Potential vs. potential energy – a subtle difference

The idea of electric potential, which we often refer to as voltage, will become like second nature to you (if it isn't already), but there are some subtleties about potential that you should understand before we go on.

The first and most important is that potential not a potential energy, in the strict sense. Electric potential is defined as the potential energy of a particle divided by its charge. Therefore a +2 Coulomb (C) charged particle at one location in an electric field has half the potential as a +1 C particle at the same location.

We define potential that way because it is intended to describe a force field, and we don't want our description of the strength of that force field to depend on the charge in any way, so we divide by charge to make sure we're always talking about a single charge, which happens (by long-standing agreement) to be a the positive value of the electron charge. Another way to view the distinction is that we need to account for the greater force that will be felt and dealt by a multiply-charged particle in the same way that we do for a singly-charged one.

Let's backtrack a bit and talk about why things move and develop the idea of a force field.


Digression: How things move

Nothing moves in the universe unless one of two things are true:

  1. An object is acted upon by a contact force, like a foot kicking a soccer ball (notice the distortion of that ball — that's cool) or

  2. An object is subject to a gradient in a force field, like a gravitational field or an electrostatic field.

In the upper photo, the ball will experience a momentary force, but after it loses contact with the foot, it can recieve no more force from it, and it continues on only under the forces of gravity and air friction.

Other forces work at a distance, like gravity or electrostatic attraction or repulsion. Think about the attraction or repulsion of two magnets. They don't have to touch to exert force on each other, and we can say that there's a force field between them. However, there can be no movement unless the forces acting on an object are unbalanced.

In the case of the skydiver, he falls because the gravitational force is weaker above than below. That is, he follows the slope or gradient of gravitational strength as he falls. If the skydiver were the same distance from two planets with equal mass, there would be no net force on him (the pulling forces would be balanced) and he wouldn't move.


Fields

Consider the two illustrations below, a positive and a negative charge. The set of circles surrounding each are a kind of topographic map describing the force that another charge would feel if placed a certain distance from the central charge. The closer the lines, the greater the force, just like how the steepness of a mountain or valley slope is shown on a topographic map.

In each diagram, I've put in two positive "test charges" with force vectors on each to indicate the direction and relative size of the force they would "feel" because of that center charge. The positive charge repels our test charges, and that repulsion is greater the closer they get, thus the longer force vectors. The negative charge attracts our test charge, and that attraction increases as we move the test charge closer.



Here is a 3-D view of the same two scenarios. Now the strength of the attractive or repulsive force is modeled by the height of the circles. Positive charges are drawn into a potential well toward the negative center charge, and they fall down a "hill"

(a gradient) away from the positive center charge. The steeper the gradient, the stronger the force. It works the same for gravity; the farther you are from Earth, the less pulling force the planet exerts on you. The analogy with a topographical map is really pretty good.



For charges all of this behavior is modeled by Coulomb's law, an inverse-square law analogous to the universal law of gravitation. Here's a comparison of the two laws, two of the most important relationships in physics:

It's worth comparing these two force laws:

  • The constants G and k, the universal gravitation constant (G = 6.674 x 10-11 N·m2Kg-2) and Coulomb's constant (k = 8.98 x 109 N·m2C-2) respectively, are proportionality constants, and you can really think of them as just being there to get the units right.

  • In the ULG, we multiply masses; in Coulomb's law it's charges. Notice that mass is always a positive quantity, while charge can be negative or positive. That means that the force between charges can be negative (by convention that's attractive) or positive (repulsive). Recall that like charges repel and opposite charges attract. On the other hand, no one ever suddenly gets ejected from the surface of a planet because gravity becomes repulsive. It's only attractive.
  • Notice the sizes of the constants: 10-11 for G and about 1010 for k. For equal masses that carry equal charge, the Coulomb force (electrostatic force) is about 40 orders of magnitude (40 zeros!) stronger than the gravitational force.

  • Both forces are inversely proportional to the square of the distance between bodies, r. That means that if the distance between particles is halved, the force increases by a factor of 22 or 4. If the distance is doubled, the force drops to a fourth of its original size.

Electric fields

We just need to do a little more thinking about electrostatice forces and electric fields before we can really understand electric potential.

Here ( → ) is a rendering of the electric field produced by a positive and negative charge moved close enough to affect one another. The arrangement is called a dipole — two "poles," positive and negative, like the north and south poles on a bar magnet.

The blue lines are called electric field lines, or just field lines. They indicate the direction of the electrostatic force that would be experienced by a single hypothetical positive charge called a test charge.

