Recall the developmental history of electrostatics.
You and I have no problem with this last idea, but back in the day it was called "action at a distance" — a rather politely worded insult. To avoid the conceptual problems of dealing with a disembodied force, Michael Faraday invented the electric field and the world was satisfied.
Well, satisfied for awhile. Then somebody pointed out that the electric field was a vector quantity and they remembered that vectors were cumbersome and difficult to work with. Conceptual comfort was gained, but practical implementation was unchanged. Damn those scientists. Always looking for the best of all possible worlds. They wanted something both conceptually satisfying and mathematically simple. Such temerity!
Believe it or not, the problem was already solved by physicists and mathematicians working on topics that had nothing to do with electricity. Water, wind, heat, and dissolved substances all flow. Some of the conceptual and mathematical tricks used to understand these subjects might also be used to understand electricity — and also to magnetism and gravity.
What are field lines if not some kind of flow pattern? Electric field lines "flow" from positive charges to negative charges. A positive charge is like an open faucet and a negative charge is like an open drain. Anyone with a working sink can make a crude model of an electric dipole in their kitchen or bathroom with the flick of the wrist. Similar analogies exist for wind, heat, and dissolved substances. (Technically, heat and dissolved substances diffuse instead of flow, so the analogies there are a bit weaker.)
Think for a moment, of the other things that flow and think of what it is that causes them to flow. This will be the answer to our next conceptual problem. Let's set up a table that compares similar phenomena. In all cases, there will be something that flows and something that causes the flow.
the flow of…  is caused by a difference in… 

a river (liquid water)  altitude 
the wind (atmospheric gases)  atmospheric pressure 
heat (internal energy)  temperature 
dissolved substances (solutes)  concentration 
In each case, the thing that's flowing can be described by a vector field (a quantity that has magnitude and direction at any location) and the thing that causes the flow can be described by a difference in a scalar field (a quantity that has magnitude only at any location).
the flow of…  is caused by a difference in… 

a vector field  a scalar field 
If we can identify the electric scalar field that causes the electric vector field, we've made all of elecricity mathematically simpler, since scalars are mathematically simpler than vectors. "Identify" probably isn't the right word. "Define" is more like it. We are going to define a quantity that serves the same role as height does for rivers, pressure does for the wind, temperature does for heat, and concentration does for solutes.
The "flow" of the electric field is "caused" by a difference in electric potential.
the flow of…  is caused by a difference in… 

electric field (test charges)  electric potential 
Now you have to ask yourself what "electric potential" is.
First of all, the second half of the term, potential, does not imply that it has the possibility of happening or something that may lead to future usefulness. The electric potential of a location in space doesn't literally "have the potential to become electric". This incorrect notion is based on a different meaning of the word potential.
The real meaning of the word potential in this context is one that is now obscure — and thus the source of potential confusion. In the context of this discussion, potential means something closer to that which gives strength, power, might, or ability. For the physicist, the noun potential is more closely related to the adjectives potent or potency. Nowadays, the word potential seems more impotent than potent. "I've got the power" is a phrase that inspires. "I've got the potential" is a phrase in search of inspiration.
Second of all, when I wrote the term electric potential, I wasn't cut off twothirds of the way through writing electric potential energy. These are two separate (but related) concepts. See if you can follow this train of reasoning. Note how I said reasoning and not logic. This isn't a proof. The mathematics will show how everything is related.
A difference in electric potential gives rise to an electric field. (This is the concept I am introducing to you in this chapter you are reading right now.) The electric field is the force per charge acting on an imaginary test charge at any location in space. (This concept was introduced in the chapter before this one.) The work done placing an actual charge in an electric field gives the charge electric potential energy. (This concept is called the workenergy theorem and was introduced a long time ago, in a chapter far, far away.) By the transitive property (I guess), electric potential gives rise to electric potential energy; and by the reflexive property (another guess), the electric potential is the energy per charge that an imaginary test charge has at any location in space.
Those are words. We need maths. We can do this the hard way (without calculus) or the easy way (with calculus). Your choice.
In any case, here are the rules for the symbols specific to this topic…
Start from the workenergy theorem. When work is done (W), energy changes (∆E).
W = ∆E  
More specifically, when work is done against the electric force (F̅_{E}_{}), electric potential energy changes (∆U_{E}). Recall that work is force times displacement (d). There's a bar over the force symbol to indicate that we'll be using the average value. This is one of the limitations of derivations done without calculus.
F̅_{E}_{}d = ∆U_{E}  
Divide both sides by charge (q).
1  F̅_{E}d =  1  ∆U_{E} 
q  q 
Rearrange things a bit.
F̅_{E}_{}  d =  ∆U_{E} 
q  q 
The ratio of force to charge on the left is called electric field (E̅). That's an old idea that was discussed earlier in this book. The only thing that's changed is we're dealing with average values right now. The ratio of energy to charge on the right is called electric potential (V). That's a new idea that's being discussed right now in this book.


