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In the mean-field dynamo theory in magnetohydrodynamics, I frequently came across a quantity; $\langle v'\times B' \rangle$, which is termed as the mean electromotive force. I want to know that why is it termed as electromotive force, if it is a vector. Everywhere else I have seen emf is just the potential difference and hence a scalar. Is this emf different than the emf used in mean-field dynamo theory?

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  • $\begingroup$ EMF is not potential difference. It is integral of total motional force acting on current in a circuit, or in any path element chosen. In special cases, it has the same magnitude as potential difference, i.e. on a perfect inductor. But even there, the EMF acts against the force due gradient of potential. $\endgroup$
    – Ján Lalinský
    Commented Feb 27, 2023 at 16:58

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$\left\langle \mathbf{v}' \times \mathbf{B}' \right\rangle$ has dimensions of electric field, rather than potential. Therefore, it is different from the standard definition of electromotive force. In a highly conductive fluid it would be equal to $-\left\langle \mathbf{E}' \right\rangle$ (by Ohm's law). It could be considered the electromotive force per unit length in the direction parallel to the vector resulting from the motion of the fluid.

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    $\begingroup$ Shouldn't that be $\langle \mathbf E' \rangle$, with electric field vector in the same frame? $\endgroup$
    – Ján Lalinský
    Commented Mar 1, 2023 at 13:14
  • $\begingroup$ Thanks @JánLalinský, the notation has been updated to show the vector quantities and reflect that electric field is in the same frame. $\endgroup$
    – FTT
    Commented Mar 2, 2023 at 20:53
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The usual EMF in circuits refers to a closed oriented path: it is the integral of net motional force per unit charge acting on current. Physical unit of this EMF is Volt. Sometimes EMF for a non-closed path is discussed, which is based on the same idea, only the integration path is not closed but has starting point and ending point.

The MHD electromotive force $\mathbf E^*$ is clearly a different concept, but still related: it is the motional electromotive force (due to motion in magnetic field) acting on current in medium, per unit charge. So the actual force on current in volume element $\Delta V$ would be

$$ \Delta \mathbf F = \rho_m \Delta V\mathbf E^* $$ where $\rho_m$ is density of mobile charge.

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Strictly speaking, you are right. However, in magnetohydrodynamics, it happens that people refer to the force per unit charge $ \frac{{\bf J}}{e}\times {\bf B}$ as emf, with the implicit assumption that its line integral over a path provides the real emf.

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