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Appendix 1C: Poynting Theorem

Appendix 1C: Poynting Theorem

Poynting theorem is nothing but a statement of conservation of energy. Since the theorem can be derived from Maxwell’s equations, it shows that Maxwell’s equations are consistent with the more general principle of energy conservation.

From Equation 1.1,

(1C.1)
H⋅∇×E=−H⋅∂B∂t,

and from Equation 1.2,

(1C.2)
E⋅∇×H=E⋅J+E⋅∂D∂t.

Subtracting Equation 1C.2 from Equation 1C.1 and making use of the vector identity

(1C.3)
H⋅∇×E−E⋅∇×H=∇⋅E×H,

we obtain

(1C.4)
∇⋅E×H=−H⋅∂B∂t−E⋅∂D∂t−E⋅J.

Integrating Equation 1C.4 over a volume V and using the divergence theorem (see Appendix 1A),

(1C.5)
∯sS⋅ds+∭VE⋅∂D∂t+H⋅∂B∂tdV+∭VE⋅JdV=0,

where s is the closed surface bounding the volume V. The second term on the LHS of Equation 1C.5 can be written as

(1C.6)
∂∂t∭V12E⋅D+12B⋅Hdv

for the case of an isotropic simple medium, since

(1C.7)
∂∂tE⋅D=E⋅∂D∂t+∂E∂t⋅D=2E⋅∂D∂t,
(1C.8)
∂∂tH⋅B=H⋅∂B∂t+∂H∂t⋅B=2H⋅∂B∂t.

Therefore, Equation 1C.6 can be rewritten as

(1C.9)
∯sS⋅ds+∂∂t∭V12E⋅D+12B⋅Hdv+∭VE⋅JdV=0.

Let us interpret Equation 1C.9 for a sourceless but lossy simple medium. If S represents the electromagnetic power density, then the first term gives the net power leaving the closed surface S. The second term is the time rate of increase of the stored electric and magnetic energy. The third term is the ohmic power loss due to the conversion of electromagnetic energy to thermal energy. The time rate of decrease in the stored energy provides for the power leaving the surface plus the inevitable power loss due to the lossy medium.

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