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Electric potential, voltage

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 r, start subscript, A, end subscript. In this discussion, Q will stay fixed in place, and q (our test charge) will move around.
Q repels q (and vice versa), with a force described by Coulomb's Law,
F, equals, start fraction, 1, divided by, 4, pi, \epsilon, start subscript, 0, end subscript, end fraction, start fraction, q, Q, divided by, r, start subscript, A, end subscript, squared, end fraction
Said another way in terms of electric field, Q establishes a field everywhere in space,
E, equals, start fraction, 1, divided by, 4, pi, \epsilon, start subscript, 0, end subscript, end fraction, start fraction, Q, divided by, r, squared, end fraction
(the direction of Q's field is along a radial line pointing away from Q).
At distance r, start subscript, A, end subscript away from q, the electric field is specifically,
E, equals, start fraction, 1, divided by, 4, pi, \epsilon, start subscript, 0, end subscript, end fraction, start fraction, Q, divided by, r, start subscript, A, end subscript, squared, end fraction
We can restate the force on q from Q in terms of electric field as,
F, equals, q, E
The little dude in this image emphasizes that something has to hold q 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, equals, m, a.
The general definition of work is "force acting through a distance" or W, equals, F, dot, d. In electric field notation, W, equals, q, E, dot, 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.
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 q closer to Q? How much work is done? To move q towards Q, we have to push on q just hard enough to overcome the repulsive electric force.
How much work is done moving q from point A to point B in an electric field?
When charges move in an electric field, something has to do work to get the charge to move. To move q, we apply a force to just barely overcome the repulsive force from Q.
Let's work it out:
The amount of work done is force times distance, W, equals, F, dot, d . The distance moved is left parenthesis, r, start subscript, A, end subscript, minus, r, start subscript, B, end subscript, right parenthesis. What is the force? This is a bit trickier, because the force changes all along the path. The closer we get to Q, the greater the force of repulsion. The closer we get, the harder we have to push to make q move. Set up some variables so we can talk about what's going on here.
  • r is the distance from Q to the current position of q.
  • start text, d, end text, r is a tiny change in distance. start text, d, end text, r 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, equals, q, E.
The external force required points in the opposite direction, F, start subscript, e, x, t, end subscript, equals, minus, q, E.
For our specific example near a point charge, the electric field surrounding Q is,
E, equals, start fraction, 1, divided by, 4, pi, \epsilon, start subscript, 0, end subscript, end fraction, start fraction, Q, divided by, r, squared, end fraction
And the external force required to move q is,
F, start subscript, e, x, t, end subscript, equals, minus, q, E, equals, minus, q, dot, start fraction, 1, divided by, 4, pi, \epsilon, start subscript, 0, end subscript, end fraction, start fraction, Q, divided by, r, squared, end fraction
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 q by a tiny start text, d, end text, r. The assumption we make is that we can make start text, d, end text, r so tiny the force is effectively constant over that distance. From the definition of work,
start text, d, end text, W, equals, minus, q, E, dot, start text, d, end text, r, equals, minus, q, start fraction, 1, divided by, 4, pi, \epsilon, start subscript, 0, end subscript, end fraction, start fraction, Q, divided by, r, squared, end fraction, start text, d, end text, r
To figure out the total work for the trip from A to B, sum up all the tiny work amounts,
W, start subscript, A, B, end subscript, equals, integral, start subscript, r, start subscript, A, end subscript, end subscript, start superscript, r, start subscript, B, end subscript, end superscript, minus, q, E, dot, start text, d, end text, r
W, start subscript, A, B, end subscript, equals, minus, start fraction, q, Q, divided by, 4, pi, \epsilon, start subscript, 0, end subscript, end fraction, integral, start subscript, r, start subscript, A, end subscript, end subscript, start superscript, r, start subscript, B, end subscript, end superscript, start fraction, 1, divided by, r, squared, end fraction, start text, d, end text, r
Solving the definite integral,
W, start subscript, A, B, end subscript, equals, minus, start fraction, q, Q, divided by, 4, pi, \epsilon, start subscript, 0, end subscript, end fraction, dot, left parenthesis, minus, start fraction, 1, divided by, r, end fraction, right parenthesis, vertical bar, start subscript, r, start subscript, A, end subscript, end subscript, start superscript, r, start subscript, B, end subscript, end superscript
The external work to bring a charge q from A to B near a point charge, Q is,
W, start subscript, A, B, end subscript, equals, start fraction, q, Q, divided by, 4, pi, \epsilon, start subscript, 0, end subscript, end fraction, left parenthesis, start fraction, 1, divided by, r, start subscript, B, end subscript, end fraction, minus, start fraction, 1, divided by, r, start subscript, A, end subscript, end fraction, right parenthesis

Electric potential energy

Question: How has q's potential energy changed?

