Formal definition of electric potential and voltage. Written by Willy McAllister.
Coulomb's Law lets us compute forces between static charges. Now we explore what happens if charges move around. We find out what it means to do work in an electric field and develop formal definitions of some new concepts.
  • electric potential energy
  • electric potential (also known as voltage)
Electric force and electric field are vector quantities (they have magnitude and direction). Electric potential turns out to be a scalar quantity (magnitude only), a nice simplification.
Let's set up a simple charge arrangement, and ask a few questions. Begin with two positive point charges, separated by some distance rAr_A. In this discussion, QQ will stay fixed in place, and qq (our test charge) will move around.
QQ repels qq (and vice versa), with a force described by Coulomb's Law,
F=14πϵ0qQrA2F = \dfrac{1}{4\pi\epsilon_0}\dfrac{q\,Q}{{r_A}^2}
Said another way in terms of electric field, QQ establishes a field everywhere in space,
E=14πϵ0Qr2E = \dfrac{1}{4\pi\epsilon_0}\dfrac{Q}{r^2}
(the direction of QQ's field is along a radial line pointing away from QQ).
At distance rAr_A away from qq, the electric field is specifically,
E=14πϵ0QrA2E = \dfrac{1}{4\pi\epsilon_0}\dfrac{Q}{{r_A}^2}
We can restate the force on qq from QQ in terms of electric field as,
F=qEF = qE
The little dude in this image emphasizes that something has to hold qq in place. In a minute, we'll put the little dude to work.

Work and potential energy

Force is any interaction that changes the motion of an object. A push or pull. F=maF = m\,a.
The general definition of work is "force acting through a distance" or W=FdW = F \cdot d. In electric field notation, W=qEdW = q E \cdot d
Energy is "the ability to do work." When an object has energy, it has the ability to do work.
When a force does work on an object, potential energy can be stored. An object with potential energy has the potential to do work. (It's not doing work right now, but it has the potential.) An object has potential energy by virtue of its position.
Work and potential energy are closely related. Additional potential energy stored in an object is equal to the work done to bring the object to its new position.
You can brush up on the concepts of work and energy in more depth here.

Field

A field is a region of space where we observe forces. Gravity, electricity, and magnetism create fields.
A field is conservative if an object travels in a closed path (makes a round trip) and no net work is done against the force associated with the field.
Gravity is conservative. When you lift a book up, you do work on the book. If you gently lower the book back down, the book does work on you. The net amount of work is zero. You can raise and lower a hundred times, and if the book ends up in the original height, the net amount of work is zero. If you move the book horizontally, the amount of work is also zero, because there is no opposing force in the horizontal direction.
A static electric field is conservative. No matter what path a charged object takes in the field, if the charge returns to its starting point, the net amount of work is zero.

Electrical potential energy resembles gravitational potential energy

The behavior of charges in an electric field resembles the behavior of masses in a gravitational field. Just like gravitational potential energy, we can talk about electric potential energy.
There are two differences between electric fields and gravity to keep in mind,
  • Electric charges can attract (like gravitational masses), but can also repel (which masses never do). Like charges move as far apart as possible.
  • Our usual experience with gravity happens close to the surface of Earth, over tiny distances relative to the size of our planet. For earthly problems, the force from gravity is often treated as a constant. For electric problems, we take 1/r21/r^2 into account. This makes the math a little more challenging, but it's not too bad.
For both gravity and electricity, potential energy differences are what's important. Wherever your book starts out, it has some potential energy. When you move the book, you add or remove potential energy relative to where it started. For moving charges, you add or subtract electric potential energy relative to where the charge started.
If you wonder if an object is storing potential energy, take away whatever might be holding it in place. If the object moves, it was storing potential energy. An apple falls from a tree and conks you on the head. It had potential energy. Let go of a charge in an electric field; if it shoots away, it was storing electric potential energy.

