# Effect of an external electric field on a electric current

I'm wondering how an external electric field influences an electric current in a conducting wire and how could we measure this effect.

Without an external electric field, the conducting medium resistance, the current passing through it, and the potential difference applied to its extremities can be related by the formula

$$\begin{equation} U=R\,i\,. \end{equation}$$

If an external electric field is applied and it opposes the movement of the charge carriers this will be perceived as an increase in the medium resistance, right? Conversely, if its orientation is favorable to the movement of the charge carriers it will be perceived as a decrease in the resistance. If $$U'$$ is the electric potential of such an external electric field, then we have that

$$\begin{eqnarray} U+U' &=& (R+R')\,i\, \\ U_{\mathrm{eff}} &=& R_{_{\mathrm{eff}}}\,i, \end{eqnarray}$$

$$\begin{eqnarray} U' = (R_{\mathrm{eff}} - R_{\mathrm{nom}})i\,, \end{eqnarray}$$

where $$R_{\mathrm{nom}}$$ is the nominal resistance, measured without the external electric field. So, with two measurements, one with and one without the external electric field, it is possible to determine the electric potential and, consequently, the external electric field amplitude and orientation. Is this correct?

Thank you very much.

$$U + U' = R i'$$