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Law of Conservation of Energy for E.M. Field

We know that by equations

J.E +1/2 [d/dt (E.D) +d/dt (H.B) +div (E x H) =0

 

For a non-conductive media J = 0. We get

 

½[d/dt (E.D.) + d/dt (H.B) ] + div (E*H) = 0

ð    ½[d/dt (E.D.) + d/dt (H.B) ] + div P = 0

ð    ½[d/dt (εE2) + d/dt (µH2) ] + div P = 0

 

Energy density = ½ (εE2) + ½ (µH2)

 

d/dt.µ + div P = 0

 

div P + du/dt = 0

 

This equation is same as equation of the continuity

 

Div J + dp/dt = 0

 

Since equation of continuity represents the law of conservation of charge. So by analogy the previous equation must represent the law of conversation of energy for electromagnetic field.