Doing work in an electric field

What happens if we move qq closer to QQ? How much work is done? To move qq towards QQ, we have to push on qq just hard enough to overcome the repulsive electric force.
In general, for electrostatics problems, we want the movement of qq to be quasi-static, so we don't have to worry about the kinetic energy of moving bodies. In a quasi-static process, we apply a net force just barely greater than qEq \vec E, or make sure FqE\vec F - q \vec E is vanishingly small. The displacement of qq happens without speeding it up, so there is no kinetic energy term. Going this slow means moving qq will take infinite time to happen. This doesn't bother us, because we are only going to talk about it, not wait for it to happen.
How much work is done moving qq from point AA to point BB in an electric field?
When charges move in an electric field, something has to do work to get the charge to move. To move qq, we apply a force to just barely overcome the repulsive force from QQ.
Let's work it out:
The amount of work done is force times distance, W=FdW = F \cdot d . The distance moved is (rArB)(r_A-r_B). What is the force? This is a bit trickier, because the force changes all along the path. The closer we get to QQ, the greater the force of repulsion. The closer we get, the harder we have to push to make qq move. Set up some variables so we can talk about what's going on here.
  • rr is the distance from QQ to the current position of qq.
  • dr\text dr is a tiny change in distance. dr\text dr is so tiny we can consider the electric force constant over this distance.
In any electric field, the force on a positive charge is F=qEF = q E.
The external force required points in the opposite direction, Fext=qEF_{ext} = -qE.
For our specific example near a point charge, the electric field surrounding QQ is,
E=14πϵ0Qr2E = \dfrac{1}{4\pi\epsilon_0}\dfrac{Q}{r^2}
And the external force required to move qq is,
Fext=qE=q14πϵ0Qr2F_{ext} = -qE = -q \cdot \dfrac{1}{4\pi\epsilon_0}\dfrac{Q}{r^2}
To deal with the problem of the force changing at every point, we write an expression for the tiny bit of work needed to move qq by a tiny dr\text dr. The assumption we make is that we can make dr\text dr so tiny the force is effectively constant over that distance. From the definition of work,
dW=qEdr=q14πϵ0Qr2dr\text d {W} = -qE \cdot \text dr = -q \,\dfrac{1}{4\pi\epsilon_0}\dfrac{Q}{r^2}\, \text dr
To figure out the total work for the trip from AA to BB, sum up all the tiny work amounts,
WAB=rArBqEdr\displaystyle W_{AB} = \int_{r_A}^{r_B} -qE \cdot \text dr
WAB=qQ4πϵ0rArB1r2dr\displaystyle W_{AB} = -\dfrac{q\,Q}{4\pi\epsilon_0} \int_{r_A}^{r_B} \dfrac{1}{r^2} \text dr
Solving the definite integral,
The integral of a power of xx is xndx=xn+1n+1\displaystyle \int x^n \text d x = \dfrac{x^{n+1}}{n+1}, for any n1n \neq -1.
In this example, n=2n = -2.
ABr2dr=x2+12+1AB=1rAB\displaystyle \int_A^{B} r^{-2} \text dr = \dfrac{x^{-2+1}}{-2+1}\,\bigg|_{A}^{B} = -\dfrac{1}{r}\,\bigg|_{A}^{B}
WAB=qQ4πϵ0(1r)rArB\displaystyle W_{AB} = -\dfrac{q\,Q}{4\pi\epsilon_0} \cdot \left (-\dfrac{1}{r} \right )\bigg|_{r_A}^{r_B}
The external work to bring a charge qq from AA to BB near a point charge, QQ is,
WAB=qQ4πϵ0(1rB1rA)\large \displaystyle W_{AB} = \dfrac{q\,Q}{4\pi\epsilon_0} \left ( \frac{1}{r_B} - \frac{1}{r_A}\right )

Electric potential energy

Question: How has qq's potential energy changed?

The change of potential energy stored in qq is equal to the work done on qq to bring it from AA to BB,
electric potential energy differenceAB=rArBqEdr=qQ4πϵ0(1rB1rA)\displaystyle \text{electric potential energy difference}_{AB} = \int_{r_A}^{r_B} -q \vec E \cdot \text dr = \dfrac{q\,Q}{4\pi\epsilon_0} \left ( \frac{1}{r_B} - \frac{1}{r_A}\right )
Like work, electric potential energy is a scalar quantity.
We now do a small manipulation of this expression and something special emerges. This line of reasoning is similar to our development of the electric field.
Multiply out the terms,
electric potential energy differenceAB=(qQ4πϵ01rB)(qQ4πϵ01rA)\displaystyle \text{electric potential energy difference}_{AB} = \left (\dfrac{q\,Q}{4\pi\epsilon_0} \frac{1}{r_B} \right ) - \left (\dfrac{q\,Q}{4\pi\epsilon_0} \frac{1}{r_A} \right )
Give the two terms a name so we can talk about them for a second. Let,
Ur=qQ4πϵ01r\displaystyle U_r = \dfrac{q\,Q}{4\pi\epsilon_0} \frac{1}{r}
UrU_r represent the electric potential energy stored in charge qq when it is distance rr away from QQ. The change in energy going from AA to BB can be written as,
electric potential energy differenceAB=UBUA\text{electric potential energy difference}_{AB} = U_B - U_A
UAU_A and UBU_B are associated with a single location in space. That is, UBU_B only depends on location BB, and UAU_A only depends on location AA. (And both depend on the values of qq and QQ.) This suggests UU can be viewed as a property of a location. We can think of electric potential energy as a field existing in the space around QQ. Potential energy is a scalar quantity, so a potential energy field is a scalar field. It has a magnitude everywhere in space, but does not have direction. (Another example of a scalar field is the temperature everywhere in a room.)
Also, notice the expression does not mention any other points, so the potential energy difference is independent of the route you take from AA to BB. This is a consequence of the conservative nature of electric fields.

Electric potential difference

With another simplification, we come up with a new way to think about what's going on in an electrical space. The equation above for electric potential energy difference expresses how the potential energy changes for an arbitrary charge, qq when work is done on it in an electric field. We define a new term, the electric potential difference (removing the word "energy") to be the normalized change of electric potential energy.
electric potentialenergydifferenceAB=UBqUAq\text{electric potential} \cancel{energy} \,\text{difference}_{AB} = \dfrac{U_B}{q} - \dfrac{U_A}{q}
Electric potential difference is the change of potential energy experienced by a test charge that has a value of +1+1.
Electric potential energy difference has units of joules.
Electric potential difference has units of joules/coulomb.

