# Definition of voltage (potential difference)?

I understand the definition to be the amount of work done in moving a charge from one point to another divided by the charge.

If you have a standard 1.5 volt battery, the charge should move easily from one end of the circuit to the other because the positive terminal attracts the electrons. Thus, it would take less work to move the charge than if the terminals were both neutral.

However, according to the formula V = W/Q if it takes less work, the voltage is less, and if it takes more work, the voltage is more, which goes against my intuitive thinking. Am I missing some piece of information?

tiny-tim
Homework Helper
hi krackers! I understand the definition to be the amount of work done in moving a charge from one point to another divided by the charge.

no, potential energy is defined as minus the work done (by a conservative force)

so voltage difference = potential energy difference per charge = minus the work done per charge: V = -∫ E.dx (it is often described as the work done against the electric field, eg in bringing a charge "from infinity", which is the same as minus the work done by the electric field)

Andrew Mason
Homework Helper
I understand the definition to be the amount of work done in moving a charge from one point to another divided by the charge.
I think your confusion may be with respect to the convention used. The convention was developed before anyone knew about electrons and their negative charge.

Potential energy between points A and B can be thought of as the work done by the electric field on a POSITIVE charge in moving from point A to point B in an electric field. It could also be thought of as -1 x the work that must be done against the electric field on a POSITIVE charge in moving it from one point to another. So in moving a positive unit charge (one Coulomb of positive charge) from the negative terminal to the positive terminal of the battery, one would have to do 1.5 Joules of work on it, so it has -1.5 Joules of potential energy at the negative terminal with respect to the positive terminal. At the positive terminal it has +1.5 Joules of potential energy relative to the negative terminal.

For negative charges, you have to change the sign, as Tiny-Tim says. So at the negative terminal, one C. of negative charge has +1.5 J. of energy relative to the positive terminal.

AM

tiny-tim
Homework Helper
Potential energy between points A and B can be thought of as the work done by the electric field on a POSITIVE charge in moving from point A to point B in an electric field. It could also be thought of as -1 x the work that must be done against the electric field on a POSITIVE charge in moving it from one point to another.

andrew, isn't it the other way round? from http://en.wikipedia.org/wiki/Voltage
Voltage is equal to the work which would have to be done, per unit charge, against a static electric field to move the charge between two points.​

Andrew Mason
Homework Helper
andrew, isn't it the other way round? from http://en.wikipedia.org/wiki/Voltage
Voltage is equal to the work which would have to be done, per unit charge, against a static electric field to move the charge between two points.​
This definition is ambiguous. It does not specify that potential difference is a measure of the potential energy of a POSITIVE unit charge; and it does not specify at which of the two points one measures the potential energy. It should say that the potential difference at B relative to A is the work done on the unit positive charge against the field in moving from A to B or -1 x the work done on the unit positive charge against the field in moving from B to A.

AM

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tiny-tim
Homework Helper
It should say that the potential difference at B relative to A is the energy done on the unit positive charge against the field in moving from A to B …

so when you say …
Potential energy between points A and B can be thought of as the work done by the electric field on a POSITIVE charge in moving from point A to point B in an electric field.

… you mean that's the PE at A minus the PE at B ? Perhaps this will help.

Point b in an electric field is said to be at a higher potential than point a in the same field if external work is done moving a positive charge from a to b.

If a positive charge is moved from a point of higher potential to one of lower potential work can be extracted from that movement.

This situation is reversed for the moving of negative charges.

The work done or extracted depends only on the position of a and b not on the path taken between them.

If you have a standard 1.5 volt battery, the charge should move easily from one end of the circuit to the other because the positive terminal attracts the electrons. Thus, it would take less work to move the charge than if the terminals were both neutral.

However, according to the formula V = W/Q if it takes less work, the voltage is less, and if it takes more work, the voltage is more, which goes against my intuitive thinking. Am I missing some piece of information?

Between the terminals of your battery there is an electric field.

The positive terminal is so designated because it is at higher potential than the negative one.

