1. Poynting’s Theorem … energy conservation John Henry Poynting (1852-1914)

2. To recap… The energy stored in an electric field E is expressed as the work needed to “assemble” a group of point charges The energy stored in an electric field E is expressed as the work needed to “assemble” a group of point charges

3. Magnetic fields also store energy Magnetic fields also store energy Total energy stored by electromagnetic fields per unit volume is… Total energy stored by electromagnetic fields per unit volume is…

4. Work done moving a charge… Use the Lorentz formula: Use the Lorentz formula: The work in time-interval dt is then: The work in time-interval dt is then: (remember: Magnetic fields do no work!)

5. Remember current density J! The moving charge or charges constitute a current density, so we can write which means the work can now be expressed as Remember current density J! The moving charge or charges constitute a current density, so we can write which means the work can now be expressed as Remember that this is a rate of change of the energy (work) and so represents power delivered per unit volume

6. Now use Maxwell’s Equations And the identity

7. The “Work-Energy” Theorem for EM Fields… Poynting’s Theorem tells us: Poynting’s Theorem tells us: Change in energy stored in the fields Energy radiated across surface by the electromagnetic fields

8. The Poynting Vector