# A Potentially Electric QuestionOctober 3, 2013 2:40 PM   Subscribe

What is "electric potential"? I've dug through the internet, gone to tutoring, asked my teacher, and nothing has helped.

I know that it's the potential energy per unit charge. I've read some semi-helpful explanations making an analogy to gravity and gravitational potential energy (which I have no problem with), but it breaks down for me when potential equals zero equidistant between two equal, but opposite, charges. How can this be? I'd assume the potential would double. Any helpful hints on how to think about potential and what it means?
posted by runcibleshaw to Science & Nature (26 answers total) 5 users marked this as a favorite

The analogy to gravitational potential breaks down when you note that gravity is always attractive, so moving one mass away from another (say, moving a weight away from the surface of the earth) always increases gravitational potential energy, and moving two masses towards each other always decreases gravitational potential energy. But electric force, unlike gravitational force, is sometimes attractive (between opposite charges) and sometimes repulsive (between like charges).

Just like moving two masses that attract each other (i.e., any two masses) apart increases gravitational potential energy, moving two electric charges that attract each other (opposite charges) apart increases electric potential energy. But for two electric charges that repel each other (like charges), moving them towards each other increases electric potential energy, and moving them apart decreases electric potential energy.

When we define electric potential as "potential energy per unit charge," keep in mind that, by convention, that means electric potential per positive unit charge. So the electric potential near a positive charge is positive (it would take energy to bring another positive charge near it) and electric potential near a negative charge is negative (energy would be released by bringing a positive charge near it). And halfway between positive and negative charges of equal magnitude, the potential is zero—to bring a positive charge from a hypothetical infinite distance to that halfway point would neither require nor release energy.
posted by DevilsAdvocate at 2:59 PM on October 3, 2013 [1 favorite]

Ok, so, one way to define the potential energy of a charge is the amount of work it takes to bring in the charge from infinity. Along the line of equidistance between two equal and opposite charges (a dipole), the direction of the electric field always points from the positive charge to the negative charge. So, if you start infinitely far away (E=F=0), but along that line, and continue to move in along that line, then the total work (dot product of F and dx) is 0, because the movement direction and the force are perpendicular to each other. So, if we define 0 as the potential at infinity, we've done no work to get it anywhere else on the equidistance line, so *boom*, zero potential everywhere. To get from that potential energy to abstract potential, just divide by charge. It's worth talking about potentials when you start to consider properties of fields, rather than properties of objects.
posted by Maecenas at 3:01 PM on October 3, 2013 [1 favorite]

I don't know how helpful this answer will be, as my understanding of potential comes from electrical safety training for non-electrician workers, but the important thing is that potential only has meaning when measured between two points, and is a measurement of how much the electrons at one point want to zap over to the other point.

So, for this zero potential when equidistant business, you're talking about measuring potential between two points that are both equidistant from the positive and negative charges.
posted by radwolf76 at 3:08 PM on October 3, 2013 [2 favorites]

You might find this simulation of potentials and electric fields helpful in visualizing the electric potential created by two or more charges. Positive charges create positive potential around them, and negative charges create negative potential around them. Add together those positive and negative potentials, and sometimes the answer is zero (white, on the simulation.)

Remember that the value of the potential at a single point tells you almost nothing. All that matters is that positive charges want to move toward lower potentials, and negative charges want to move toward higher potentials.
posted by BrashTech at 3:18 PM on October 3, 2013 [1 favorite]

I guess where I'm getting messed up is that if I were to actually put a test charge there, it wouldn't do what you would expect, which is just sit there, it would rocket off away from the positive charge and toward the negative charge. If the gravitational potential energy per unit mass were zero you'd expect that mass to just sit there like a lump.
posted by runcibleshaw at 3:19 PM on October 3, 2013

I think you're mixing up potential and force. Bringing a charge from infinity to that midpoint requires no net energy but the location resembles the point where a mountain and a canyon meet - yes, you are at zero elevation from sea level, and the net energy getting there from someplace else at sea level is zero, but what will cause you to roll down into the canyon is not your potential (zero) but the *slope* of the potential (sharply downward). Ditto that midpoint - value is zero, slope (not zero) is what generates the force.
posted by range at 3:32 PM on October 3, 2013 [3 favorites]

Picture a hill. Nice round-topped conical hill. Halfway up that hill, someone's painted a circle around the hill. Pick any two points on that circle. The likelihood that "gravity" would pull something from point A on that circle to point B on that circle is zero, because all the points on that circle are the same altitude. Doesn't stop things from wanting to roll downhill.
posted by radwolf76 at 3:34 PM on October 3, 2013

For the gravity analogy to work you need a positive and a negative gravity. So let's say you have a planet that attracts mass and an anti-planet that repels mass. There is strong anti-gravity near the anti-planet (gravity < 0). There is strong gravity near the planet (gravity > 0). Somewhere in between the two, it's balanced out so net gravity is zero, but mass still moves towards the planet, because both gravity and antigravity create force in that direction.
posted by PercussivePaul at 3:53 PM on October 3, 2013