The electric field is the force on a test charge divided by its charge for every location in space. Because it's dervied from a force, it's a vector field. The electric potential is the electric potential energy of a test charge divided by its charge for every location in space. Because it's dervied from an energy, it's a scalar field. These two fields are related.
The electric field and electric potential are related by displacement. Field times displacement is potential…
E̅d = ∆V  
…or field is potential over displacement, if you prefer.
E̅ =  ∆V 
d 
In fancy calculus language, field is the gradient of potential — because the real world is fancy, by which I mean threedimensional. Gradient is the three dimensional equivalent of a slope. An ordinary slope is onedimensional, because a line is onedimensional (even if it's not straight). There's only one decision to make when moving along a curve. Do I go ahead, or do I go back? In normal euclidean space, we have three options. Up or down? Left or right? Forward or backward?
Start from the workenergy theorem. When work is done (W), energy changes (∆E).
W = ∆E  
More specifically, when work is done against the electric force (F_{E}), electric potential energy changes (∆U_{E}). Recall that work is the forcedisplacement integral.
−  ⌠ ⌡ 
F_{E} · dr = ∆U_{E}  
Divide both sides by charge (q).
−  1  ⌠ ⌡ 
F_{E} · dr =  1  ∆U_{E} 
q  q 
Rearrange things a bit.
−  ⌠ ⌡ 
F_{E}  · dr =  ∆U_{E} 
q  q 
The ratio of force to charge on the left is called electric field (E). That's an old idea that was discussed earlier in this book. The ratio of energy to charge on the right is called electric potential (V). That's a new idea that's being discussed right now in this book.


