# Why is voltage described as potential energy per charge?

Voltage is often called an electromotive force since it causes a flow of charge. However, it is described in terms of Joules per Coulomb or Potential Energy per Charge.

Question: How does the potential energy associated with charge contribute to its effect on the flow of charge?

High voltage, or high electromotive force, causes high current. So this means charge with high PE will cause high current. This doesn't seem to make sense to me. Why does potential energy affect current?

I know of the PE gradient explanation, but this doesn't make sense to me. In most cases of diffusion, there is an explanation as to why it occurs: particles diffuse from high to low concentration DUE to random particle movement. Things fall from high to low gravitational PE DUE to the force of grav.

Question: Charge moves from high to low PE in a circuit, but why? What is the driving force?

Actually, voltage is a difference in potential energy per unit charge. Specifically, the voltage between two points A and B is the difference between how much PE a unit charge would have at A, and how much PE it would have at B. It's important that you always have two points in mind when talking about voltage, since it's technically meaningless to talk about the voltage at a single point.

Now, you presumably know that the difference in potential energy per unit charge between two points is related to the gradient of potential energy per unit charge between those points. $$V_B - V_A \equiv \Delta V_{BA} = \int_A^B \vec{\nabla} V(\vec{x})\cdot\mathrm{d}\vec{x}$$ where $V$ is potential energy per unit charge (a.k.a. "electric potential"), $\Delta V$ is voltage, and the integral is a line integral along the path from A to B. Well, the gradient of $V$ here is physically the same thing as the electric field. And as you probably know, when a charge is placed in an electric field, the field causes it to experience a force. That force is precisely $-q\vec{\nabla} V(\vec{x})$.

Given a path, if the difference $\Delta V_{BA}$ is large, then there will be a large gradient somewhere along the path, thus a large electric field, which causes a large force pushing the charged particles from A to B, which in turn makes the particles accelerate faster and move faster, producing a larger current.

• The implication here is that it's not the potential energy (or voltage at a point, or electric potential at a point) that causes the electron to move. It's the gradient of those quantities. – garyp Jun 23 '18 at 12:30
• I've heard of this gradient explanation before, but it seems lacking to me, because all other examples of diffusion have an explanation as to why they happen. Diffusion of particles occurs as a large mass of particles gradually separates DUE to random particle movement. Decrease in PE of an object happens as it falls to the ground DUE to the force of grav. I know that in a circuit charge moves from high to low PE, but DUE to what? I know of diffusion, but in most cases there is an explanation as to why it occurs, it isn't simply stated as the sole reason for an event. – Peter Blood Jun 24 '18 at 17:11
• @PeterBlood Gravity is the closes analogue. You say objects fall because of gravity force. Well, charges move due to electric force, which is very simmilar. $\vec{F}_g=-\vec{\nabla}{V_g}$, $\vec{F}_E=-\vec{\nabla}{V_E}$, – FGSUZ Jun 24 '18 at 17:37
• @PeterBlood I edited the answer, see if that helps. – David Z Jun 24 '18 at 21:41
• @DWade64 Thanks for catching that. I edited to address the first thing, but I think getting into the issue of potential itself being relative to a reference point would overcomplicate the answer. I just tried not to include anything that would imply otherwise. – David Z Jun 24 '18 at 22:06

Potential energy is just a way to talk about which way a force is pushing. That is why the force always pushes from high to low potential energy.

Think of a ball on a shelf. There is a lot of gravitational potential energy stored. Was it at the ground, there would be less potential energy stored. And the ball wants to fall as far down as possible - could it roll off the shelf and down, then it would.

The reason is that there is a force pulling downwards. From points higher to lower potential energy. This is always the case.

In electric circuits, if the total electric force at a point pulls leftwards, then the potential energy is lower when the charge moves leftwards.

In other words: potential energy is just another way to talk about how much forces pull. It is another way to indicate which way stuff will move if allowed to do so.

The reason for defining something called potential as potential energy per charge, is just an easier way to compare different points. Voltage is then a name for potential difference.

