## Monday, April 18, 2011

### Fields: Mrs. and Electric

The electric field and electric potential are easily the most difficult subjects that I teach each year in the honors physics class.  They're conceptually more difficult than anything in the AP Mechanics C class, too, I'd wager.

The root issue is that, by the time we're to electric potential, we're about two levels away from reality (energy's not real, and V is the potential energy per unit charge) and and three levels from student experience (... and all of this electron and proton business is somewhat on the small side).

Fortunately, a good study break is all we need.  Let's surf on over to ohnuts.com and grab some pecans!  You know, for sandies.
Ah, yes... it's a tale of two pecans.  Well, maybe two bags of pecans.  You see, I have a bag of cinnamon pecans (bag A) and a bag of regular pecans (bag 1).  Which one cost more?

...

...

...well?

That's not fair of me, of course.  You don't know which cost more until I tell you the weight of each bag.  The price listed on the website, then, isn't really a price - it's something more abstract.  It's the price per unit weight.  The "per" part, even in this mundane example, is the hard bit.  We spend a whole year on "per" - it's called calculus.

Don't assume that your students really understand any sort of rate intuitively - that has to be part of the training that you're giving them!

So, each type of pecan has a price per unit weight (rho) and each bag of pecans has a weight (W).  Only at this point do we have a price (P): the cost of a bag is the product of the price per unit weight and the weight.
I called the price per unit weight rho, which seemed about as good as any symbol.  It's a cost density, in a way.  Anyway, it's the concept, rather than the notation, that's super-important here.

Hey, did you notice how the units told you exactly what to do there?
I mean, what else could you do to find the price from the price per unit weight?!

Let's shift this pecan business over to electrostatics:
 Quantity Gravity Electricity Field (gen. by point mass/charge) $\bg_white \frac{GM}{r^2} (=g)$ $\bg_white \frac{kQ}{r^2} (=E)$ Force (between points masses/charges) $\bg_white \frac{GMm}{r^2} (=mg)$ $\bg_white \frac{kQq}{r^2} (=qE)$ Potential (near Earth vs. due to point charge) $\bg_white gh$ $\bg_white \frac{kQ}{r} (=V)$ Potential Energy (near Earth vs. between point charges) $\bg_white mgh$ $\bg_white \frac{kQ}{r} (=qV)$

That's a mouthful of physics right there!  We'll come back to potential on another day, but the big idea to communicate is that:

The electric field is the force per unit charge, just as the pecans have a price per unit weight.

You don't know the price of a bag of pecans until you know its weight.  Plenty of the information that goes into determining the price is already there, though, in the price per unit weight.  The shipping, growing, and labor costs are all right there (and they pretty much are all per weight costs!), but there's one critical piece of information missing: how much are you buying?

Once you know the weight, the price is the product of the weight and the price per unit weight.

The same goes for the electric force.  The electric field at a point in space is the result of the influence of all of the charges in the area (technically, in the universe) - their sizes and layout in space.  That's a big amount of information, but you can't have a force until there's a charge put into that location.

Once you know the charge that you're putting at some location in space, the force on it is the product of the charge and the force per unit charge at that location in space.

That sentence is pretty awkward, so we name the "force per unit charge" electric field, and it becomes:

Once you know the charge that you're putting at some location in space, the force on it is the product of the charge and the electric field strength.

It seems a little awkward, but we've actually been doing it for some time.  The lovable old g is the gravitational field strength (near the Earth's surface).  How do you determine the gravitational force (weight) on something?  Well, you don't know until you know its mass, but information about the Earth's size and mass is packed into g.  Take the product of the gravitational field strength and the mass and you get... mg!  Same deal, different symbols.

1. Understanding F = Eq and E = Q/r^2 is definitely important, but since the most interesting aspects of electromagnetic fields deal with electromagnetic waves, which carry momentum and energy and can propagate in a vacuum, it might also be good to point out that the Coulomb law is only a very simple approximation, perhaps drawing an analogy to the difference between hydrostatics (water seeks its own level) and hydrodynamics (water sloshes back and forth, supports waves, etc.) Sometimes students who haven't been introduced to this distinction begin to think that electric fields are merely a trick for calculating forces.

2. Hmmm... That's something to think about. Once I get to magnetism, we do discover the connections among the constants and c, which is my (so far only) nod to EM waves. Your analogy's interesting, though!

3. I agree potential is hard for students, even for the university students. I often use gravity analogies, including "mgh" has an analogy for electric potential energy; and "gh" as the mass potential. I also try to talk about potential in terms of going up and down hill, where with you just have to remember that electrons oddly want to roll up hill.

Not sure how much it helps, but it gives them something to hang their hat on.

4. I hope that my students are reading! It's nice to hear it from someone other than me. :) That's a pretty go-to analogy for me, too. Maybe I should emphasize the concept of rolling down the PE hill back in energy, since it's a really important bit here (and is common for electrons and protons).