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Lets say we have a closed circular loop of wire in a constant magnetic field in the (-z) direction. If I suddenly make the circular loop smaller (meaning a smaller surface area), I will induce a current traveling in the clockwise direction.

Since voltage is the electric potential difference between two points, what does the induced voltage of the wire represent? Is there two points along the circular loop that have a potential difference? Doesn't the current move uniformly throughout the wire?

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  • $\begingroup$ $\mathbf E =-\nabla \phi - \partial \mathbf A /\partial t$ $\endgroup$
    – Mauricio
    Mar 9, 2023 at 13:45

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The sentence "since voltage is the electric potential difference between two points" is not correct. Note that the electric field has two contributions. The conservative part, associated with electric charge accumulations (whose integration between two points gives place to the potential difference and do not exist in this problem) and the non-conservative contribution or the induction part, associated with the time derivative of the vector potential. The voltage between two points corresponds to the sum of the two line integrals between those points. The first one is independent of the chosen path while the second one depends. In this problem, we have a conducting loop equipotential and the current is uniform and flows in the clockwise direction. A voltmeter connected between the top (positive terminal) and the bottom (negative terminal) of the loop will read a positive voltage if the wires of the voltmeter are on the right hand side and out of the loop area. However, it will read a negative voltage if the wires of the voltmeter are on the left hand side and also out of the loop area. Moreover, if the wires form a rectilinear path between the top and the bottom points, the voltmeter will mark zero, but if the wires do not form a rectilinear path between these points the voltmeter will no longer read zero.

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Induced voltage is a misleading and ambiguous term. By voltage, some people mean net effect of induced EMF and potential difference, some people reserve it for potential difference (which we can't easily determine here, as potential depends on details not stated, such as how the loop gets deformed, geometry and conductive properties of the source of magnetic field, etc).

It is better to talk explicitly about induced EMF in this case. Induced EMF is the line integral of net induced electric force acting on the current, over the whole closed circuit:

$$ \mathscr{E}_i = \oint_{circuit} \mathbf E_i \cdot d\mathbf s. $$

It is the work the induced electric field would do on a unit charge if the charge got displaced along the circuit and made one complete turn.

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