Principles of Electromagnetic Waves and Materials, 2nd Edition

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)$$H\cdot \nabla \times E=-H\cdot \frac{\partial B}{\partial t},$$

and from Equation 1.2,

(1C.2)$$E\cdot \nabla \times H=E\cdot J+E\cdot \frac{\partial D}{\partial t}.$$

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

(1C.3)$$H\cdot \nabla \times E-E\cdot \nabla \times H=\nabla \cdot \left(E\times H\right),$$

we obtain

(1C.4)$$\nabla \cdot \left(E\times H\right)=-H\cdot \frac{\partial B}{\partial t}-E\cdot \frac{\partial D}{\partial t}-E\cdot J.$$

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

(1C.5)$${\displaystyle \underset{s}{∯}S\cdot \text{ds}+}{\displaystyle \underset{V}{\iiint }\left[E\cdot \frac{\partial D}{\partial t}+H\cdot \frac{\partial B}{\partial t}\right]\text{d}V+}{\displaystyle \underset{V}{\iiint }\left(E\cdot J\right)\text{d}V=0,}$$

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