The electric field is the force on a test charge divided by its charge for every location in space. Because it's dervied from a force, it's a vector field. The electric potential is the electric potential energy of a test charge divided by its charge for every location in space. Because it's dervied from an energy, it's a scalar field. These two fields are related.
The electric field and electric potential are related by a path integral that works for all sorts of situations. My advice when working with a path integral is to always pick the easiest path to work with. Electricity is a conservative force, so the work done by it doesn't depend on the path taken. This equation says something more astounding. The integral on the left is so path independent that its value depends only upon the electric potential at the beginning and end of the path. If you can find those two numbers and subtract them, you've done the whole integral. If more integrals worked this way, students wouldn't get so hung up on calculus.
−  ⌠ ⌡ 
E · dr = ∆V  
Electric field and electric potential are also related by a derivative that works for one dimensional situations only — situations with spherical, cylindrical, or planar symmetry.
E = −  d  V r̂ 
dr 
In fancier calculus terms, field is the gradient of potential — because the real world is fancier than a onedimensional problem. The gradient is the equivalent of a derivative in higher dimensions (in this book, two and three dimensions). This relationship works for all kinds of symmetry and nonsymmetry.
E = −∇V  
The Greek letter delta looks like a triangle pointing upward (∆). An inverted delta is called a del (∇). The delta and del symbols are examples of mathematical devices called operators — symbols that indicate that an operation needs to be performed on a variable. The delta operator has been discussed numerous times throughout this book. The del operator is a bit more rare.
The delta operator is used whenever the change or difference of a quantity is needed. Jump back a bit to the equation that relates electric field to electric potential through a path integral.
−  ⌠ ⌡ 
E · dr = ∆V  
Here, ∆V means a difference in electric potential between two points — usually a starting or initial location (indicated in this book with a subscript zero) and an ending or final location (indicated in this book without any subscript).
r  
−  ⌠ ⌡ 
E · dr = V − V_{0}  
r_{0} 
In cartesian coordinates, the del operator is the sum of the partial derivatives in the three unit vector directions. (In noncartesian coordinates, the del is a bit more complicated).
∇ = î  ∂  + ĵ  ∂  + k̂  ∂ 
∂x  ∂y  ∂z 
When the del operator is applied to a scalar field, the resulting operation is known as a gradient. Jump back a bit. The equation that says the electric field is the gradient of the electric potential…
E = −∇V  
…looks like this when the del operator is expanded…
E = − î  ∂  V − ĵ  ∂  V − k̂  ∂  V 
∂x  ∂y  ∂z 
…and like this when the terms are rearranged so that scalars precede vectors…
E = −  ∂  V î −  ∂  V ĵ −  ∂  V k̂ 
∂x  ∂y  ∂z 
Maybe now you can see why the del symbol was invented. The compact equation has 5 symbols in it (not counting spaces). The expanded equation has 23 (counting "hats", but not counting spaces).
Hold it now. What's the deal with all these minus signs? Let me explain… later.
What's new in this chapter of this book? The whole notion of electric potential. I introduced electric potential as the way to solve the evils of the vector nature of the electric field, but electric potential is a concept that has a right to exist all on its own. Electric potential is the electric potential energy on a test charge divided by the charge of that test charge.
ΔV =  ΔU_{E} 
q 
Old stuff. SI is an abbreviation for le Système International d'Unités in French or the International System of Units in English. The SI unit of energy is the joule, named for James Joule, the English brewer turned physicist who determined that heat and electricity were forms of energy equivalent to other forms of mechanical energy like gravitational potential energy and kinetic energy. The SI unit of charge is the coulomb, named for CharlesAugustin Coulomb, the French nobleman and soldier turned physicist who discovered the inverse square rule of the electrostatic force. The SI unit of displacement (or distance) is the meter — a word named after nobody and ultimately derived from the Greek word for measure (μετρον, metron).
New stuff. The SI unit of electric potential is the volt, named for the Italian nobleman turned physicist Alessandro Volta, whose full name is the astonishingly long Conte (Count) Alessandro Giuseppe Antonio Anastasio Volta. The joke today is that Volta's full name was so long that when they cut it down to the name of a unit, they went too far and chopped off the final "a". The unit of electric potential should rightly be called the volta instead of the volt (a joke appreciated only by the pedantic among us). Count Volta is best known as the inventor of the electrochemical cell — what we now mistakenly call a battery (another distinction appreciated only by the pedantic — a battery is a collection of electrochemical cells). For those that care about the important stuff, a volt is a joule per coulomb.
⎡ ⎣ 
V =  J  ⎤ ⎦ 
C 
Electric potential is a way to explain a "difficult" vector field in terms of an "easy" scalar field. By definition, the electric field is the force per charge on an imaginary test charge.
E =  F_{E} 
q 
By means of a long explanation, the electric field is also the gradient of the electric potential (the rate of change of electric potential with displacement).
E =  ∆V 
d 
Set the two quantities equal…
F_{E}  =  ∆V 
q  d 
…and then set their units equal.
⎡ ⎣ 
N  =  V  ⎤ ⎦ 
C  m 
The newton per coulomb and the volt per meter are equivalent units for the electric field. The volt per meter is more frequently used by those who actually measure things because the volt (which can be measured with a voltmeter) and the meter (which can be measured with a ruler of any size, including the appropriately named meter stick) are much easier to measure than force (which could be measured with a spring scale or strain gauge attached to a charged object, I suppose) and charge (which could be measured with no device I know of).
Robert Millikan (1865–1953) United States
apparatus
findings