Poynting's theorem: differential and integral forms
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Proof of Poynting's theorem
Poynting's theorem: differential and integral forms
- 1. Course: Electromagnetic Theory paper code: EC 501 Course Coordinator: Arpan Deyasi Department of Electronics and Communication Engineering RCC Institute of Information Technology Kolkata, India Topic: Poynting’s Theorem 20-08-2025 Arpan Deyasi, EC501 1 Arpan Deyasi Electromagnetic Theory
- 2. Poynting’s Theorem It states that time rate at which e.m energy in a finite volume decreases, is equal to the rate of dissipation of energy in the form of Joule heat by the e.m field, plus the rate at which energy is flowed out through the surface Arpan Deyasi Electromagnetic Theory
- 3. Proof Work done dW F.dl = ( ) dq E u B .dl = + ( ) dq E u B .udt = + ( ) dqE.udt dq u B .udt = + ( ) dqE.udt dq u B .udt = + ( ) dv E.udt = Arpan Deyasi Electromagnetic Theory
- 4. dW E.Jdvdt = Proof V dW P E.Jdv dt = = Power delivered From Maxwell’s equation at free space 0 0 0 E B J t = + ( ) 0 0 1 E J B t = − Arpan Deyasi Electromagnetic Theory
- 5. Proof ( ) 0 0 1 . . . E E J E B E t = − ( ) ( ) 0 0 1 . . . E B E E B E t = − − ( ) 0 0 1 . . . B E B E B E t t = − − − Arpan Deyasi Electromagnetic Theory
- 6. Proof ( ) ( ) ( ) 0 0 0 1 . . . . 2 2 E J B H E B D E t t = − − − ( ) ( ) ( ) 0 V 0 0 V V dW B.H dv dt 2 t 1 . E B dv D.E dv 2 t = − − − Arpan Deyasi Electromagnetic Theory
- 7. Proof ( ) ( ) ( ) 0 V 0 0 S V dW B.H dv dt 2 t 1 E B .ds D.E dv 2 t = − − − ( ) ( ) ( ) 0 0 V V 0 S V B.H dv D.E dv 2 t 2 t 1 E B .ds E.Jdv − − = + Arpan Deyasi Electromagnetic Theory
- 8. Proof Electrostatic energy 2 0 0 E V V W E dv E.Ddv 2 2 = = Total energy stored 2 B 0 0 V V 1 W B dv B.Hdv 2 2 = = Magnetostatic energy 0 EB E B 0 V V W W W E.Ddv B.Hdv 2 2 = + = + Arpan Deyasi Electromagnetic Theory
- 9. ( ) EB 0 S dW dW 1 E B .ds dt dt − = + Proof Integral form of Poynting’s theorem LHS describes time rate of decrease of total energy in e.m field 1st term in RHS denotes power delivered through the motion of free charges 2nd term in RHS signifies rate at which energy is carried out of volume Arpan Deyasi Electromagnetic Theory
- 10. Proof Poynting’s vector is defined as ( ) 0 1 S E B = Energy per unit time per unit area transported by the field Arpan Deyasi Electromagnetic Theory
- 11. Proof EB EB V dW d U dv dt dt = M V dW d U dv dt dt = Electromagnetic energy density Mechanical energy density Arpan Deyasi Electromagnetic Theory
- 12. Proof ( ) EB M 0 V V S d d 1 U dv U dv E B .ds dt dt − − = EB M V V S d d U dv U dv S.ds dt dt − − = EB M V V V d d U dv U dv .Sdv dt dt − − = Arpan Deyasi Electromagnetic Theory
- 13. Proof ( ) EB M V V d U U dv .Sdv dt − + = ( ) EB M .S U U t = − + Differential form of Poynting’s theorem Arpan Deyasi Electromagnetic Theory