Sourcefree Electromagnetic Fields Exist In A Localized Region Of Space

Source-free electromagnetic fields exist in a localized region of space. Consider the various conservation laws that are contained in the integral of ∂αMαβγ = 0 over all space, where Mαβγ is defined by (12.117).  

(a) Show that when β and γ are both space indices conservation of the total field angular momentum follows.

(b) Show that when β = 0 the conservation law is

dX/dt = c2Pem/Eem

where X is the coordinate of the center of mass of the electromagnetic fields, defined by

X ∫ ud3x = ∫ xu d3x

where μ is the electromagnetic energy density and Eem and Pem are the total energy and momentum of the fields.

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