parts, bearing in

mind the assumption that the fields are zero at the surface of volume V introduced for the

mode expansion, one finds that JR can be written as a sum of two terms:

[itex]J = L + S , [/itex]

given by

[itex]L = ε_0 ∑_{j=(x,y,z)} ∫d^3r Ej (r, t)(r × ∇)Aj (r, t) ,[/itex]

[itex]S = ε_0 ∫d^3r E(r, t) × A(r, t) [/itex]

"

Would you please help me derive them?

Thank you

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# The Orbital and Spin Momenta of Light

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