poynting.md

Poynting Vector: System-Independent Form

Energy Density in Electromagnetic Field

Let's calculate the electric potential energy by: ^[1,3]

Use this trick, where electric potential follows: ^[1]

and Gauss's law is: ^[2]

Then we get: ^[3]

Analogously, the magnetic potential energy is: ^[3]

Apply the magnetic vector potential: ^[1]

and Ampère's circuital law: ^[2,3]

then we get: ^[3]

Finally, we get the energy density in electromagnetic field: ^[4]

Poynting's Theorem

Let's focus on the time-varying of the energy density:

Apply Ampère-Maxwell equation and Maxwell-Faraday equation: ^[5]

then we get:

Apply the trick: ^[6]

then we get Poynting's Theorem: ^[1]

where S is the Poynting vector and represents the directional energy flux (the energy transfer per unit area per unit time) or power flow of an electromagnetic field:

The momentum density of an electromagnetic field is:

In all unit systems, the dimension of S is M T^-3 and the dimension of g is M L^-2 T^-1. In natural units, the dimension of S and g is 4 (eV⁴).

Notes

1. These always hold in all unit systems.
2. Refer to (5.1) and (5.6).
3. These only hold in electrostatic or magnetostatic situation.
4. However, we assume this holds in time-varying fields.
5. Refer to (5.6) and (5.5).
6. Refer to the first page in Jackson 1999.