So if I move 1 coulomb of electrons from the positive terminal to the negative one I must do 1*1.5 = 1.5 joules of work on the electrons.

If I move 1 coulomb of electrons from the the negative terminal to near the positive terminal I can extract 1.5 joules of work.
This second scenario is what happens if we consider electron current in a resistor connected across the battery terminals. The work is extracted as heat.

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Andrew Mason
Homework Helper
so when you say …

… you mean that's the PE at A minus the PE at B ? Yes. (I meant to say "work done" not "energy done").

The potential difference is the difference between the potential energies of a unit (positive) charge at each point measured relative to a common point (such as infinity). But that is not really very useful. Do we really care what the potential of the positive terminal of the battery is relative to any point other than the negative terminal?

AM

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I understand the definition to be the amount of work done in moving a charge from one point to another divided by the charge.

If you have a standard 1.5 volt battery, the charge should move easily from one end of the circuit to the other because the positive terminal attracts the electrons. Thus, it would take less work to move the charge than if the terminals were both neutral.

However, according to the formula V = W/Q if it takes less work, the voltage is less, and if it takes more work, the voltage is more, which goes against my intuitive thinking. Am I missing some piece of information?

You are missing the internal chemical potential. There is an internal chemical potential in the electrolytic solution of the battery that forces the electrons to pile up in the negative electrode. The chemical reactions in the battery are forcing the electrons to move against the electric field. Outside the battery, the internal chemical potential is zero.
There is a “pseudoforce” on electrons that has nothing to do with electric charge. This chemical force is quantum mechanical in nature. It is not directly associated with an electric field. It has to do with chemical bonding and the Pauli exclusion principle. The pseudoforce results in a chemical analog to electrostatic potential. This chemical analog is referred to as the chemical potential.
The difference in potential that you are talking about is the electric potential from one electrode to the other. The electric potential is caused by the electric field that goes through the electric conducting wire, avoiding the electrolytic solution. However, there is also a chemical potential. This is basically caused by the chemical reactions that are occurring along the path from one electrode and the other through the electrolytic solution.
An electron that was forced to go through the electrolytic solution would have to pass through a gradient of ions that are maintained because of chemical bonding. The potential is caused by the fact that some atoms and ions attract electrons, or repel them, more than others. Thus, the electron is prevented from going through the electrolytic solution by the chemical potential.
The study of how chemical reactions generate electric current is called electrolytic chemistry. Electrolytic chemistry explains how electric currents move through batteries and other voltage generating structures. Electrolytic chemistry is also important in the study of corrosion, because electric current can speed up oxidation.
If you move 1 coulomb of electrons from the negative terminal to the positive terminal, on a path through the battery instead of the circuit, you will cause chemical reactions that charge the electric battery. Your question is equivalent to asking how to charge an electric battery of 1.5 V.
You would need a voltage difference in excess of 1.5 volts to recharge a battery that ordinarily provides 1.5 volts. The current, going in the "wrong" direction through the battery, will cause chemical reactions in reverse of the ones that create the voltage in the battery.

http://en.wikipedia.org/wiki/Chemical_potential
“The abstract definition of chemical potential given above—total change in free energy per extra mole of substance—is more specifically called total chemical potential. If two locations have different total chemical potentials for a species, some of it may be due to potentials associated with "external" force fields (Electric potential energy differences, gravitational potential energy differences, etc.), while the rest would be due to "internal" factors (density, temperature, etc.) Therefore the total chemical potential can be split into internal chemical potential and external chemical potential:"

Inside the electrolytic solution of the battery, there are two types of chemical potential. One is just the chemical potential caused by the electric field that you apply to the electrodes. This is the external chemical potential. The other chemical potential is caused by the chemical reactions and diffusion through the fluid. This is the internal chemical potential.
The external chemical potential is the voltage that the battery applies to the outside circuit. The internal chemical potential is the pseudoforce on the ions caused by the chemical reactions.