"Zero" has next to no meaning for potential or potential energy. We prefer to set up potentials to go to zero at infinity, because this tidies things up mathematically, but if you're looking at particular (not-at-infinity) point, it doesn't matter if the potential there has a value of zero, or 100 V, or a million V, or -3.26 V at a particular point. All that matters is what the other values of the potential are, near that point. Potential and potential energy can be negative as well as positive, and the zero point isn't special in any physical way.
posted by BrashTech at 4:03 PM on October 3, 2013 [1 favorite]

Yeah, I guess I still don't understand what potential represents then. "The amount of force required to move something from infinity" is pretty abstract. Is this stuff actually this hard to understand, or am I just being dumb?
posted by runcibleshaw at 4:19 PM on October 3, 2013

If the gravitational potential energy per unit mass were zero you'd expect that mass to just sit there like a lump.

Not if it could move somewhere with negative gravitational potential. As radwolf76 gets at, potential is relative. We can define potential to be zero at some specific point—usually at a place that makes the math work out conveniently—and measure other potentials relative to that. Depending what we define "zero" to be, some potentials may be negative.

Near the surface of the earth, gravitational potential energy can be defined as mgh, where m is the mass of the object, g is the acceleration due to gravity (9.8m/s2), and h is the height of the object. Which makes gravitational potential equal to gh.

But wait...when I said "height," I meant height relative to what? If you're doing a high school physics experiment, you might decide that h is the height above the surface of the table you're working on, or you might say it's the height above the floor. For larger scale (but still near the surface of the earth) projects you might decide that "height" is "height above sea level."

But it doesn't matter which you choose as your reference height, as long as you're consistent about it, because all that matters is the difference between gravitational potential between one point and another. And depending on what you choose, the potential may be negative at some points. If you choose the surface of the table you're working on as your reference "zero" height, than the floor it's sitting on has negative height and negative gravitational potential. A ball sitting on the table has zero gravitational potential energy, if that's your reference, but it can still roll off the table and onto the floor, because the floor has negative gravitational potential, compared to your reference. Masses don't just want to move to zero gravitational potential, they want to move to the lowest possible gravitational potential, and negative numbers are less than zero.

Now, if you're working with things like planets and stars and rockets, it turns out that it's convenient to define a hypothetical infinite distance away as your "zero" gravitational potential—and when you do that, every point within a finite distance of a mass has negative gravitational potential!

Similarly, when you're dealing with electric potential, you can choose whatever "zero" you want. If you have a 9-volt battery and want to assign electric potentials to the terminals, you can say the positive terminal is +9V and the negative terminal is 0V, or you can say the positive terminal is 0V and the negative terminal is -9V, or you can say the positive terminal is +4.5V and the negative is -4.5V, or you can say the positive terminal is -13.3V and the negative terminal is -22.3V, because the important thing is the difference between the terminals. The math all works out the same in the end, but some of the choices will make the math easier to work out in the middle than other choices.

Just like gravity in outer space, it turns out to often be convenient to define the electric potential an infinite distance away as zero; and if you do that, the potential at the point halfway between two equal but opposite charges is also zero. That doesn't mean that charges placed at that point don't move; a positive charge placed there will move towards the negative charge, because points closer to the negative charge have negative potential, and positive charges want to move towards the lowest possible electrical potential—which in this case is less than zero. Similarly, a negative charge placed at that halfway point will move towards the positive charge, because negative charges want to move towards the highest possible electric potential, and the electric potential is highest closest to the positive charge.
posted by DevilsAdvocate at 4:20 PM on October 3, 2013 [1 favorite]

Is this stuff actually this hard to understand, or am I just being dumb?

Sometimes these concepts just don't translate well to analogy for some people. Make sure you understand the math, and the intuitive understanding will come secondary to that.
posted by no regrets, coyote at 4:27 PM on October 3, 2013

"The amount of force required to move something from infinity" is pretty abstract.

It isn't even the amount of force, it's the integral of the force with respect to the distance. So it's even abstracter than you thought.

The concept of work and potential is something every physics student struggles mightily with. Apart from a few really exceptional people, I think successful Physics 101 students make peace with some kind of analogy of potential being like a hill. I don't think people grasp work or the true idea behind potential on a very deep level until upper-level physics classes, if then. (I didn't "get it" well enough to teach it confidently until after I finished my PhD.)
posted by BrashTech at 4:41 PM on October 3, 2013

You're probably familiar with the 2-D "rubber sheet" analogy for gravitation (usually in cheesy graphics of black holes). That is just the gravitational potential.

The electric potential is the same except that it can go both "up" and "down" (in the 2D rubber sheet model.) So if you have a "well" and a "hill" of the same size: halfway between them, is it intuitively clear that the sheet is at the 0 level?
posted by phliar at 5:03 PM on October 3, 2013 [1 favorite]

I guess where I'm getting messed up is that if I were to actually put a test charge there, it wouldn't do what you would expect, which is just sit there, it would rocket off away from the positive charge and toward the negative charge.

Potential is a scalar field. The test charge reacts to the electric field, which is a vector field (the gradient of the potential field).

Think a topographic map. Potential would be height, the electric field would be slope, and the test charge a rolling ball.