Electric potential

We can give a name to the two terms in the previous equation for electric potential difference. We can say there is an electrical potential everywhere in space surrounding QQ, expressed as,
electric potential =Urq\text{electric potential } = \dfrac{U_r}{q}
It might seem strange to think about this as a property of space. (But no stranger than the notion of an electric field.) It is basically saying if we put a unit test charge at this location, it would have this potential energy. Take away the unit charge, and the property of space still remains.
We can use the concept of electric potential to run this whole discussion in reverse. Suppose we know what the electric potential looks like in some region of space. We can figure out the work required to move a charged object between two locations by,
  1. Subtracting the starting potential from the ending potential to get the potential difference, and
  2. Multiplying potential difference by the actual charge of the introduced object.

Electric potential near a point charge

Near a point charge, we can connect-the-dots between points with the same potential, showing equipotential contours. Remember, for a point charge, only the difference in radius matters, so the equipotential contours are circles centered on the the charge creating the potential field, in this case, QQ.

Voltage

Electric potential difference gets a very special name. We call it voltage, measured it in units of volts, in honor of Alessandro Volta, the inventor of the battery. The voltage between points AA and BB is,
voltageAB=electric potentialdifferenceAB=UBqUAq\text{voltage}_{AB} = \text{electric potential}\,\text{difference}_{AB} =\dfrac{U_B}{q} - \dfrac{U_A}{q}

Absolute voltage

Up to now the equations have all been in terms of electric potential difference. We talk about the potential difference between here and there. Can we come up with a concept of an absolute potential difference (an absolute voltage)? Yes, we can, in a sense. An established convention is to define voltage=0\text{voltage}=0 at a point infinity away. With this convention, a meaning of absolute voltage emerges by setting the starting location to rA=r_A = \infty. Then the voltage at a location rr away from a point charge is,
Vr=(Q4πϵ01r)(Q4πϵ01)0\displaystyle \text{V}_{r} = \left (\dfrac{Q}{4\pi\epsilon_0} \frac{1}{r} \right ) - \cancel{\left (\dfrac{Q}{4\pi\epsilon_0} \frac{1}{\infty} \right )}^{\Large 0}
The term with 1/1/\infty goes to zero. The absolute voltage at a location is then defined as the external work required to bring a unit test charge from infinity up to some location.
Vr=Q4πϵ01r\displaystyle \text{V}_{r} = \dfrac{Q}{4\pi\epsilon_0} \frac{1}{r}
There isn't any magic here. It's just a turn of phrase. It means the same thing as saying the voltage at location xx is the potential difference between xx and infinity. This works because we share the assumption that the reference point for zero voltage is out at infinity.
The concept of absolute gravitational potential energy has the same requirement for a conventional reference point. We can only measure gravitational potential energy differences between two points. Usually, the reference point from which we measure (floor, table top, sea level, center of the Earth) is understood from the surrounding context. We assume cultural conventions for the reference point when asking questions like, "How tall are you?" "How high is that mountain?" "How much does the Earth's gravity pull on the Moon?"
As we learn about electrical potential, we use an analogy to gravity. Oddly (for me, anyway), when talking about gravitational potential, there isn't a word corresponding to "voltage". We just call it "gravitational potential", without honoring a scientist.

Wrap up

The terms we've been tossing around can sound alike, so it is easy for them to blur.
  • Electric potential energy is a property of a charged object, by virtue of its location in an electric field. Electric potential energy exists if there is a charged object at the location.
  • Electric potential difference, also known as voltage, is the external work needed to bring a charge from one location to another location in an electric field. Electric potential difference is the change of potential energy experienced by a test charge that has a value of +1+1.
  • Electric potential exists at one location as a property of space. A location has electric potential even if there is no charged particle there.
  • Absolute voltage at a location is something we can talk about as long as everyone agrees that zero volts is way out at infinity. The concept of absolute voltage is sort of a verbal sleight-of-hand. It is always safe to stick with the definition: voltage is a potential difference.
The concept of voltage was developed here using a fixed point charge QQ as the source of electric field. We derived an exact expression for voltage in the space surrounding QQ. The whole idea of electric potential and voltage is valid for any kind of charge arrangement. Of course, there is a different specific solution each time (the equation above for U=...U = ... changes, but everything after that using UU is still correct). The power of the voltage concept is that it describes space with a scalar field. We don't have to keep track of vector directions. This significantly simplifies the math.

What is a volt ?

You may have noticed something missing so far. We have not provided any details on the unit of voltage: the volt. The volt is a so-called "SI derived unit". The article on Standard electrical units covers the definition of the volt in detail.

References

Kip, A. H. (1969), Fundamentals of Electricity and Magnetism (2nd edition, McGraw-Hill)
This article is licensed under CC BY-NC-SA 4.0.
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