# Matter & Interactions II, Week 11

More with circuits, and this time capacitors, and the brilliantly simple description M&I provides for their behavior. In chapter 19, we see that traditional textbooks have misled students in a very serious way regarding the behavior of capacitors. Those “other” textbooks neglect fringe fields. Ultimately, and unfortunately, this means that capacitors should not work at all! The reason becomes obvious in chapter 19 of M&I. We see that in a circuit consisting of a charged capacitor and and a resistor, it’s the capacitor’s fringe field that initiates the redistribution of surface charge that, in turn, establishes the electric field inside the wire that drives the current. The fringe field plays the same role that a battery’s field plays in a circuit with a flashlight bulb and battery. It initiates the charge redistribution transient interval. As you may have already guessed, the capacitor’s fringe field is what stops the charging process for an (initially) uncharged capacitor in series with a battery. As the capacitor charges, its fringe field increases and counters the electric field of the redistributed surface charges, thus decreasing the net field with time. If we want functional circuits, we simply cannot neglect fringe fields.

Ultimately, the M&I model for circuits amounts to the reality that a circuit’s behavior is entirely due to surface charge redistributing itself along the circuit’s surface in such a way as to create a steady state or a quasisteady state. It’s just that simple. You don’t need potential difference. You don’t need resistance. You don’t need Ohm’s law. You only need charged particles and electric fields.

One thing keeps bothering me though. Consider one flashlight bulb in series with a battery. The circuit draws a certain current $i_1$ for example. Now, consider adding nothing but a second, identical flashlight bulb in parallel with the first one. Each bulb’s brightness should be very nearly the same as that of the original bulb. The parallel circuit draws twice the current of the original lone bulb $i_2 = 2i_1$ but that doubled current is divided equally between the two parallel flashlight bulbs. That’s all perfectly logical, and I can correctly derive this result algebraically. I end up with a factor of 2 multiplying the product of either bulb’s fliament’s electron number density, cross sectional area, and electron mobility.

$i_2 \propto 2nAu$

My uneasiness is over the quantity to which we should assign the factor of 2. A desktop experiment in chapter 18 that establishes we get a greater current in a wire when the wire’s cross sectional area increases. Good. However, in putting two bulbs in parallel is it really obvious that the effective cross sectional area of the entire circuit has doubled? It’s not so obvious to me because the cross sectional area can possibly only double by virtue of adding an identical flashlight bulb in parallel with the first one. Unlike the experiment I mentioned, nothing about the wires in the circuit change. Adding a second bulb surely doesn’t change the wire’s mobile electron number density; that’s silly. Adding a second bulb also surely doesn’t change the wire’s electron mobility; that’s equally silly. Well, that leaves the cross sectional area to which we could assign the factor of 2, but it’s not obvious to me that this is so obvious. One student pointed out that the factor of 2 probably shouldn’t be thought of as “assigned to” any particular variable but rather to the quantity $nAu$ as a whole. This immediately reminded me of the relativistic expression for a particle’s momentum $\vec{p} = \gamma m \vec{v}$ where, despite stubborn authors who refuse to actually read Einstein’s work, the $\gamma$ applies to the quantity as a whole and not merely to the mass.

So, my question boils down to whether or not there is an obvious way to “assign” the factor of 2 to the cross sectional area. I welcome comments, discussion, and feedback.