A defining equation of electrochemical cells is that the EMF, $\mathcal{E} = E_{red/cat} - E_{an/ox}$, equals the electric potential of the cathode minus that of the anode.
This means that if a cell is discharging, $\mathcal{E}$ is positive and if it is charging, $\mathcal{E}$ should be negative. This is consistent with the relation $\Delta G = -nF\mathcal{E}$, since charging should be non-spontaneous and as such requires a negative EMF.
The EMF from A to B may be expressed as an integral
$$\mathcal{E} = -\int_{A}^{B} \vec{E} \cdot d\vec{l}$$
Electrochemically, A should always be at the anode and B always at the cathode (in a redox sense).
However, Wikipedia states that
The path is taken from the negative terminal to the positive terminal to yield a positive emf
Indeed, I've only seen positive values of volts labelled next to cells no matter their orientation. That is, in circuit theory, the EMF appears to be given as e.g. $+3V$ no matter whether the cell is charging or discharging.
This doesn't seem correct, since couldn't and shouldn't the EMF be negative during charging/when the cell is connected in the opposite direction relative to the positive current? I don't know how to reconcile the electrochemical definition of EMF with that given in Physics.
Why is it that in Physics, EMFs are only ever expressed as positive numbers - for instance, in labels on cells in circuit diagrams - even when the actual EMF of a cell is negative?
