[Physics] Electrostatic notion of voltage as it applies to circuits


I have a question that's been bothering me about electric fields, voltage, and circuit analysis.

Initially, I came to understand voltage as it was taught in the context of electrostatics – through energy analysis. The concept of the electric field as a conservative field makes a lot of sense. No matter what path a charge takes, if it travels from a positive charge to a negative charge in any way (in a dipole situation), it has the same difference in voltage, namely, the voltage between the positive and negative charge.

This concept of voltage seems to take a huge leap when we jump to considering circuits.

The terminals of a battery create an electric field, which tends to move charge carriers from the higher to the lower voltage.

But how can the circuits be of absolutely any shape? If I have an extremely long, convoluted wire, how can we analyze voltage as simply as we can? Imagine a practical situation of a lamp plugged into a wall outlet far across the room. Circuit analysis says that the wall outlet is "at 120V" (AC) and the voltage drops through all resistance in the circuit on its way back to the ground wire. I just have trouble matching up this notion of voltage with the notion of voltage at a point in an electric field. On the other hand, the only way the charges in a circuit move is because of an electric field. Is the electric field just conveniently shaped such that it moves charges along the circuit at every point? What if the wire goes "uphill" and turns momentarily back toward the positive battery terminal but then turns around again. Wouldn't the charges get stuck in the local potential energy well?

A simple circuit

Best Answer

The plates of a charged battery create an electric field in the space around the circuit and in the circuit itself. At every point in the wire, this field "instructs" charged particles to redistribute themselves. This redistribution gives rise to another electric field that eventually combines with the battery's field to create a net uniform electric field inside the wire. It is this final uniform electric field which drives the current (causes mobile charge carriers within the wire to move). This feedback process is driven by surface charge redistributing itself on the wire's surface. The most astounding thing is that this entire process happens every time the wire's shape changes, and it happen VERY quickly. See the relevant chapters of Matter & Interactions by Chabay and Sherwood (3rd edition, Wiley, 2011) for details. This the only introductory text I know of that deals with surface charge gradients in circuits, and thus is the only one that correctly explains the underlying physics.

By the way, one shouldn't use the word "voltage" because it can mean either electric potential or electric potential difference. The two are very similar, and are related, but are not the same thing. You would never ask someone their "yearage" would you? It's never appropriate to use a unit in place of the quantity that carries that unit. Voltage shouldn't be used. Use either electric potential or electric potential difference for clarity.

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