- #1

Twigg

Gold Member

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## Main Question or Discussion Point

Hi all,

I derived a formula last night for the Poynting vector I have never seen before, and wanted some verification and perhaps insight on.

I started with the definition for the Poynting vector in free space: $$\vec{S} = \frac{1}{\mu_{0}} \vec{E} \times \vec{B}$$ and substituted the potential formalism for radiation in free space: $$\vec{E} = -\frac{\partial \vec{A}}{\partial t}$$ $$\vec{B} = \nabla \times \vec{A}$$

Much algebra later, I wind up with the following formula in index notation using the standard Cartesian basis:

$$S_{i} = \frac{1}{\mu_{0}} \frac{\partial A^{j}}{\partial t}(\frac{\partial A_{i}}{\partial x^{j}} - \frac{\partial A_{j}}{\partial x^{i}})$$

As far as verification, I don't see any obvious signs that it's false. The units appear to add up (A has units of ##T*m## and ##\mu_{0}## has units of ##T^{2} * m^{3} / J##, so the whole thing has units of ##J / m^{2} * s##, which is the units of S). Also, the formula is independent of ##\nabla \cdot \vec{A}##, since the ##\frac{\partial A_{i}}{\partial x^{j}} - \frac{\partial A_{j}}{\partial x^{i}}## term is traceless, which makes the formula gauge-invariant. All I can tell is that it doesn't break any of the core rules, although it's not a covariant formula (since Lorentz transformations would transform some of the Poynting vector into momentum density, IIRC). I'm still not 100% convinced though. Does anyone have a reference to confirm/disprove this expression?

I derived a formula last night for the Poynting vector I have never seen before, and wanted some verification and perhaps insight on.

I started with the definition for the Poynting vector in free space: $$\vec{S} = \frac{1}{\mu_{0}} \vec{E} \times \vec{B}$$ and substituted the potential formalism for radiation in free space: $$\vec{E} = -\frac{\partial \vec{A}}{\partial t}$$ $$\vec{B} = \nabla \times \vec{A}$$

Much algebra later, I wind up with the following formula in index notation using the standard Cartesian basis:

$$S_{i} = \frac{1}{\mu_{0}} \frac{\partial A^{j}}{\partial t}(\frac{\partial A_{i}}{\partial x^{j}} - \frac{\partial A_{j}}{\partial x^{i}})$$

As far as verification, I don't see any obvious signs that it's false. The units appear to add up (A has units of ##T*m## and ##\mu_{0}## has units of ##T^{2} * m^{3} / J##, so the whole thing has units of ##J / m^{2} * s##, which is the units of S). Also, the formula is independent of ##\nabla \cdot \vec{A}##, since the ##\frac{\partial A_{i}}{\partial x^{j}} - \frac{\partial A_{j}}{\partial x^{i}}## term is traceless, which makes the formula gauge-invariant. All I can tell is that it doesn't break any of the core rules, although it's not a covariant formula (since Lorentz transformations would transform some of the Poynting vector into momentum density, IIRC). I'm still not 100% convinced though. Does anyone have a reference to confirm/disprove this expression?