Poynting Theorem

Poynting 's theorem is a theorem that describes the law of conservation of electromagnetic field energy . The theorem was proved in 1884 by John Henry Pointing . It all comes down to the following formula:

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

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

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

Where $u$ ${\ displaystyle u}$ $u$ - energy density : $u={\frac {1}{2}}\left(\varepsilon _{0}\mathbf {E} ^{2}+{\frac {\mathbf {B} ^{2}}{\mu _{0}}}\right)$ ${\ displaystyle u = {\ frac {1} {2}} \ left (\ varepsilon _ {0} \ mathbf {E} ^ {2} + {\ frac {\ mathbf {B} ^ {2}} {\ mu _ {0}}} \ right)}$ ${\ displaystyle u = {\ frac {1} {2}} \ left (\ varepsilon _ {0} \ mathbf {E} ^ {2} + {\ frac {\ mathbf {B} ^ {2}} {\ mu _ {0}}} \ right)}$ ;

\varepsilon _{0}

{\ displaystyle \ varepsilon _ {0}}

\ varepsilon _ {0}

- electric constant

\mu _{0}

{\ displaystyle \ mu _ {0}}

\ mu _ {0}

- magnetic constant ;

\nabla

{\ displaystyle \ nabla}

\ nabla

- operator nabla ; S is the Poynting vector ;

J is the current density and E is the electric field strength .

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

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

{\ 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 $\partial V$ ${\ displaystyle \ partial V}$ $\ partial V$ - surface limiting volume $V$ ${\ displaystyle V}$ $V$ .

In the technical literature, the theorem is usually written as follows ( $u$ ${\ displaystyle u}$ $u$ - energy density):

\nabla \cdot \mathbf {S} +\varepsilon _{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

{\ displaystyle \ nabla \ cdot \ mathbf {S} + \ varepsilon _ {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}

\ nabla \ cdot {\ mathbf {S}} + \ varepsilon _ {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 $\varepsilon _{0}\mathbf {E} \cdot {\frac {\partial \mathbf {E} }{\partial t}}$ ${\ displaystyle \ varepsilon _ {0} \ mathbf {E} \ cdot {\ frac {\ partial \ mathbf {E}} {\ partial t}}}$ $\ varepsilon _ {0} {\ mathbf {E}} \ cdot {\ frac {\ partial {\ mathbf {E}}} {\ partial t}}$ Is the energy density of the electric field, ${\frac {\mathbf {B} }{\mu _{0}}}\cdot {\frac {\partial \mathbf {B} }{\partial t}}$ ${\ displaystyle {\ frac {\ mathbf {B}} {\ mu _ {0}}} \ cdot {\ frac {\ partial \ mathbf {B}} {\ partial t}}}$ ${\ frac {{\ mathbf {B}}} {\ mu _ {0}}} \ cdot {\ frac {\ partial {\ mathbf {B}}} {\ partial t}}$ Is the energy density of the magnetic field and $\mathbf {J} \cdot \mathbf {E}$ ${\ displaystyle \ mathbf {J} \ cdot \ mathbf {E}}$ ${\ mathbf {J}} \ cdot {\ mathbf {E}}$ - power of joule losses per unit volume.

Conclusion

The theorem can be deduced using two Maxwell equations (for simplicity, we assume that the medium is a vacuum (μ = 1, ε = 1); for the general case with an arbitrary medium, we need to assign ε and μ to the formulas for each ε ₀ and μ ₀ ) :

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

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

Multiplying both sides of the equation by $\mathbf {B}$ ${\ displaystyle \ mathbf {B}}$ we get:

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

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

We first consider the Maxwell-Ampère equation:

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

{\ displaystyle \ nabla \ times \ mathbf {B} = \ mu _ {0} \ mathbf {J} + \ varepsilon _ {0} \ mu _ {0} {\ frac {\ partial \ mathbf {E}} { \ partial t}}.}

Multiplying both sides of the equation by $\mathbf {E}$ ${\ displaystyle \ mathbf {E}}$ we get:

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

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

Subtracting the first from the second, we get:

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

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

Finally:

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

{\ displaystyle - \ nabla \ cdot \ (\ mathbf {E} \ times \ mathbf {B}) = \ mu _ {0} \ mathbf {E} \ cdot \ mathbf {J} + \ varepsilon _ {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 $\mathbf {S}$ ${\ displaystyle \ mathbf {S}}$ defined as:

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

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

this is equivalent to:

\nabla \cdot \mathbf {S} +\varepsilon _{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.

{\ displaystyle \ nabla \ cdot \ mathbf {S} + \ varepsilon _ {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. }

Summary

The mechanical energy of the theorem described above

{\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),

{\ displaystyle {\ 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 kinetic energy of density in the system. It can be described as the sum of the kinetic energy of particles α

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

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

$\mathbf {S_{m}}$ ${\ displaystyle \ mathbf {S_ {m}}}$ - energy flow, or “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)).

{\ displaystyle \ 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)).}

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,

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

Alternative Forms

Other forms of Poynting's theorem can be obtained. Instead of using a stream vector $\mathbf {S} \propto \mathbf {E} \times \mathbf {B}$ ${\ displaystyle \ mathbf {S} \ propto \ mathbf {E} \ times \ mathbf {B}}$ can choose the shape of Abraham $\mathbf {E} \times \mathbf {H}$ ${\ displaystyle \ mathbf {E} \ times \ mathbf {H}}$ Minkowski form $\mathbf {D} \times \mathbf {B}$ ${\ displaystyle \ mathbf {D} \ times \ mathbf {B}}$ , or some other.

Poynting Theorem

Conclusion

Summary

Alternative Forms

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