# 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.

# Matter & Interactions II, Week 10

Chpater 18. Circuits. You don’t need resistance. You don’t need Ohm’s law. All you need is the fact that charged particles respond to electric fields created by other charged particles. It’s just that simple.

When I took my first electromagnetism course, I felt stupid becuase I never could just look at a circuit and tell what was in series and what was in parallel. And the cube of resistors…well I still have bad dreams about that. One thing I know now that I didn’t know then is that according to traditional textbooks, circuits simply should not work. Ideal wires don’t exist, and neither do ideal batteries nor ideal light bulbs. Fringe fields, however, do indeed exist and capacitors just wouldn’t work without them. So basically, I now know that the traditional textbook treatment of circuits is not just flawed, but deeply flawed to the point of being unrealistic.

Enter Matter & Interactions. M&I’s approach to circuits invokes the concept of a surface charge gradient to establish a uniform electric field inside the circuit, which drives the current. This was tough to wrap my brain around at first, but now I really think it should be the new standard mainstream explanation for circuits in physics textbooks. the concept of resistance isn’t necessary. It’s there, but not in its usual macroscopic form. M&I treats circuits from a purely microscopic point of view with fundamental parameters like mobile electron number density, electron mobility, and conductivity and geometry in the form of wire length and cross sectional area. Combine these with charge conservation (in the form of the “node rule”) and energy conservation per charge (in the form of the “loop rule”) and that’s all you need. That’s ALL you need. No more “total resistance” and “total current” nonsense either. In its place is a tight, coherent, and internally consistent framework where the sought after quantities are the steady state electric field in each part of the circuit and the resulting current in each part. No more remembering that series resistors simply add and parallel resistors add reciprocally. Far more intuitive is the essentially directly observable fact that putting resistors in series is effectively the same as increasing the filament length and putting resistors in parallel is effectively the same as increasing the circuit’s cross sectional area. It’s so simple, like physics is supposed to be.

Of course, in the next chapter (chapter 19) the traditional “Ohm’s law” model of circuits is seen to be emergent from chapter 18’s microscopic description, but honestly, I see no reason to dwell on this. Most of my students are going to become engineers anyway, and they’ll have their own yearlong circuit courses in which they’ll learn all the necessary details from the engineering perspective. For now, they’re much better off understanding how circuits REALLY work and if they do, they’ll be far ahead of me when I was in their shoes as an introductory student, and will have the deepest understanding of anyone else in their classes after transferring. That’s my main goal after all.

Feedback welcome.

# Conceptual Understanding in Introductory Physics XIX

This is a very quick post addressing a frequently asked conceptual question. Maybe it my heightened awareness, but I’ve also seen this question asked a lot on various physics Q&A sites lately. It’s a question that gets to the heart of how vectors are often defined, loosely and incorrectly, in introductory physics. Here’s the question.

Given that current (either electron current or conventional current) has both magnitude and direction, why is it not, and indeed cannot be, defined as a vector quantity whereas the closely related quantity current density is defined as a vector quantity.

The answer, I think, lies in one single simple property of vectors that does not apply to current. What do your students say?

# Conceptual Understanding in Introductory Physics XIV

This post continues this series into second semester introductory calculus-based physics, usually electromagnetic theory. This question addresses basic DC circuits. In a traditional introductory e&m course, circuits are presented with so many idealizations that according to such treatments simple circuits shouldn’t work at all! Two of the most important idealizations are that potential differences along wires are neglected so there are effectively no wires, and that fringe fields associated with real capacitors are neglected so capacitors effectively don’t work. These same treatments are also based on the concepts of potential difference and resistance, which is fine for an engineering perspective. However, all of DC circuit analysis can be done more realistically by adopting a surface charge gradient model that emphasizes some deceptively fundamental physics: charged particles act in response to local electric fields. That’s the whole story really.

Here are some conceptual questions that may be quite difficult for students of the traditional approach to circuits.

(a) Two circuits have identical wires, identical batteries, and identical light bulbs. The only difference between the circuits is that one has two bulbs in series with the battery and the other has only one bulb in series with the battery. Why does the single bulb glow brighter than either of the two bulbs in the other circuit? Do not invoke the concept of resistance (traditional treatments say that adding a second identical bulb doubles the circuit’s effective resistance, but do not use this concept in your explanation).

(b) Two circuits have identical wires, identical batteries, and identical light bulbs. The only difference between the circuits is that one has two bulbs in parallel with the battery and the other has only one bulb in parallel (really in series I suppose) with the battery. Why does the single bulb glow with the same brightness as either of the two bulbs in the other circuit? Do not invoke the concept of resistance (traditional treatments say that adding a second identical bulb doubles the circuits effective resistance, but do not use this concept in your explanation).

(c) What effect on a circuit does adding two identical bulbs in series, compared to having just one bulb in series, have?

(d) What effect on a circuit does adding two identical bulbs in parallel, compared to having just one bulb, have?

(e) What effect on a circuit does adding two identical capacitors in series, compared to having just one capacitor in series, have?

(f) What effect on a circuit does adding two identical capacitors in parallel, compared to having just one capacitor, have?

Have fun!