Velocity of charges or bounding curve features in motional EMF?

[etotheipi]

The motional EMF is$$\mathcal{E}_{\text{motional}} = \oint_{\partial \Sigma} (\vec{v} \times \vec{B}) \cdot d\vec{x} = \int_{\Sigma} \frac{\partial \vec{B}}{\partial t} \cdot d\vec{S} - \frac{d}{dt} \int_{\Sigma} \vec{B} \cdot d\vec{S}$$(that's because Maxwell III integrates to $\mathcal{E}_{\text{transformer}} = \oint_{\partial \Sigma} \vec{E} \cdot d\vec{x} = -\int_{\Sigma} \frac{\partial \vec{B}}{\partial t} \cdot d\vec{S}$ and the flux rule is $\mathcal{E} = \oint_{\partial \Sigma} \left( \vec{E} + \vec{v} \times \vec{B} \right) \cdot d\vec{x} = -\frac{d}{dt} \int_{\Sigma} \vec{B} \cdot d\vec{S}$, with the thin wire assumption and where $\mathcal{E} = \mathcal{E}_{\text{transformer}} + \mathcal{E}_{\text{motional}}$).

Does $\vec{v}$ refer to the velocity of the charge carriers, or to the element $d\vec{x}$ on the boundary $\partial \Sigma$? I suspect it will be the velocity of the charged medium, which will work because e.g. in the case where the loop is stationary, $\vec{v} \parallel d\vec{x}$ and the triple product is zero, in accordance with $\mathcal{E}_{\text{motional}} = 0$. Also, is it right to say that motional EMF is only defined for instances in which we are integrating around an actual physical conductor (and not just a mathematical boundary $\partial \Sigma$), because otherwise it's not clear how $\vec{v}$ is defined? Thanks!

 

[vanhees71]

Written in this way it's the velocity of the boundary $\partial \Gamma$. Often this type of the Reynolds transport theorem is applied in this context such that the surface and its boundary is moving with the moving parts of the wires/coils etc. Since the drift velocity of the charge carriers in usual "household conditions" is in the range $\lesssim 1 \text{mm}/\text{s}$ the velocity of the moving parts of a circuit (like a generator coil) is the velocity of the charge carriers.

 

[etotheipi]

Thanks! I hadn't considered that $\vec{v}_{\mathrm{charge}/\mathrm{lab}} = \vec{v}_{\mathrm{charge}/\mathrm{d\vec{x}}} + \vec{v}_{\mathrm{d\vec{x}}/\mathrm{lab}} \approx \vec{v}_{\mathrm{d\vec{x}}/\mathrm{lab}}$, because the drift velocity is small. In that case I won't worry about it, because it will hardly make any difference

Whilst I was reading yesterday I also found a really interesting discussion in Feynman Vol. II Chap. 17 which gave some exceptions to the 'flux rule'. In that whilst $\oint_{\partial \Sigma} \vec{E} \cdot d\vec{x} = -\int_{\Sigma} \frac{\partial \vec{B}}{\partial t} \cdot d\vec{S}$ is a fundamental law which always holds for any surface $\Sigma$ and bounding curve $\partial \Sigma$, the 'flux rule' $\oint_{\partial \Sigma} \left( \vec{E} + \vec{v} \times \vec{B} \right) \cdot d\vec{x} = -\frac{d}{dt} \int_{\Sigma} \vec{B} \cdot d\vec{S}$ will only work if $\partial \Sigma$ coincides with the real conducting material! There was an example here of where the flux rule fails (here because the current is not restricted to a thin curve, but moves within an extended volume!):

 

[vanhees71]

Yes, in such cases it's also better to use the local (macroscopic) laws. In such cases you have to use also the full Ohm's Law, (in SI units)
$$\vec{j}=\sigma(\vec{E}+\vec{v} \times \vec{B}).$$
The usually neglected term with the magnetic field takes into account the Hall effect. Feynman is among the best books concerning these issues of E&M though it's not true that any law doesn't hold. It only has to be applied correctly. He is right in saying that the complete set of equations are given by the Maxwell equations and the Lorentz force.

There's also a thorough discussion on many issues in relativistic electromagnetism in Sommerfeld, Lectures on Theoretical Physics, vol. 3 and in Abraham & Becker. Usually the relativistic formulation leads to a better understanding. Very good are the books by Landau&Lifthitz vol. 2 (microscopic classical electrodynamics) and vol. 8 (macroscopic classical electrodynamics).

For examples concerning some issues which are often confusing in the literature (at least to me), see my Insights articles on the homopolar generator and the relativistic treatment of the DC current along a cylindrical straight wire.