The change of potential energy stored in q is equal to the work done on q to bring it from A to B,
start text, e, l, e, c, t, r, i, c, space, p, o, t, e, n, t, i, a, l, space, e, n, e, r, g, y, space, d, i, f, f, e, r, e, n, c, e, end text, start subscript, A, B, end subscript, equals, integral, start subscript, r, start subscript, A, end subscript, end subscript, start superscript, r, start subscript, B, end subscript, end superscript, minus, q, E, with, vector, on top, dot, start text, d, end text, r, equals, start fraction, q, Q, divided by, 4, pi, \epsilon, start subscript, 0, end subscript, end fraction, left parenthesis, start fraction, 1, divided by, r, start subscript, B, end subscript, end fraction, minus, start fraction, 1, divided by, r, start subscript, A, end subscript, end fraction, right parenthesis
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,
start text, e, l, e, c, t, r, i, c, space, p, o, t, e, n, t, i, a, l, space, e, n, e, r, g, y, space, d, i, f, f, e, r, e, n, c, e, end text, start subscript, A, B, end subscript, equals, left parenthesis, start fraction, q, Q, divided by, 4, pi, \epsilon, start subscript, 0, end subscript, end fraction, start fraction, 1, divided by, r, start subscript, B, end subscript, end fraction, right parenthesis, minus, left parenthesis, start fraction, q, Q, divided by, 4, pi, \epsilon, start subscript, 0, end subscript, end fraction, start fraction, 1, divided by, r, start subscript, A, end subscript, end fraction, right parenthesis
Give the two terms a name so we can talk about them for a second. Let,
U, start subscript, r, end subscript, equals, start fraction, q, Q, divided by, 4, pi, \epsilon, start subscript, 0, end subscript, end fraction, start fraction, 1, divided by, r, end fraction
U, start subscript, r, end subscript represent the electric potential energy stored in charge q when it is distance r away from Q. The change in energy going from A to B can be written as,
start text, e, l, e, c, t, r, i, c, space, p, o, t, e, n, t, i, a, l, space, e, n, e, r, g, y, space, d, i, f, f, e, r, e, n, c, e, end text, start subscript, A, B, end subscript, equals, U, start subscript, B, end subscript, minus, U, start subscript, A, end subscript
U, start subscript, A, end subscript and U, start subscript, B, end subscript are associated with a single location in space. That is, U, start subscript, B, end subscript only depends on location B, and U, start subscript, A, end subscript only depends on location A. (And both depend on the values of q and Q.) This suggests U can be viewed as a property of a location. We can think of electric potential energy as a field existing in the space around Q. 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 A to B. 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, q 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.
start text, e, l, e, c, t, r, i, c, space, p, o, t, e, n, t, i, a, l, end text, start cancel, e, n, e, r, g, y, end cancel, start text, d, i, f, f, e, r, e, n, c, e, end text, start subscript, A, B, end subscript, equals, start fraction, U, start subscript, B, end subscript, divided by, q, end fraction, minus, start fraction, U, start subscript, A, end subscript, divided by, q, end fraction
Electric potential difference is the change of potential energy experienced by a test charge that has a value of plus, 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 Q, expressed as,
start text, e, l, e, c, t, r, i, c, space, p, o, t, e, n, t, i, a, l, space, end text, equals, start fraction, U, start subscript, r, end subscript, divided by, q, end fraction
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, Q.