In the lower figure, three locations of our test charge are shown along with vectors representing the forces applied to them. The red vectors represent the repulsive force of the positive pole. Note that the red vectors are longer (representing a stronger force) closer to the positive pole, and always point away from it. Likewise, the black vectors are the attractive forces due to the negative pole. The blue vector is the sum of these for each location, the net force on the test charge. These net force vectors form the field lines an give us their direction.

We use the test charge method to map out all electric fields, and we draw in the resulting field lines so that they're close together where the force is high, farther apart where it is low.

We use the test charge method to map out all electric fields, and we draw in the resulting field lines so that they're close together where the force is high, farther apart where it is low.


Gradients

Finally, let's look at a 3D view of that dipole below. The bottom axis represents the position of the test charge along the axis connecting the two charges. The right axis represents position at a 90˚ angle to that, and on the vertical axis we plot the force; up is repulsive, and down is attractive. This plot makes clear the force felt by our test charge and how it would move if we placed it somewhere and let go.

The blue areas of the plot are fairly flat, so the test charge would accelerate (remember that forces produce acceleration, F = ma) only slowly if placed in those regions. Other regions are steep. We say that there is a steep force field gradient in these regions, and the test charge would accelerate rapidly there.


Force field gradients accelerate particles. If there is no gradient, there is no net force, so there can be no acceleration. The steeper the gradient, the greater the acceleration.


Force → potential energy → electric potential

Now we need to replace force with potential energy, and that's pretty easy. Remember that the work done to achieve a certain potential energy (PE) is equal to that PE. Work is force multiplied by distance:

w = F·d   so   PE = F·d

Now if we make a plot of the potential energy of our test charge with respect to distance (or distances in the case of our 3-D plot above), the slope, PE/d is just the force.

The slope of a potential energy graph along some direction is the force along that direction.

Ok, now that's fine. We know how to calculate force between charged particles, but as we saw from Coulomb's law, force is directly proportional to charge.

Thus any given arrangement of charges will exert a proportionally bigger force on a larger charge — larger than, say, our test charge.

We need to make sure we're all speaking the same language when talking about fields, so we always agree to use a +1 test charge. That means that practically, when measuring the strength of a field with a real charge, we need to divide by the magnitude of the charge to make sure we normalize the field to a +1 charge. For example, if we measure or in any way determine a property of some field using a +3 charge, we need to divide by 3 so we don't over-express its strength.

Electric potential is potential energy divided by the size of the charge involved, so strictly speaking electric potential is not a potential energy. Electric potential is often referred to as "voltage."


"Voltage" ... or how lousy language persists


This is more of a peeve of mine than anything else: I don't like how we append the suffix "age" onto words to make them mean what can be better expressed by another perfectly fine word. When we say "voltage" we really mean electric potential. When we say "square footage" we really mean "area", and when say "percentage" we really mean “fraction” (which can be expressed as percent).

Oh, and never, ever say "amperage" when you mean current or "ohmage" when you mean resistance. That's strictly bush-league.


Fields and calculus

The force on a particle (particle is a generic word for "object" in physics) in a force field is the slope of the field or potential energy function in some direction. When potential energy functions get more complicated, like the mock-2D potential below, we generalize that concept of slope to the gradient, slope that has to be specified by more than one direction.

For a function f(x) of one variable, the slope is given by

df/dx equation.

For functions of two variables, we use the gradient operator, given the symbol ∇ ("nabla"), often read "grad".

In this equation the derivative of the 2D function with respect to x is taken by treating y like a constant. Likewise, the second term is the derivative of the 2D function with respect to y, treating x like a constant. i and j are unit vectors in the x- and y-directions, respectively. More properly, we should write partial derivative symbol     instead of d.

The gradient operator is easily generalized to any number of dimensions. Even though you might not be able to conceive of them, many problems are solved in many more than three dimensions.

The figure below shows a 2D function with a gradient field below it. Each arrow in the gradient field indicates the direction and magnitude of the slope at that location. In this figure the arrows point up-slope.



Ohm's law and the flow of current

In most applications of electricity, what we're really interested in is regulating the flow of electric current to do useful things. We'd like to make the flow of current predictable and to be able to manipulate it.

It turns out the the flow of current through a material is directly proportional to the potential, and it's inversely proportional to the resistance of the material to the flow of current. That's known as Ohm's law, and it's written like this:

Ohm's law is one of the most important relationships in all of the field of electricity and magnetism.

If we want more current, we can either

  • increase the potential (voltage) or
  • decrease the resistance.

Later you will see electric circuits in which the potential, current and resistance are fixed, and those in which each can depend upon other things, such as frequency of switching current on and off. All of this leads to all of the wonderful electronic devices to which we've become so accustomed.

Ohm's law is usually written on one line as a product, V = IR.

Ohm's law

Ohm's law: Potential is current multiplied by resistance.

V   =   IR

 

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