Going back to your two charges thought experiment, imagine swapping the charges. The potential in the center would be unchanged. But the electric field would be reverses and the test charge would zip off in the opposite direction.
posted by ryanrs at 5:05 PM on October 3, 2013

The concept of work and potential is something every physics student struggles mightily with.

That's because it is taught to students before they have learned vector calculus.
posted by ryanrs at 5:10 PM on October 3, 2013

Actually, I should pay more attention when quoting people. What I should have said is that vector calculus is the necessary tool to understand electromagnetism. Without it, you struggle with not-quite-right analogies. Vector calculus is to electromagnetics as calculus is to mechanics.
posted by ryanrs at 5:22 PM on October 3, 2013

Okay, I still don't understand potential, and now I think I may have now de-understood gravity. Crap.

Let me write it out, because maybe it will help.

Electric potential is potential energy per unit charge. In the case of the volt, the charge is 1 Coulomb.
So far, so far.
Potential energy is... um... hold on... well it's the force required to move something from one point to another?

So, how can you pick just one point and say that it has potential energy per unit charge? Don't you need two points?

Everything I'm reading is talking about electric potential difference between two points...
posted by runcibleshaw at 5:52 PM on October 3, 2013

Not the force to move something from one place to another. The energy. Which is the integral of the force along the path from point A to point B.

So here's the thing about moving something to point B that is "infinitely far away": all points "infinitely far away" are the pretty much same. When you get very far away, all the positive and negative charges in the universe more or less cancel out and look like a single point charge equal to the net charge of the universe. God damn that is a lot of handwaving, but you can see how any point infinitely far away is much like any other and you don't need to specify the exact position. That's your second point.
posted by ryanrs at 6:12 PM on October 3, 2013 [1 favorite]

So, how can you pick just one point and say that it has potential energy per unit charge? Don't you need two points?

Pretty much, yeah. It's the difference in potential between two points that's important. Sometimes that's the difference between two actual points (like the terminals of a battery), sometimes between an actual point and some theoretical construct (like a point an infinite distance away from everything else, let's call it point ∞).

Why would you care about the difference between an actual point and a theoretical construct? Well, if you know how to calculate the difference between point A and point ∞, and the difference between point B and point ∞, it's easy to calculate the difference between point A and point B.

And if some theoretical construct such as point ∞ becomes very widely used as one of the two points for measuring the difference between it and other points, eventually you reach the point where people just say "the electrical potential of point A" to mean "the difference in electrical potential between point A and point ∞."
posted by DevilsAdvocate at 6:15 PM on October 3, 2013

Note that in engineering, we don't use an infinitely distant point. We pick a convenient nearby point, label it "ground", and reference all potentials to that ground. (Since we measure potentials with a voltmeter and most test leads are only 3 feet long, such accomodations had to be made.)
posted by ryanrs at 6:22 PM on October 3, 2013 [1 favorite]

Seriously, the "infinitely distant point" construct is not a more fundamental reference point than say, the water pipe the grounds your house. But it's a lot easier to concisely state in a textbook physics problem. When your homework problem takes place in an otherwise empty universe with 2 point charges, you can't very well introduce a water pipe earth ground. The infinitely distant point is used for the same reason you use frictionless surfaces and massless springs.
posted by ryanrs at 6:40 PM on October 3, 2013

In the gravity analogy, potential is like elevation from sea level - the actual movement of objects depends on the slope of the hill, or the friction with the environment, or all kinds of stuff, but each point on the earth has an elevation, and by knowing that we can begin to calculate what will happen when you put an object somewhere.

It's like elevation, but it's not exactly the same. Here's what it exactly is: you've got a unit mass at sea level, and you raise it up one meter. The potential is the energy you spent raising it.

It's a lot easier, as with elevation, to figure out the difference between two points than to figure out their absolute elevation from sea level. Who the heck knows where sea level is? But it's easy to measure how much energy it takes to raise a weight from the bottom to the top of a building.

This "infinite point" business is just trying to figure out where sea level is. It's more of a theoretical question than anything else - like you say, any actual measurement is going to involve measuring the difference between two points. When engineers use "earth" as a reference point for zero, what they're really saying is that the earth is a big enough object that the charges in it probably even out to zero. That's a close enough approximation for, well, really anything except physicists, who like to define things exactly.

(It took me until mid-university to properly understand what electric potential was, and I think it's confusingly named, taught, and badly understood by most of the people that claimed to me to understand it.)
posted by spielzebub at 4:50 AM on October 4, 2013

As ryanrs says, you kind of need to know vector calculus to understand potential well. We used Swokowski's book in college for that and I thought it was pretty good.
posted by Monday, stony Monday at 5:53 AM on October 4, 2013

Thanks for all your answers, (non gender-specific) dudes. I have been so busy with school, work, and life that I didn't get a chance to pop back in since my last comment. I still don't really have a clue on the whole electric potential being zero/electric field not being zero thing, but we've moved on to other stuff, so I guess I'll just have to deal with it. Thanks.
posted by runcibleshaw at 8:16 PM on October 11, 2013

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