Setting the Stage
In my last post I wrote about a theoretical approach to having students discover the concept of electric flux for a point charge. While they struggled with explaining what the proportionality constant for their Electric field vs. enclosed area graphs meant (aka, the flux), I was still satisfied with the results. What turned out to be the most useful question was asking them why I might have chosen a spherical surface to draw around the point charge as opposed to some other shape. After all, it’s a completely imaginary surface, so why this one?
They didn’t have much to say until we spent some additional time refining the equation we’d come up with:
Those of you more familiar with the concept of flux will recognize the missing cos(theta), so my next goal was to have them choose to add that term to the equation.
I started the discussion by holding up a whiteboard and told them to imagine a steady stream of wind flowing with a constant speed directly at me. I asked At what angle should I hold the whiteboard so that I could “harness” the wind stream as best I can?
It only took a few moments for them to realize that angling it completely perpendicular would allow them to maximize their “use” of the wind stream and that holding the board parallel would minimize it. I then asked about the wind then blowing at some in-between angle and put this picture up to help them visualize.
They first decided that this would yield some level of “usefulness” that was in-between the maximum and minimum values based on the angle between the vectors and the surface. Asking which “part” of the vector contributed to the “usefulness” led them to conclude that the perpendicular component of the vector was the only part that contributed. They were unable to see a difference between using the sin of the angle between the vertical and the vector and the cos of the angle perpendicular to the surface, which is fine because there isn’t one. I said that we should use the angle with the perpendicular only because that’s the convention, which will come in handy later. Their equation was now this:
Revisiting my original question
My original question was this: Why did I choose to draw a spherical surface around the point charge? Why not any other surface?
I put this picture back up on the board to help with visualization. I’m discovering more and more that the difficulties with flux and Gauss’ Law are primarily with visualizing the various surfaces and charge distributions, so any chance I have to aid in that I take.
They quickly determined that the field lines were perpendicular at each point of the circular surface, which was easily extended to a spherical one. Since each point was perpendicular, cos(theta) is just 1, thus the flux calculation is almost trivial.
Making the flux calculation trivial is one of the primary ways to effectively utilize Gauss’ Law. My purpose here was to continue their development of seeing spatial symmetries and using those symmetries to simplify otherwise complicated calculations.
Finalizing the flux equation
The equation they’d come up with was more generalized, but still not as general as it needs to be for both using and understanding Gauss’ Law at a deeper conceptual level. I also continue to emphasize that the more general a scientific model is, the more useful it becomes. So the issue with their equation is that it assumes a surface in which the field lines are perpendicular at each point of the surface. What if that’s not possible? How can we generalize the equation to work for any surface?
I drew a square around the point charge to help jump start the discussion.
The question now is this: How can we calculate the flux through the square?
I’m careful to clarify that I’m not looking for them to jump to a mathematical equation, but rather I want them to brainstorm on a conceptual way to do it. We’ll worry about the mathematics later. Just talk me through how you might calculate this.
I should mention that this isn’t the first time I’ve had them do this kind of exercise. I first exposed them to this when calculating electric fields of extended bodies, pre-flux/Gauss’ Law as a way to get them thinking in “calculus-mode.” Even though all of my students are in AP Calc BC, they haven’t gotten to integrals yet, but that doesn’t mean they can’t start “thinking calculus.” Writing this post now has me thinking that I’ll write up my approach to that. So I’ll do that at some point.
Anyway, when they got stuck, I asked them how they could get the contribution to the flux due to an individual field line and then reminded them that the total flux was the contribution due to all the field lines. They came up with this:
We just need to know the E at each point on the square, multiply it by the cos of the angle perpendicular to the square, add all that up, and then multiply by the area of the square.
I then asked them to write that out mathematically yielding this:
While that may cause a mathematician to cringe (and the Asquare part isn’t fully correct), I wasn’t worried about the correctness of their notation. The essence is there. This also let them see if E was the same at each point of the square and if cos(theta) was always 1, then those things could be pulled out of the sum, which will be useful once they get to Gauss’ Law. I then introduce the more formal definition of flux and made sure to show how each element of their equation translated to this definition:
I also briefly discuss the dot product, though I honestly haven’t done a great job of helping them build their understanding of it. I’m not sure that I care right now… but we’ll see if that comes back to bite me later one. I’m careful to emphasize that the only real differences between their equation and this one is that of notation. I explain what the circle on the integral means along with the S below the integral.
Next up: Gauss’ Law
My students still weren’t exactly sure why we we were spending so much time on this concept of flux. I’d mentioned several times that this would help make deriving expressions for electric fields much easier than the brute force method they’d done previously, but that doesn’t mean they see why this will be a path to that.
One student had alluded to looking at the relationship between the flux and the charge inside the surface, which is exactly what Gauss’ Law is, so I brought that point back up for discussion. In my next post, I’ll talk about how I had them approach investigating that and how they used it to come up with Gauss’ Law.