Why proton test charges and not electrons?

 Sci Advisor Thanks P: 2,438 This is just historical. For some reason one has defined what's now known to be carried by protons as the positive charge. There is a lot of confusion in the literature, talking about "technical direction of current" vs. "real direction of current" and similar gibberish. It's way easier to just use the vector-field concept to understand these issues. The best is to use even relativistic four-dimensional notation right away. E.g., the electric four-current of a fluid with number density $n_0$ of charge carriers is given by $$j^{\mu}=q n_0 u^{\mu}=q n_0 \gamma \begin{pmatrix}c \\ \vec{v} \end{pmatrix}, \quad \gamma=\frac{1}{\sqrt{1-\vec{v}^2/c^2}}.$$ Here, $q$ is the charge of one particle ($+e >0$ for protons, $-e<0$ for electrons), $c$ the speed of light, $n_0$ the density of the fluid as measured in the local rest frame of the fluid cell, and $\vec{v}$ the flow-velocity field. The sign of the total current through a cross section then is uniquely defined by the spatial components of this current-density vector and the orientation of the cross-sectional area: $$I=\int_{A} \mathrm{d}^2 \vec{A} \cdot \vec{j}.$$