Andrew Mason
Homework Helper
There is a “pseudoforce” on electrons that has nothing to do with electric charge. This chemical force is quantum mechanical in nature. It is not directly associated with an electric field. It has to do with chemical bonding and the Pauli exclusion principle. The pseudoforce results in a chemical analog to electrostatic potential.
Can you explain why this is a pseudoforce? There has to be a real electric field (ie. a real electric force on charges) inside the battery. The field is caused by the chemical separation of ions within the battery. The forces that keep the ions separate is chemical, which means it is electrical. Even if it is a quantum effect, the force is still electrical is it not?

AM

Can you explain why this is a pseudoforce? There has to be a real electric field (ie. a real electric force on charges) inside the battery. The field is caused by the chemical separation of ions within the battery. The forces that keep the ions separate is chemical, which means it is electrical. Even if it is a quantum effect, the force is still electrical is it not?

AM

I apologize. I am using the word "pseudoforce" here for any process that can't be explained by classical mechanics. I may be using the word out of context.
The concept of "force" is not fundamental in quantum mechanics. There are effects caused by what could be called wave interference that move the position of the body without a corresponding change in another body. There are odd constructs in quantum mechanics such as "exchange force" or "chemical potential". They allow Newton's Laws to be treated as phenomenological laws. However, they are not entirely consistent with Principia.
"Momentum" is conserved even in quantum mechanics. However, this is a generalized momentum that is partially uncoupled from position by the uncertainty principle.
The simplest example of what I am talking about is the covalent bond between two atoms. The covalent bond is caused when two unpaired electrons "pair". This pairing would take place even if the electrons had no electric charge. The best physical picture that I can find is that the corresponding waves of the two electrons constructively interfere so the maximum intensity of the wave appears between the two nucleii. The probability of the particle "appearing" at any instant is at a maximum where the wave amplitude is at a maximum, which is between the two nuclei.
Once the position of the "appearing" electron is calculated, then classical mechanics can be used in a phenomenological sort of way. The electron has a negative charge. The nuclei have positive charges. Therefore, both nuclei are pulled in the direction of this electron. Because the electron is right between them, the two nuclei are effectively pulled toward each other. So that is a covalent bond.
Note that the "classical" forces don't explain why the electron is right between the two nuclei. That is a "wave" effect. Even better. The classical explanation doesn't explain why the electron doesn't crash into either nuclei and stick.
Physicists have constructed effective forces that explain why the two electrons pair. However, they have to solve Schroedinger's equation first and then construct a "force" that is consistent with the "pairing".
I heard you think "magnetic dipoles". Yes, the pair force to some approximation acts like a magnetic dipole. However, the tendency to pair is much stronger than can be explained using the measured magnetic dipole of the free electrons.
I call it a pseudoforce because one can't really make a simple explanation for it consistent with Principia. The force pulling the two electrons together is a bit like the epicycles in Ptomelic theory. Basically, you don't get a universal explanation of covalent bonding without quantum mechanics.
I feel obliged to raise a flag whenever I talk about "chemical forces". I use the word "pseudoforce" to remind the reader that he isn't in Principia anymore!

Andrew Mason
Homework Helper
I call it a pseudoforce because one can't really make a simple explanation for it consistent with Principia. The force pulling the two electrons together is a bit like the epicycles in Ptomelic theory. Basically, you don't get a universal explanation of covalent bonding without quantum mechanics.
I feel obliged to raise a flag whenever I talk about "chemical forces". I use the word "pseudoforce" to remind the reader that he isn't in Principia anymore!
I think you are going back too far. After all, Newton did not deal with electrical forces so if the test is: "it isn't in the Principia", you are leaving out classical electromagnetic theory.

Should we not just stick to the four fundamental forces of nature but dispense with the concept of well defined objects with well defined positions and movements at the atomic and sub-atomic level?

AM

I think you are going back too far. After all, Newton did not deal with electrical forces so if the test is: "it isn't in the Principia", you are leaving out classical electromagnetic theory.

