# How to choose the surface for the magnetic flux in case of an open circuit (Faraday Law)?

Each time I've applied Faraday Law ($$EMF = -\frac{d\phi(B)}{dt}$$), such magnetic field crossed a loop conductor (for instance a coil). So, the surface that should be considered for the flux $$\phi(B)$$ computation was the area inside that loop.

In an open circuit, it's known that a time variation of magnetic field flux causes an induced voltage, but obviously not an induced current (let's think for instance at the secondary circuit of a transformer without a load).

But, in this situation, which surface should we consider to calculate the flux? Let's see some examples:

1. Partial conducting circular loop

Which is the induced voltage $$V_A-V_B$$? Well, we may imagine to close that partial loop in several ways: with a straight line (first picture), with a circular arc (secondpicture), or with whatever you want strange curve (third picture):

Obviously the enclosed area will be different, and so the induced voltage. But I'd say that, given a time - varying magnetic field, the induced voltage $$V_A-V_B$$ will have a unique definite value (or time - dependent expression). So, which is the surface that allows us to evaluate the EMF?

1. There are several situations (unfortunately with not simple explanations) in which I have seen the computation of the inductance of an open circuit. For instance, the inductance of the wire. Or maybe also, the inductance of a dipole antenna (see page 2 if you want to read details), which is simply a couple of wires left open.

In those situation, those authors evaluated the magnetic flux $$\phi(B)$$ across a specific surface around the device (the wire, the dipole antenna etc), without explaining how can we decide the surface.

It seems to me that the point is: if a there exists a time - varying magnetic field, there is some energy due to the fact that such a time - varying magnetic field flows across a surface.

But if this statement is true, we should each time choose an infinite surface: all the space!

1. I do not understand why this topic is so ignored by all (at least, my) textbooks. I don't think it describes a so strange and unreal situation.

Let's think about a couple of straight wires used as an antenna (receiving dipole antennas).

An incident electromagnetic wave will induce voltage between its arms that represents the received signal. Well, how can we evaluate such a voltage if we don't know which surface should we consider?

Obviously the enclosed area will be different, and so the induced voltage.

But I'd say that, given a time - varying magnetic field, the induced voltage VA−VB will have a unique definite value (or time - dependent expression). So, which is the surface that allows us to evaluate the EMF?

The area will be different, but the "induced voltage" has nothing to do with these imaginary areas. Voltage between two such points is determined by response of the wire to induced EMF in the wire.

In other words, you are misunderstanding Faraday's law. It does not say anything directly about voltages; it only says change of magnetic flux through any closed loop will produce net EMF for that loop. Voltage between two points (=difference of the Coulomb electric potential) that may result in response to that EMF depends on other things, such as whether the circuit is closed or open, resistance of the wire. If resistance is zero, voltage between A and B will have the same magnitude as EMF from A to B; if resistance if very big, voltage will be much lower.

EMF for a piece of wire A to B is defined as integral of induced electric field from A to B. This integral has definite value at any time, but it can't be determined from Faraday's law alone; positions of points A and B are important. So (in general case) there is no point in completing the wire into a closed loop and calculating flux through that loop - the loop completion and magnetic flux is arbitrary, as you correctly observe. But EMF for the wire A to B isn't arbitrary, it has definite value

$$\text{EMF}_{AB} = \int_A^B \mathbf E_i \cdot d\mathbf r.$$

This cannot be determined using Faraday's law only; we have to measure induced electric field $$\mathbf E_i$$ at all points of the wire, or determine $$\mathbf E_i$$ by calculation from other known things.

However, there are special cases where this integral has the same value as loop EMF for some particular loop, and this can be used to calculate this integral; see below.

There are several situations (unfortunately with not simple explanations) in which I have seen the computation of the inductance of an open circuit. For instance, the inductance of the wire [...] if a there exists a time - varying magnetic field, there is some energy due to the fact that such a time - varying magnetic field flows across a surface [...] But if this statement is true, we should each time choose an infinite surface: all the space!

I do not understand why this topic is so ignored by all (at least, my) textbooks. I don't think it describes a so strange and unreal situation.

Calculating self-inductance is simple only in simple cases such as solenoidal or toroidal inductor. The relevant magnetic flux and closed loop are chosen in such a way that the loop passes at least partially through the wires/conductor boundaries so that the EMF for the loop is relevant to induced EMF in those conductors. It is hard to give general description, but in all valid cases the loop goes through conductors somewhere, and where it does not, it is very far away from the system (infinity) or has some symmetry so contributions there can cancel each other and so be ignored.

Let's think about a couple of straight wires used as an antenna (receiving dipole antennas). An incident electromagnetic wave will induce voltage between its arms that represents the received signal. Well, how can we evaluate such a voltage if we don't know which surface should we consider?

Voltage between end-points is a result of (or measure of) displacement of electric charges in the antenna. This displacement is a result of finite electric current flowing for some non-zero time. In ohmic conductor, this current flows only if net electric field exists in the conductor. This can be due to external electric field of the wave. Electric current may increase in time, but the speed is damped by induced electric field of the antenna itself (self-inductance effect).

This self-inductance effect is related to Faraday's law: imagine any closed loop whose segment goes through the linear antenna (two rod-like conductors of non-zero thickness) but then turns right at 90 degree angle and continues to infinity on both sides of the antenna. There are infinity of such loops of different size and for each of them Faraday's law is valid. Different loop will have different change of magnetic flux and correspondingly different loop EMF. However due to geometry of the conductor, induced electric field everywhere will be nearly parallel to the antenna. So only the segment parallel and close with the antenna contributes appreciably to the loop EMF (the other parallel segment is very far away and induced field there is negligible). If we choose the loop in such a way that this segment ends where the antenna conductor ends, then loop EMF has the same value as the sought EMF for the antenna segment. We can calculate the loop EMF using Faraday's law for the infinite loop and this is approximately giving EMF for the antenna.

In the limit of zero resistance, this EMF has the same value as voltage. In reality, voltage is weaker, as some electric field has to be left in the conductor to drive the motion of mobile charges.

• Self-inductance is due to EMF for line segment coincident with the dipole antenna, stretching from one endpoint of antenna to the other. It can be expressed in terms of flux $\Phi^*$ for the particular loop which is relevant to antenna EMF. Here the only relevant loop is that described, whose loop EMF equals antenna EMF. Other loops are not usable because they either have greater EMF due to contributions outside the antenna which does not act on the antenna, or they have smaller EMF because they do not include part of the antenna. – Ján Lalinský Dec 1 '20 at 13:28
• The formula $L = \Phi / I$ is not valid for every loop. It is only valid for those loops that have the same value of EMF the conductor experiences. – Ján Lalinský Dec 1 '20 at 13:31
• Part of the loop has to coincide with the conductor, the rest has to contribute zero or some known amount so it can be subtracted. – Ján Lalinský Dec 1 '20 at 15:53
• Beware that this method of calculating inductance is only approximate. There is inherent problem with these calculations that non-zero dimensions of the wires themselves (thickness) have to be accounted for to prevent infinite magnetic flux, and also skin effect makes the current concentrate near the conductor-nonconductor boundary so the space region relevant for finding EMF is not simply a line going through the center of the conductor, but it is the whole region containing the electric current. All this makes calculations much harder and the only general viable method is to use computers. – Ján Lalinský Dec 1 '20 at 15:59
• .. to simulate EM field interaction with the antenna and infer effective inductance from that. – Ján Lalinský Dec 1 '20 at 16:01