# Voltage vs. EMF for a varying magnetic flux against an ohmic circuit

Suppose there is a circular wire, whose material is ohmic with uniform resistivity.

If an increasing magnetic flux is applied to this "circuit", electric current will flow in one direction. Then by Ohm's Law, there will be voltage decreasing in the same direction as the current. Since the wire is a closed circuit, this contradicts with Kirchoff's Voltage Law.

Yet my textbook says that so-called "electromotive force"(EMF) is induced in this case. Is EMF a separate concept from voltage despite that both are measured in Volts? If so, how can the two be distinguished?

If an increasing magnetic flux is applied to this "circuit", electric current will flow in one direction.

Yes, that's right.

Then by Ohm's Law, there will be voltage decreasing in the same direction as the current.

This is a backwards way to think about it. Your presentation to here roughly says:

• There is an increasing magnetic flux
• We take it as an axiom that this starts a current flowing
• By hypothesis, the material is ohmic
• We conclude there is a voltage change when you go around in a circle in the loop of wire

It makes more sense to think of it like this:

• There is an increasing magnetic flux
• We take as an axiom Faraday's Law, which says that a changing magnetic flux causes a voltage drop when going around in a circle around that flux
• By hypothesis the material is ohmic
• Because the material is ohmic and because Faraday's law says there is a voltage drop, a current runs through the loop

Faraday's law is one of Maxwell's equations, which are better basic starting points for thinking about classical electromagnetism than Ohm's law, or the axiom that a changing magnetic flux induces a current.

Since the wire is a closed circuit, this contradicts with Kirchoff's Voltage Law.

Correct. Kirchoff's voltage law is not true when there are time-varying magnetic fields.

Yet my textbook says that so-called "electromotive force"(EMF) is induced in this case.

Yes, this is another name for the voltage drop we know about due to Faraday's law.

Is EMF a separate concept from voltage despite that both are measured in Volts?

In this case, I think of them as the same thing. In some cases, I would say there is an EMF but no voltage drop. For example, in your setup, I presume the magnetic flux is changing because you're making the magnetic field stronger or weaker over time, or changing its direction. In this case, there is a voltage drop around the circuit and that voltage drop is the EMF.

Another possibility is that the magnetic field is constant in time, but it's weaker in some places and stronger in others. You take the circuit and move the circuit around. If you move it to some place where the magnetic field is stronger, there is more flux through the circuit. However, although current runs through the wire, there is not actually any voltage drop around the wire in this case. Instead, the electrons in the wire are driven around by the Lorentz force. In this case, I would say there is an EMF, but not a voltage.

That is, I'd identify "voltage change" or "voltage drop" around a circuit as the line integral of the electric field around the circuit. However, I'd identify the emf around a circuit as the line integral of the electromagnetic force per unit charge a charged particle feels when traveling around the circuit, either due to electric force or the Lorentz force. However, I can't guarantee that every source distinguishes the words "voltage" and "emf" in exactly this way.

• If Kirchoff's Voltage Law is false, does that mean the wire's electric potential difference against the ground (e.g. voltage at infinite points) is undefined? – Dannyu NDos Oct 15 '18 at 13:41
• Yes............. – Mark Eichenlaub Oct 15 '18 at 14:00
• @Mark Eichenlaub I'll leave aside the moving circuit to be short. To me the situation is clear: because of changing $B$, in the circuit there is an electric field (otherwise charges would not move, in presence of resistivity). Ohm's law holds, in the sense that $V=RI$ when $V$ is the line integral of $E$ along a tract of wire, $R$ its resistance, $I$ the current. But $E$ is not conservative, so that its line integral along the whole loop is not zero, and this is the emf. However no "voltage" (an ugly word, btw) exists as a well defined quantity in all points of the wire. – Elio Fabri Oct 15 '18 at 16:00
• I'm just using the definition of a voltage difference as the line integral of the electric field, like my answer says. Of course, that means that the voltage difference between two points is not a fixed number, but a function of the path you choose between the points, but so be it. It sounds like you just want to quibble that I'm not allowed to use the word "voltage" that way. I don't know of any authoritative source on how we're allowed to use words. You use it your way. I'll use it my way. I don't think anything in my answer is physically wrong or not stated clearly. – Mark Eichenlaub Oct 15 '18 at 18:53
• > "I'm just using the definition of a voltage difference as the line integral of the electric field," And that is incorrect and lead you to incorrect statements. Using your definition, voltage difference for two close points depends on the path used in the integral and can be made arbitrary low or high. That integral is properly named electromotive force, while voltage is reserved for difference of electrostatic potential. – Ján Lalinský Oct 17 '18 at 11:25

I wish to add a supplement in order to better explain my criticism to the term "voltage". There are two different issues:

• Is it correct to speak of voltage in the example we are discussing?
• Is "voltage" an acceptable physical term?

(1) @Mark Eichenlaub writes in his answer:

In this case, I think of them as the same thing. In some cases, I would say there is an EMF but no voltage drop. For example, in your setup, I presume the magnetic flux is changing because you're making the magnetic field stronger or weaker over time, or changing its direction. In this case, there is a voltage drop around the circuit and that voltage drop is the EMF.

and in his comment:

I'm just using the definition of a voltage difference as the line integral of the electric field, like my answer says. Of course, that means that the voltage difference between two points is not a fixed number, but a function of the path you choose between the points, but so be it.

In my opinion such an approach is not tenable. If you speak of a "voltage difference" between two points of the circuit, say A and B, you are assuming that is the difference of values of a physical quantity (the "voltage" defined in each point: there should be a "voltage at A" $$V(\rm A)$$ and a "voltage at B" $$V(\rm B)$$. Nor can we speak of a "voltage drop", for the same reason. That would mean that something "drops", i.e. changes its value, when going from A to B.

