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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!

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!

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