Poynting's theorem

Poynting's theorem is a statement due to John Henry Poynting about the conservation of energy for the electromagnetic field. Poynting's theorem takes into account the case when the electric and magnetic fields are coupled---static or stationary electric and magnetic fields are not coupled. It relates the time derivative of the energy density, u to the energy flow and the rate at which the fields do work. It is summarised by the following formula:

${\frac {\partial u}{\partial t}}+\nabla \cdot \mathbf {S} =-\mathbf {J} \cdot \mathbf {E}$

where S is the Poynting vector representing the flow of energy, J is the current density and E is the electric field. Energy density u is (symbol ε₀ is the electric constant and μ₀ is the magnetic constant):

$u={\frac {1}{2}}\left(\epsilon _{0}\mathbf {E} ^{2}+{\frac {\mathbf {B} ^{2}}{\mu _{0}}}\right).$

Since the magnetic field does no work, the right hand side gives the negative of the total work done by the electromagnetic field per second·meter³.

Poynting's theorem in integral form:

${\frac {\partial }{\partial t}}\int _{V}u\ dV+\oint _{\partial V}\mathbf {S} \ d\mathbf {A} =-\int _{V}\mathbf {J} \cdot \mathbf {E} \ dV$

Where $\partial V\!$ is the surface which bounds (encloses) volume $V\!$.

In electrical engineering context the theorem is usually written with the energy density term u expanded in the following way, which resembles the continuity equation:

$\nabla \cdot \mathbf {S} +\epsilon _{0}\mathbf {E} \cdot {\frac {\partial \mathbf {E} }{\partial t}}+{\frac {\mathbf {B} }{\mu _{0}}}\cdot {\frac {\partial \mathbf {B} }{\partial t}}+\mathbf {J} \cdot \mathbf {E} =0$

Where $\mathbf {S}$ is the energy flow of the electromagnetic wave, $\epsilon _{0}\mathbf {E} \cdot {\frac {\partial \mathbf {E} }{\partial t}}$ is the density of reactive power driving the build-up of electric field, ${\frac {\mathbf {B} }{\mu _{0}}}\cdot {\frac {\partial \mathbf {B} }{\partial t}}$ is the density of reactive power driving the build-up of magnetic field, and $\mathbf {J} \cdot \mathbf {E}$ is the density of real power dissipated by the Lorentz force acting on charge carriers.

Derivation

The theorem can be derived from two of Maxwell's Equations. First consider Faraday's Law:

$\nabla \times \mathbf {E} =-{\frac {\partial \mathbf {B} }{\partial t}}$

Taking the dot product of this equation with $\mathbf {B}$ yields:

$\mathbf {B} \cdot (\nabla \times \mathbf {E} )=-\mathbf {B} \cdot {\frac {\partial \mathbf {B} }{\partial t}}$

Next consider the Ampère-Maxwell law equation:

$\nabla \times \mathbf {B} =\mu _{0}\mathbf {J} +\epsilon _{0}\mu _{0}{\frac {\partial \mathbf {E} }{\partial t}}$

Taking the dot product of this equation with $\mathbf {E}$ yields:

$\mathbf {E} \cdot (\nabla \times \mathbf {B} )=\mathbf {E} \cdot \mu _{0}\mathbf {J} +\mathbf {E} \cdot \epsilon _{0}\mu _{0}{\frac {\partial \mathbf {E} }{\partial t}}$

Subtracting the first dot product from the second yields:

$\mathbf {E} \cdot (\nabla \times \mathbf {B} )-\mathbf {B} \cdot (\nabla \times \mathbf {E} )=\mu _{0}\mathbf {E} \cdot \mathbf {J} +\epsilon _{0}\mu _{0}\mathbf {E} \cdot {\frac {\partial \mathbf {E} }{\partial t}}+\mathbf {B} \cdot {\frac {\partial \mathbf {B} }{\partial t}}$

Finally, by the product rule, as applied to the divergence operator over the cross product (described here):

$-\nabla \cdot \ (\mathbf {E} \times \mathbf {B} )=\mu _{0}\mathbf {E} \cdot \mathbf {J} +\epsilon _{0}\mu _{0}\mathbf {E} \cdot {\frac {\partial \mathbf {E} }{\partial t}}+\mathbf {B} \cdot {\frac {\partial \mathbf {B} }{\partial t}}$

Since the Poynting vector $\mathbf {S}$ is defined as:

$\mathbf {S} ={\frac {1}{\mu _{0}}}\mathbf {E} \times \mathbf {B}$

This is equivalent to:

$\nabla \cdot \mathbf {S} +\epsilon _{0}\mathbf {E} \cdot {\frac {\partial \mathbf {E} }{\partial t}}+{\frac {\mathbf {B} }{\mu _{0}}}\cdot {\frac {\partial \mathbf {B} }{\partial t}}+\mathbf {J} \cdot \mathbf {E} =0$

Generalization

The mechanical energy counterpart of the above theorem for the electromagnetical energy continuity equation is

${\frac {\partial }{\partial t}}u_{m}(\mathbf {r} ,t)+\nabla \cdot \mathbf {S} _{m}(\mathbf {r} ,t)=\mathbf {J} (\mathbf {r} ,t)\cdot \mathbf {E} (\mathbf {r} ,t),$

where u_m is the mechanical (kinetic) energy density in the system. It can be described as the sum of kinetic energies of particles α (e.g., electrons in a wire), whose trajectory is given by $\mathbf {r} _{\alpha }(t)$:

$u_{m}(\mathbf {r} ,t)=\sum _{\alpha }{\frac {m_{\alpha }}{2}}{\dot {r}}_{\alpha }^{2}\delta (\mathbf {r} -\mathbf {r} _{\alpha }(t)),$

$\mathbf {S_{m}}$ is the flux of their energies, or a "mechanical Poynting vector":

$\mathbf {S} _{m}(\mathbf {r} ,t)=\sum _{\alpha }{\frac {m_{\alpha }}{2}}{\dot {r}}_{\alpha }^{2}{\dot {\mathbf {r} }}_{\alpha }\delta (\mathbf {r} -\mathbf {r} _{\alpha }(t)).$

Both can be combined via the Lorentz force, which the electromagnetical fields exert on the moving charged particles (see above), to the following energy continuity equation or energy conservation law :

${\frac {\partial }{\partial t}}\left(u_{e}+u_{m}\right)+\nabla \cdot \left(\mathbf {S} _{e}+\mathbf {S} _{m}\right)=0,$

covering both types of energy and the conversion of one into the other.

See also

• Poynting vector

External links

• Eric W. Weisstein "Poynting Theorem" From ScienceWorld--A Wolfram Web Resource.