But if you agree that potential difference is not a fixed number, it is no difference at all. It is the line integral of electric field, but the names "voltage difference" or "voltage drop" shuld be avoided.

Worse still, this "voltage difference" cannot be measured. If you attach a voltmeter's terminals between A and B, you will read a different number according to where the voltmeter is physically placed and which path the voltmeter's wires follow. It looks peculiar to speak of a "voltage difference" that a voltmeter cannot measure.

This is not quibbling, and I can't agree that anyone may use words his own way. Not when talking of science. Scientific language exists to allow people to communicate one with another. One of the first tasks a science teacher has to set himself is to get his pupils used to employ words according precise meanings, often different from the ones in everyday language.

(2) I dubbed the word "voltage" ugly. It's an opinion, but with some reasons. I'm aware this word is common in electricity writing and speaking, at least in english. Not in other european languages, AFAIK.

One argument is that if you like to call "voltage" the physical quantity measured in volt, you ought to be coherent, and also say "metrage", "wattage", "joulage", and so on. I can't see why a special linguistic treatment should be reserved to the particular quantities I (and many others around the world) prefer to name "electric potential" or "potential difference".

The second argument is more scientific. "Voltage" comes from "volt", the name of unit in SI. It isn't a good practice to mix the concept of a physical quantity with that of its unit. First, because units can change (there were, and still are, several systems of units) whereas the p.q. stays the same. Second, and more fundamental, because when we reason, as in our case, about what happens in a physical system, our main interest are p.q.'s and not their units. We have a varying magnetic field (a p.q.) which induces an emf (a p.q.) in the circuit and causes an electric field (another p.q.) and an electric current (a further p.q.). A good teacher must insist that p.q.'s are what matters in the first place, that physical laws are about p.q.'s, etc.

All this is well known, but some practices (like naming "voltage" a potential difference) go against good teaching. At least, I repeat, this is my opinion. Nothing more.

Then by Ohm's Law, there will be voltage decreasing in the same direction as the current.

Ohm's law has different forms, which are not always equivalent. You refer to the simple form

$$I = \frac{U}{R}$$

where $$U$$ is difference of potential of two points, called voltage between the two points, and in this form it does link current to that voltage. This relation is obeyed only in the simple cases where the concept of voltage makes sense and EM induction is negligible everywhere on the wire. For example, it works for DC current or low frequency current in a straight wire.

It does not work when EM induction is taking place in the wire, such as AC current changing in a wire winded into a solenoid. In other words, the above form of Ohm's law is not valid in a solenoid. This is because in EM induction setups, voltage, although it may be defined, is not indicative of the electromotive force $$\int_A^B \mathbf E\cdot d\mathbf s$$ pushing the electrons along the wire from A to B.

Ohm's law has a more general form

$$\mathbf j = \sigma \mathbf E$$ meaning, when electric field inside the material medium is varying, electric current density at that point is always proportional to it.

This form of Ohm's law is working well even if EM induction is taking place. It can be integrated over the cross-section of the wire $$S$$ and restated as

$$I = \sigma S E_n$$ where $$E_n$$ is a component of electric field in the direction of the wire.

If the current is constant along the wire of length $$L$$ (the usual case for regular wires with constant cross-section $$S$$), we can rewrite it as

$$I = \frac{\sigma S}{L} E_n L$$

or

$$I = \frac{\mathscr E}{R}$$ where

$$\mathscr E = \int \mathbf E \cdot d\mathbf s$$ is one example of total electromotive force (in this case, due to total electric field), and $$R = \frac{L}{\sigma S}$$ is Ohmic resistance of the wire. In other setups there may be several electromotive force (such as those due to thermal or chemical nonequilibrium inside the medium), in which case we add them algebraically.

Since the wire is a closed circuit, this contradicts with Kirchoff's Voltage Law.

The KVL was inferred from experiments with usual circuits where potential is defined on every terminal of every component so the concept of voltage can be introduced. It makes no sense to apply the KVL to a ring hanging in space.

In your setup where the ring just hangs in space and is not connected to any body of known potential, there is no voltage to be talked about, because the potential of the ring is not guarranteed by anything. The potential is not guarranteed to drop inside the ring in the direction of the indcued current. Ohm's law in the simple form is not valid for the ring, just as it is not valid for a solenoid.

Yet my textbook says that so-called "electromotive force"(EMF) is induced in this case. Is EMF a separate concept from voltage despite that both are measured in Volts? If so, how can the two be distinguished?

Of course, they are different things. EMF is an integral of force acting on the charge carriers, along some piece of wire, or along some path in space. In this case, the force is due to electric field, so

$$EMF_A^B = \int_{A, through~wire}^B \mathbf E\cdot d\mathbf s.$$ EMF depends on the path, not only the endpoints. Thus the EMF for a path entirely inside wire may be different than EMF for a path that goes outside the wire. For Ohm's law, we need EMF for path inside the wire.

Voltage is a difference of electrostatic potentials of two points, or integral of electrostatic component of electric field:

$$U_A^B = \int_A^B \mathbf E_{elstat}\cdot d\mathbf s.$$ It does not depend on that path, only on the endpoints.

Often the EMF can be simplified into voltage difference (no induction). But if induction is taking place, or other electromotive forces are present, voltage becomes secondary and one must use total electromotive force in the Ohm law.

In cases such as AC current in solenoid, electromotive force for path inside the wire has different sign and magnitude from the voltage between the terminals, which is best example that they are different things.