Should we not just stick to the four fundamental forces of nature but dispense with the concept of well defined objects with well defined positions and movements at the atomic and sub-atomic level?

AM

"After all, Newton did not deal with electrical forces so if the test is: "it isn't in the Principia", you are leaving out classical electromagnetic theory."
When I said classical, I meant without quantum mechanics. Many physicists refer to relativity as classical physics. In fact, general relativity starts with the notion that there are no "pseudoforces." The law of equivalence says that the gravity is locally equivalent to inertia. So I really did mean what I say.
There are all sorts of artificial kluges that make quantum mechanics look more classical. For example, many physicists have worked on developing "pseudopotentials" for atoms. That is, they develop a force law for atoms that they can use in Newtonian style computer simulations. The reason that it is a pseudopotential rather than a potential is that they have to solve Schroedinger's equation to get the force law. I am borrowing the word pseudoforce to describe the forces taht are associated with pseudopotentials. I don't mean any harm!
You are right about one thing. Principia does have certain limitations with respect to electromagnetic forces. It works fine with a certain type of electromagnetic problem. However, there is an important limitation of Principia with respect to electromagnetic theory.
Newton wrote the third law of motion in present tense. His mathematical analysis is consistent with the third law of motion as written in present tense. This means that his analysis is good only for forces that act instantaneously. This means that his analysis only works for contact forces and forces that propagate with infinite speed.
When Newton derives the speed of sound, he hypothesizes contact forces between fluid elements. His hypothesized forces are local, working only on contact. He uses the bulk modulus, which pretty much means elastic forces that work on contact. So his "speed of sound" has nothing to do with the speed of the forces between objects at a distance. So in effect, the speed of sound is independent of the speed of the forces involved. In his analysis of sound, there is no delay between action and reaction.
Principia works very well for forces that are effectively "acting at a distance". If there were no delay between action and reaction, then the dynamics will satisfy the famous three laws of motion with no difficulty.
If the delay between action and reaction in the third law is negligible, then for all practical purposes the force is instantaneous. Hence, Principia can be used with almost no modification as long as the electromagnetic fields are effectively static. Principia can be used in the limit of the near field approximation even if there is some time variation in the electromagnetic fields.
Benjamin Franklin and Greene managed to analyze electrostatic problems by assuming that electric charges obeyed an analogous law to gravity. They did not postulate any deviation from the three laws of motion. Their type of analysis was classical electromagnetic theory at its most classical.
There is a delay in electromagnetic forces that is related to the speed of light in a vacuum. However, the third law of motion was written in the present tense. If the distances in a problem are large enough that the delay time is significant, then the problem is beyond Principia's range of validity.
This little problem with delay time was what motivated H. A. Lorentz to develop his "Theory of Electrons" (1915). Of course, it was also crucial when A. Einstein developed relativity. Einstein started with different hypotheses, but finished with the importance of the delay time.
Which is irrelevant to chemical potential. Chemical potential works for getting a certain amount of "classical mechanics" into the calculations. For instance, the laws of thermodynamics work fine with chemical potential. If ones sticks to chemistry, and not think about the physical basis of chemical potential, then one can solve all sorts of problems without difficulty. You don't need a deep understanding of quantum mechanics to do chemistry. If you want to know why "chemical potential", then you will need a deeper understanding of quantum mechanics.
Don't get me wrong. One needs at least a shallow understanding of quantum mechanics to do chemistry. However, a very deep understanding of quantum mechanics may slow you down. I find it helpful to think of chemical potential as a pseudopotential. Something that allows one to use classical mechanics, sometimes.

Forgive me, Darwin, but haven't you put in a great deal of way off topic writing here?

Look again at the thread title and the level at which it is being asked.

Perhaps this will help.

Point b in an electric field is said to be at a higher potential than point a in the same field if external work is done moving a positive charge from a to b.

If a positive charge is moved from a point of higher potential to one of lower potential work can be extracted from that movement.

This situation is reversed for the moving of negative charges.

