# Induced voltage as a difference of potentials on a ring

If you put a ring (Picture 1) of wire into changing magnetic field, the process called magnetic induction creates induced voltage. Voltage is by definition (Definition 2) the difference in electric potential between two points. But where exactly on a ring are this two points of mentioned electric potential? If they were stationary, it would not be possible to measure induced current in the looped piece of wire, but we can absolutly do that. So my question is... What is induced voltage on a ring? Where are this two points of electric potential? Does the definiton of voltage make any sanse in this case?

""Voltage, electric potential difference, electromotive force (emf), electric pressure or electric tension is the difference in electric potential between two points"" (Definiton 2; from https://en.wikipedia.org/wiki/Voltage)

Picture 1

(a) "Voltage is by definition (Definition 2) the difference in electric potential between two points." No, you are defining potential difference, a concept that can't be usefully applied to this situation.

(b) If there is a potential difference between two points A and B then a test charge taken from one point to the other will have work done on it by an electric field of an amount independent of the path taken from A to B. This is clearly not the case here. If you choose any two points on the ring, positive work will be done on the test charge if you take it from A to B in one sense round the ring, and negative work if you take it in the other sense. The concept of electric potential is inapplicable.

(c) The electric field generated in the ring if we continuously increase or decrease the magnetic flux linked with the ring is, we say, a non-conservative field. It isn't like the electric field due to static charges, which is a conservative field, to which we can apply the concepts of potential and potential difference.

(d) If the ring were made of metal and had a gap in it, then we would be able to talk about the potential difference between the cut ends. This is because mobile charges are urged round the ring by the induced electric field so one end gets a positive charge and the other a negative. These (essentially static) charges give rise to a conservative electric field.

(e) Back to 'voltage'. This is a more general term meaning work done per unit charge on a test charge. So we can talk about the voltage induced in the ring, meaning the line integral of the electric field strength as we go once round the ring. Specifically, this is the emf induced in the ring.

Magnitude of the voltage induced in the Conductor also depends upon the angle between the magnetic field lines and the conductor.

Changing magnetic field lines will make different angles with each point on the rings and therefore will cause different voltages at different points and that will make the current flow in the ring.

Once $$\partial {\bf B}/{\partial t}\ne 0$$ the electric field cannot be written as $$-\nabla V$$ with a single valued $$V$$, so there is no "potential." The idea of an electric potential is a concept from electrostatics.

Indeed if you take a voltmeter and try to read a potential difference between two points on the ring the reading will depend on how you dispose the leads from the probes to the meter. This is because there will be a changing flux in the loop made of the bit of ring between the probes, the leads and the meter.

The concept of electric potential goes out the window for changing magnetic fields, as $$\nabla\times\mathbf{E}=-\frac{\partial \mathbf{B}}{\partial t} \neq \mathbf{0},$$ or, equivalently, the line integral of the electric field is dependent on the path taken. This means that the potential difference between two points is no longer uniquely defined, as it depends on the path taken.

We must therefore turn to a new concept: emf. The emf associated with a loop $$C$$ is the line integral around $$C$$ of the force per unit charge. In electromagnetism, $$\mathcal{E} = \oint_{C}{(\mathbf{E}+\mathbf{v}\times\mathbf{B})\cdot\,d\mathbf{l}},$$ where $$\mathbf{v}$$ denotes the velocity of the charges as they move around the loop. The flux rule says that the emf induced in any loop is the negative time rate of change of the flux of the magnetic field through a surface bounded by the loop: $$\mathcal{E}=-\frac{d\Phi_B}{dt}.$$ This notion of emf has none of the problems that you discussed. Only the loop as a whole, not two particular points in the loop, factors into the calculation.

Summary:

• The electric potential is not well-defined when the magnetic field is changing. The potential is only useful in electrostatic situations.
• The emf associated with a loop is the line integral around the loop of the force per unit charge. This concept frees us to consider the entire loop instead of two points on it.