Matter & Interactions II, Week 6

I’m writing this a whole week late due, in part, to having been away at an AAPT meeting and having to plan and execute a large regional meeting of amateur astronomers.

This week was all about the concept of electric potential and how it relates to electric field. I love telling students that this topic is “potentially confusing” becuase the word “potential” comes up in two different contexts. The first is in the context of potential energy. Potential energy, which I try very hard to call interaction energy, is a property of a system, not of an individual entity. There must be at least two interacting entities to correctly speak of interaction energy. Following Hecht [reference needed], I like to think of energy, and thus interaction energy, as a way of describing change in a system using scalars rather than vectors. Conservative forces, like gravitational and electric forces, can be described with scalar energies and fortunately, these forces play a central role in introductory physics. The second context is that of electric potential, a new quantity that is the quotient of a change in electric potential energy and the amount of charge that gets moved around as a result of an interaction. The distinction between the two contexts is subtle but very important.

Oh and speaking of potential or interacting energy, Matter & Interactions is the only textbook I know of that correctly shows the origins of The World’s Most Annoying Negative Sign (TWMANS) and how it relates to potential energy. When you write the total change in your system’s energy, you can attribute it to work done by internal forces and work done by external forces. When you rearrange this expression to put all the internal terms on the lefthand side and all the external terms on the righthand side, you pick a negative sign that goes on to become TWMANS. This term with the negative sign, which is nothing more than the oppositve of the work done by forces internal to the system, is DEFINED to be the change in potential energy for the system. It’s just that simple, but this little negative sign caused me so much grief in both undergrad and graduate courses. Some authors explicitly included it, other didn’t, and instead flipped the integration limits on integrals to account for it. Chabay and Sherwood include it explicitly and consistently and there should be no trouble in knowing when and where it’s needed.

There is also some interesting mathematics in this chapter. Line integrals and gradients are everywhere and we see they are intimately related. In fact, they are inverses of each other. I want to talk about one mathematical issue in particular, though, and that is within the context of the following problem statement:

Given a region of space where there is a uniform electric field \vec{E} and a potential difference \Delta V between two points separated by displacement \Delta \vec{r}, calculate the magnitude of the electric field \lVert \vec{E}\rVert.

This problem amounts to “unwrapping” a dot product (in this case \Delta V = -\vec{E}\bullet\Delta\vec{r} ), something the textbooks, to my knowledge, never demonstrate how to do. My experience is that student inevitably treat the dot product as scalar multiplication and attempt to divide by \Delta\vec{r} and of course dividing by a vector isn’t defined in Gibbsian vector analysis. I think the only permanent cure for this problem is to take a more formal approach to introducing vectors and dot products earlier in the course but I tend to think I’m in the minority on that, and I don’t really care. The problem needs to be addressed one way or the other. Solving either a dot product or a cross product for an unknown vector requires knowledge of two quantites (the unknown’s dot product with a known vector and the unknown’s cross product with a known vector OR the unknown’s divergence and curl) as constraints on the solution. Fortunately, at this point in the course we’re dealing with static electric fields, which have no curl (\nabla\times\vec{E}=0) or equivalently (I think) \vec{E} is collinear with \nabla V (differeing in signs because gradient points in the direction of increasing potential (I don’t like saying that for some reason…) and electric field points in the direction of decreasing potential) so we can find something about \vec{E} from just a dot product alone. So, students need to solve \Delta V = -\vec{E}\bullet\Delta\vec{r} for \lVert\vec{E}\rVert. Here’s the beginning of the solution. The first trick is to express the righthand side in terms of scalars.

\Delta V = -\vec{E}\bullet\Delta\vec{r}

\Delta V = -\lVert\vec{E}\rVert \lVert\Delta\vec{r}\rVert\cos\theta

\lVert\vec{E}\rVert = -\dfrac{\Delta V}{\lVert\Delta\vec{r}\rVert\cos\theta}

We have a slight problem, and that is the lefthand side is a vector magnitude and thus is always positive. We must ensure that the righthand side is always positive. I see two ways to do this. If \vec{E} and \Delta\vec{r} are parallel (\theta=0) then \Delta V must represent a negative number and TWMANS will ensure that we get a positive value for the righthand side, and thus also for the lefthand side. If \vec{E} and \Delta\vec{r} are antiparallel (\theta=\pi) then \Delta V must represent a positive number and TWMANS, along with the trig function, will ensure that we get a positive value for the righthand side, and again also for the lefthand side. I want to install this kind of deep, geometric reasoning in my students but I’m finding that it’s rather difficult. Their approach is to simply take the absolute value of the righthand side.

\lVert\vec{E}\rVert = \left\lvert-\dfrac{\Delta V}{\lVert\Delta\vec{r}\rVert\cos\theta}\right\rvert

It works numerically of course, but bypasses the physics in my opinion. There’s one more thing I want students to see here, and that is the connection to the concept of gradient. Somehow, they need to see

\lVert\vec{E}\rVert = -\dfrac{\Delta V}{\lVert\Delta\vec{r}\rVert\cos\theta}

as

E_x = -\dfrac{\partial V}{\partial x}

and I think this can be done if we think about the role of the trig function here, which tells us how much of \Delta\vec{r} is parallel to \vec{E}, and remembering that the component label x is really just an arbitrary label for a particular direction. We could just as well use y, z, or any other label. We must be careful about signs here too, because the sign of E_x must be consistent with the geometry relative to the displacement.

As an aside, it kinda irks me that position vectors seem to be the only vectors for which we label the components with coordinates. I don’t know why that bothers me so much, but it does. Seems to me we should use r_x rather than just x and there’s probably a deep reason for this, but I’ve yet to stumble onto it. Perhaps it’s just as simple as noticing that a position’s components are coordinates. Is it that simple?

As always, feedback is welcome.



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