Charges don't posses potential energy. To be more accurate a system possesses potential energy. In electrostatics, you have to distinguish between potential, potential difference and potential energy.

First of all potential energy of a system refers to the amount of energy spent in assembling the system from infinity. Electric potential at a point refers to change in potential energy of the system when a unit charge is brought to that point. Difference in electric potential at two points is called potential difference. What drives current is potential difference.

Second, in mechanics, force is caused by change in potential energy (more precisely gradient in potential energy). When two points have high potential difference and a charge goes from high potential to low potential, there is tremendous decrease in potential energy. This causes larger force on the charges, since force is proportional to gradient of potential energy, and consequently higher current.

Suppose you have a positively charged plate of 10 coulombs and a positively charged plate of 5 coulombs, then the plate with 10 coulomb charge will have the more potential to attract or repel the nearby charges or you can say that the charges do have more potential energy in this case to move towards or away from the plate. 10 coulomb charged plate will exert more energy or energy per coulomb as compared to the 5 coulomb charged plate on the nearby charges and the charges will move faster towards the 10 coulomb charged plate as compared to the 5 coulomb charged plate and the current you know is the rate of flow of charge.

Just handle voltage and gravity the same. The "voltage force" is the electrostatic force, and "static electricity" drives the current in circuitry. Whenever voltage is non-zero, we have a static-electric force upon any electrons in the wires.

One way to understand this is: take a conceptual approach with "alternate mental toolbox," where the Volt isn't defined as a Joule per Coulomb. Instead, define the Joule in terms of Volts and Coulombs.

In that case, any altered energy of the system is proportional to the transport of a small charge Q across a potential-difference V. In other words, moving a small charged object against a potential-hill is the cause of potential energy. Knowing the voltage-pattern in space, we can calculate the PE of any moving charges. This is typically demonstrated by "charging" a parallel plate capacitor: by taking charges from one plate and forcing them against electrostatic repulsion, moving them to the other plate, which injects energy into the capacitor.

But but... what then does "voltage" mean?

In that case, voltage is a mathematical concept called "Potentials." It is not defined in terms of potential energy. Instead, it's the line-integral of the e-field flux. Instead, it is a way to describe e-fields, even if no test-charges are being moved, with no potential energy associated with test-charges. The e-field is a thing alone, and it still exists even when test-charges are all removed. Imagine an e-field hanging in empty space.

We can view fields as being made of thin fibers, "lines of force" or flux lines. But we can also view fields as being made of equipotential planes, like stacked pages of a book. If flux is the group of imaginary lines in an e-field, what are Potentials? They're the imaginary stacked-layers in an e-field. Either approach is valid, and these voltage-membranes are just as "real" and important as lines of force. We habitually describe e-fields in terms of flux lines because they're easy to draw on flat paper, while potential planes are difficult 3D objects. (Note that the flux lines are always perpendicular to the equipotential planes which they penetrate. E.g. flux lines around a charged particle are radial, while equipotential layers around the same particle will appear as nested spheres, like an onion.)

What is voltage? It's a stand-alone concept: "Potentials." Michael Faraday and JC Maxwell made much of this mathematical concept, back when the rest of the physics community shunned and rejected it. After all, fields didn't exist, and proper physicists only believed in Instant Action At A Distance. Faraday birthed the EM-fields concept, but was ignored. Maxwell put it on a firm mathematical foundation, and finally the fields could be considered as genuine physics-entities; strange objects hanging in empty space. The math concept of Potentials or "Voltage," is one approach to describing these entities.

In other words, voltage isn't a joule per coulomb. Voltage instead is one face of e-fields, when no infinitesimal test-charges are present.

PS, not convinced? If not, then ask yourself, can magnetic potential-fields exist in space, even when no test-poles are being moved around to create PE? And, is there still some gravity in the space above the ground, even when no boulders are being lifted (are gravity potential-fields really made out of Joules per Boulder? What if there is no boulder?) The potential-field still is there above the dirt, and the Volts are still hanging in empty space between the capacitor plates. Well, only if you reject the Distant-Action beliefs of the pre-Maxwell physics community, and allow fields to have genuine existence.