The work done or extracted depends only on the position of a and b not on the path taken between them.

Between the terminals of your battery there is an electric field.

The positive terminal is so designated because it is at higher potential than the negative one.

So if I move 1 coulomb of electrons from the positive terminal to the negative one I must do 1*1.5 = 1.5 joules of work on the electrons.

If I move 1 coulomb of electrons from the the negative terminal to near the positive terminal I can extract 1.5 joules of work.
This second scenario is what happens if we consider electron current in a resistor connected across the battery terminals. The work is extracted as heat.

Thanks for the clarification. I understood the moving of the electron from the lower potential to the higher. However, you mention moving a positive charge. As I understand, there is no such thing as a "positive charge". What we call positive is only the absence of electrons, allowing the positive charge of the proton to be felt. With that in mind, then how is it possible to move the positive charge? Does the proton itself move?

@Darwin123, thank you for your effort, but I'm not ready to go that deep yet :P.

Of course there is positive charge.

Any postive ion can carry current in solution (or in a beam such as a beam of alpha particles).

Equally, and this is important, if I attached some positive or negative charge to the end of a lollipop and moved it from near one terminal to the other of the battery through the air around the terminals, I would have to do the work stated.

However that is not the point.

Movement of positive charge towards higher potential is equivalent to movement of negative charge towards lower potential. It's just the way the original definitons panned out when they were made, so all the equations of electricity (electrodynamics, electrostatics, electromagnetism etc etc) are all conventionally written in terms of 'postive' charge.

But its all relative and all just a model of reality that gets the desired results.

Here is my ruler analogy.

"I have an 18inch ruler and a 6 inch ruler.

That is the length of ruler A is 18 inches and the length of ruler B is 6 inches.

But the length difference between the two rulers is 12 inches.

Both length and length difference are measured in inches.

So it is with EMF and Potential Difference which are both measured in volts."

If you like we can take this further.

However, you mention moving a positive charge. As I understand, there is no such thing as a "positive charge". What we call positive is only the absence of electrons, allowing the positive charge of the proton to be felt. With that in mind, then how is it possible to move the positive charge? Does the proton itself move?

@Darwin123, thank you for your effort, but I'm not ready to go that deep yet :P.

The proton doesn't have to move. In a solid metal, the protons don't move very much.
Imagine a metal crystal where all the atoms except one are neutral. That one atom has an electron missing. Therefore, that one atom, atom A, has a positive charge.
All nuclei except nucleus A has a number of electrons around it equal to the number of protons in the nucleus. One nucleus has one less electron than the number necessary to keep the atom neutral.
Imagine an electron jumping from atom B to atom A. Now atom B is positive while atom A is neutral. Now imagine an electron jumping from atom C to atom B. Now atom C is positive and atom B is neutral.
Notice that in this case, the positive charge is moving while the nuclei are standing still. One may call this hypothetical positive charge a hole.

Note that "the protons" could move in an electrolytic solution. In a salt water solution with an electric current, the positive charges are carried by sodium ions. The sodium ions do move.

I just wanted to share. You say that you are not ready for this. So I will back off now.

The proton doesn't have to move. In a solid metal, the protons don't move very much.
Imagine a metal crystal where all the atoms except one are neutral. That one atom has an electron missing. Therefore, that one atom, atom A, has a positive charge.
All nuclei except nucleus A has a number of electrons around it equal to the number of protons in the nucleus. One nucleus has one less electron than the number necessary to keep the atom neutral.
Imagine an electron jumping from atom B to atom A. Now atom B is positive while atom A is neutral. Now imagine an electron jumping from atom C to atom B. Now atom C is positive and atom B is neutral.
Notice that in this case, the positive charge is moving while the nuclei are standing still. One may call this hypothetical positive charge a hole.

Note that "the protons" could move in an electrolytic solution. In a salt water solution with an electric current, the positive charges are carried by sodium ions. The sodium ions do move.

I just wanted to share. You say that you are not ready for this. So I will back off now.

No, that was perfect :D.