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## Main Question or Discussion Point

i need an interpretation for the poynting vector, and its derivation for EM waves (sinusoidal)

- Thread starter Ahmad Kishki
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i need an interpretation for the poynting vector, and its derivation for EM waves (sinusoidal)

- #2

ShayanJ

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Consider the two "curly" Maxwell's equations!(
,
)

Dot the first one with [itex]\mathbf H[/itex] and the second one with [itex] \mathbf E [/itex] and then subtract the second from the first. Using some vector calculus identities, You'll get the following:

[itex]

\frac{\partial}{\partial t} \frac 1 2 (\mathbf E \cdot \mathbf D+\mathbf B \cdot \mathbf H) + \mathbf \nabla \cdot (\mathbf E \times \mathbf H)=-\mathbf E \cdot \mathbf J_f

[/itex]

which is of the form [itex] \frac{\partial u}{\partial t}+\mathbf \nabla \cdot \mathbf K=G [/itex], i.e. is a continuity equation.

Where u is the density of "something", [itex] \mathbf K [/itex] is the current density of that "something"(amount of "something" passing from a unit cross section in the unit of time) and G is the generation of "something" per volume.

For the present case, that something is energy and so the equation is describing the (non-)conservation of electromagnetic energy(which can appear or disappear since its only one of the energy forms present!).

As you can see, the quantity [itex] \mathbf S=\mathbf E \times \mathbf H [/itex](Poynting vector) is playing the role of [itex] \mathbf K [/itex] and so is the electromagnetic energy current density.

Dot the first one with [itex]\mathbf H[/itex] and the second one with [itex] \mathbf E [/itex] and then subtract the second from the first. Using some vector calculus identities, You'll get the following:

[itex]

\frac{\partial}{\partial t} \frac 1 2 (\mathbf E \cdot \mathbf D+\mathbf B \cdot \mathbf H) + \mathbf \nabla \cdot (\mathbf E \times \mathbf H)=-\mathbf E \cdot \mathbf J_f

[/itex]

which is of the form [itex] \frac{\partial u}{\partial t}+\mathbf \nabla \cdot \mathbf K=G [/itex], i.e. is a continuity equation.

Where u is the density of "something", [itex] \mathbf K [/itex] is the current density of that "something"(amount of "something" passing from a unit cross section in the unit of time) and G is the generation of "something" per volume.

For the present case, that something is energy and so the equation is describing the (non-)conservation of electromagnetic energy(which can appear or disappear since its only one of the energy forms present!).

As you can see, the quantity [itex] \mathbf S=\mathbf E \times \mathbf H [/itex](Poynting vector) is playing the role of [itex] \mathbf K [/itex] and so is the electromagnetic energy current density.

Last edited:

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Alternatively, the Poynting vector is associated with the momentum of the EM field.

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