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All the questions that have asked relating to this topic have been too advanced for my high school-physics mind, but I am still interested in knowing it so here goes:

I always thought that current in a circuit would flow from negative to positive, since negative electrons would move to the positive terminal in order to 'balance' out the charge (sorry for my lack of scientific terminology). However, I recently learned that current flows from +ve to -ve. Why is this the case? It doesn't make any since to me, and when I tried to find a reason online, it said that there was even some debate as to whether this is the case and/or why it is the case. Is this true? If so, why? Could anyone explain this to me please.

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Both are true. Electrons come out of the negative part of the battery and negative potential and flow to the positive potential.

However, current density is defined by $J = \rho v $ where $\rho$ is the charge density. The charge of electrons is negative so by definition, the current (density) J is in the opposite direction of the velocity of the particles $v$.

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You have to be clear of what you're talking about.

The particle current is generally from negative voltages to positive voltages, in the strictest sense. This is because an electron loses electric potential energy as voltage increases. This sense of "current" also stops on a capacitor, charging it up.

However sometimes, in many semiconductors, there are a bunch of electron states "filled up to the brim" with electrons, except for a very few states which are missing electrons. The filled states no longer interact much with each other because of Pauli exclusion, but these "holes" in the electron states can be treated as their own particles, with positive charge, moving through the semiconductor. These will also stop on a capacitor. Technically it's the electrons moving the opposite way, but if you can relax your mind on what you think a particle is, you will do better to think of holes as particles driving the charge-transport.

Finally, there is electric current. Electric current does not stop at a capacitor, it travels through it until the capacitor charges up. Even though a capacitor is literally a break in a circuit and particles cannot travel across it, current can. It can do this because electric current can be carried both by charged particles (and has magnitude of (particles / second / cross-sectional area) * (cross-section area) * (charge / particle)) and changing electric fields (where it has magnitude of (electric constant) * (change in electric field / second) * (cross-section area)). In a capacitor, an electric field is increasing in the direction that electric current is flowing, and this electric field carries the current even though the particles cannot flow through it.

With that said, this definition of electric current cannot stop and accumulate anywhere, which is why circuits must always be circuitous: they must always loop around on themselves. This definition of electric current cannot stop anywhere so it must always flow in loops. But electric charge routinely accumulates in places, so it is sometimes helpful to just consider the particle current times the charge-per-particle, and call that the "electric current", and then just accept that it doesn't "really" pass through a capacitor but only "virtually" does via this displacement current, leading to the buildup of charge on the capacitor. I have seen both interpretations being used by professors and researchers at various times.

Anyway: electrons have negative charge, so that (charge / particle) term is negative, and the electric-current for electrons points opposite to their particle-current. So we would say that the "current" goes from the + terminal to the - terminal even though that's really electrons going from - to +.

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