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 A and B is,
start text, v, o, l, t, a, g, e, end text, start subscript, A, B, end subscript, equals, start text, e, l, e, c, t, r, i, c, space, p, o, t, e, n, t, i, a, l, end text, start text, d, i, f, f, e, r, e, n, c, e, end text, start subscript, A, B, end subscript, equals, start fraction, U, start subscript, B, end subscript, divided by, q, end fraction, minus, start fraction, U, start subscript, A, end subscript, divided by, q, end fraction

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 start text, v, o, l, t, a, g, e, end text, equals, 0 at a point infinity away. With this convention, a meaning of absolute voltage emerges by setting the starting location to r, start subscript, A, end subscript, equals, infinity. Then the voltage at a location r away from a point charge is,
start text, V, end text, start subscript, r, end subscript, equals, left parenthesis, start fraction, Q, divided by, 4, pi, \epsilon, start subscript, 0, end subscript, end fraction, start fraction, 1, divided by, r, end fraction, right parenthesis, minus, start cancel, left parenthesis, start fraction, Q, divided by, 4, pi, \epsilon, start subscript, 0, end subscript, end fraction, start fraction, 1, divided by, infinity, end fraction, right parenthesis, end cancel, start superscript, 0, end superscript
The term with 1, slash, infinity 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.
start text, V, end text, start subscript, r, end subscript, equals, start fraction, Q, divided by, 4, pi, \epsilon, start subscript, 0, end subscript, end fraction, start fraction, 1, divided by, r, end fraction
There isn't any magic here. It's just a turn of phrase. It means the same thing as saying the voltage at location x is the potential difference between x and infinity. This works because we share the assumption that the reference point for zero voltage is out at infinity.

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 plus, 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 Q as the source of electric field. We derived an exact expression for voltage in the space surrounding Q. 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, equals, point, point, point changes, but everything after that using U 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.

Want to join the conversation?

  • mr pants teal style avatar for user yash.kick
    I can't understand why we have a section of absolute voltage, I mean voltage itself means potential difference so then what do we mean by "absolute voltage" and "voltage"?
    (3 votes)
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    • spunky sam orange style avatar for user Willy McAllister
      Go back to the equation for Electric Potential Energy Difference (AB) in the middle of the section on Electric Potential Energy. That equation tells you how electric potential energy changes when you move a test charge from point A to point B. For example, you could be moving your test charge towards or away from some charged object. To use this equation you have to put in two locations, A and B. There are just a few oddball situations that give us some trouble... What if I told you where B was but did not mention A? I might say it this way: "What is the potential energy of a test charge when you place it at B"? Well, you need an A to answer that question. If I don't give it to you, you have to make one up. A common choice that lots of engineers and scientists make is "A is infinity away from the charged object." When we make that choice, we say we are determining the absolute potential energy, or the absolute voltage. It's the same voltage as usual, but with the assumption that the starting point is infinity away. In almost all circuits, the second point is provided and this absolute idea isn't needed.
      (13 votes)
  • mr pants teal style avatar for user yash.kick
    Willy said-"Remember, for a point charge, only the difference in radius matters", WHY??
    (3 votes)
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  • leaf green style avatar for user Abhinay Singh
    Sir just for shake of awareness Does moving charge also create Electric field ?
    & Does static charge also create Magnetic field ?
    (3 votes)
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    • spunky sam orange style avatar for user Willy McAllister
      Yes, a moving charge has an electric field.
      No, a static charge does not create a magnetic field.

      A moving charge creates a magnetic field. You can see this if you pass a current through a wire (moving charge) and hold it near a compass. The compass will tip to the side.

      The science of moving charges is called electrodynamics or electromagnetism.
      (4 votes)
  • blobby green style avatar for user skusecam9
    how much voltage is there in a electric fence.
    (2 votes)
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    • purple pi purple style avatar for user APDahlen
      It depends on the fence...

      Agricultural fence chargers are designed to produce several thousand volts. It is important to note that the voltage is NOT continuous. Instead the voltage is pulsed for short amount of time. The result is akin to a static electric shock.

      Regrettably there are some fences in the world designed to deliver lethal shock. I'd rather not think about those...

      Regards,

      APD
      (5 votes)
  • primosaur seed style avatar for user Bhagyashree U Rao
    In the 'Doing work in an electric field section',
    The equation we get for the small amt. of work done is
    dW=−qE⋅dr
    I get that F(ext.) is -ve since it acts opp. to the electrostatic force on q. But then why isn't 'dr' also -ve?

    Also, for the movement of q for that small distance 'dr' is in the same direction as that of the F(ext.). So, shouldn't 'dW' be +ve?

    Please do reply.
    (2 votes)
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    • male robot hal style avatar for user Andrew M
      Work is positive if the force is in the same direction as the displacement, negative if it's not.
      You have to pay attention to which force you are talking about: The force by the field or the force by an external force.
      It's really better not to try to handle this by manipulating a bunch of signs. Just calculate the amount of work that's done and then ask yourself if the force was pushing in the direction of the displacement or opposite to the displacement.
      (4 votes)
  • duskpin tree style avatar for user Maiar
    So, basically we said that Fex=-qE=Fe because the difference between them is negligible, but actually speaking, the external force is a little greater than the the electrostatic force ? Am I getting this right?
    Got another question, what's exactly this "external force" that does work on the charge to move it, in case of charges moving in a circuit, what causes these charge to move from the source in the first place ?
    (2 votes)
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    • spunky sam orange style avatar for user Willy McAllister
      If you want to actually move a charge, you have to apply an ever-so-slightly greater force to the charge to get it to start moving. It is important not to push too long or too hard because we don't want the charged particle to accelerate. (If it accelerates then all sorts of new physics starts to happen involving magnetism, which at the moment is way over our heads.) For now we make our charges sit still (static) or we move them super slow where they move but they don't accelerate, a condition called "pseudo-static".

      In this general definition of of electric potential you can imagine any external force you want pushing the charge. If I do the experiment with a charged balloon, I can use my hand to push the charged object towards another. In a circuit, charges move because they are repelled from like charges and attracted to unlike charges. A chemical reaction near the negative terminal of a battery releases an excess number of electrons in the vicinity (in the attached wire). Electrons in the wire are repelled from this concentration and move away, towards the positive battery terminal. There is a different chemical reaction happening at the positive terminal that absorbs electrons, creating a deficit of electrons for a net positive charge in the region near that terminal. Nearby electrons are attracted and move toward the positive terminal.
      (3 votes)
  • male robot donald style avatar for user V
    I understand the term of electric potential difference between two particles , but how do we define the electric potential difference between two charged plates that are fixed ? Like I know the equation Delta V = Ed , but can someone explain it ? Thanks
    (2 votes)
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  • blobby green style avatar for user HI
    I know that electrical potential formula is V=kq/r. How is this related to columb's law?
    (2 votes)
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  • blobby green style avatar for user shivangshukla884
    In house switches, they declare a specific voltage output. How voltage is constant if voltage is dependent on distance from reference point as mentioned in the formula voltage = electric potential difference ab, where electric potential difference is inversely proportional to distance from the reference point. Will the voltage not decrease from the increase of distance from the power generation site to my house (according to the formula)
    (1 vote)
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    • spunky sam orange style avatar for user Willy McAllister
      The formal definition of voltage is based on two positive charges near each other. One charge is in a fixed location and a second test charge is moved toward and away from the other. The farther away the test charge gets the lower its potential and the lower its voltage.

      However, power generated by the electric utility is not using this simple setup. They don't collect a big pile of electrons as the fixed charge and repel some free electrons into the transmission lines. If that was the case then yes, the voltage would drop if you live farther away from the power station.

      What they actually do is continuously raise electrons to a high voltage and force them into the transmission lines, actively pushing the wires full of electrons.

      It's similar to the difference between pouring water from a height (high voltage) onto the ground (low voltage) compared to flowing water constantly through a garden hose. No matter how long the garden hose the water comes out the end at a high pressure (high voltage). It pretty much does not matter how long the hose is.
      (2 votes)
  • blobby green style avatar for user Joffer Piton
    So, if the electric potencial measures the field produced by one charge, like the explanations above. One could ask what we do really measures when we have for exemplo 220v? It would be a bunch of electrons? Tks
    (1 vote)
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    • spunky sam orange style avatar for user Willy McAllister
      Electric potential measures the force on a unit charge (q=1) due to the electric field from ANY number of surrounding charges. When we define electric "potential" we set the test charge to 1 and allow the other charge in Coulomb's Law to be any value. Voltage is defined in terms of the potential of the q=1 unit charge.
      (2 votes)