Which law defines the AC inducing voltage in the inductor?

[Dario56]

If we connect an inductor without ohmic resistance to the alternating voltage source, voltage should induce in the inductor because of the Faraday's law. Voltage is induced by changing magnetic flux through the inductor which is accomplished by alternating current through it. Therefore, in order for us to find the induced voltage, we need to know what that current is, $I(t)$.

Which law defines this current? If we say, Faraday's law (or Kirchoff's voltage law in the context of the lumped model), we're already assuming that the voltage on the inductor is equal to the source voltage or that we know what the voltage is. But, we can't know this prior to knowing what the current, $I(t)$ is, as voltage is induced by changing the current through the inductor. If you ask me, there is a circularity in this reasoning which is often used in textbooks.

I'd highly appreciate your thoughts as I'm confused.

 

[vanhees71]

It's Faraday's Law. You have
$$L \dot{i}=U(t)=U_0 \cos(\omega t),$$
which gives by simple integration
$$i(t)=\frac{U_0}{\omega L} \sin(\omega t)=\frac{U_0}{\omega L} \cos(\omega t-\pi/2),$$
i.e., the current lags behind the voltage sourece by a phase shift of $\pi/2$.

In complex notation, where one puts $U(t)=\mathrm{Re} U_c(t)$ with $U_c(t)=U_0 \exp(\mathrm{i} \omega t)$ (sign convention in the exponential as by most electrical-engineering textbooks) you have
$$L \dot{i}_c = U_0 \exp(\mathrm{i} \omega) \; \Rightarrow \; i_c=\frac{U_0}{\mathrm{i} \omega L} \exp(\mathrm{i} \omega t),$$
from which you get immediately the right phase shift, because $1/\mathrm{i}=\exp(-\mathrm{i} \pi/2)$, and the impedance for the ideal inductance,
$$Z_L=U_c/i_c=\mathrm{i} \omega L.$$

 

[DaveE]

The voltage on/in your inductor isn't induced, it's applied by your voltage source. You have defined it. That determines all of the other parameters (flux, current, etc). I guess you could argue that voltage is induced in each coil to force sharing in each, but they must sum to the applied source voltage.

 

[Dale]

If we say, Faraday's law (or Kirchoff's voltage law in the context of the lumped model), we're already assuming that the voltage on the inductor is equal to the source voltage or that we know what the voltage is. But, we can't know this prior to knowing what the current, I(t) is, as voltage is induced by changing the current through the inductor.

Frankly, this is backwards. Where do you think that Faraday’s law or Kirchoff’s laws come from? We know that those models work because we have well over a century of experience that backs them up. We have measured the voltages and currents many many many times. In fact, we measured them and validated those assumptions before making the models. The models were developed precisely to match those assumptions which we had already validated experimentally.

So of course we can make that assumption. The statement that we cannot know how it behaves simply ignores a vast body of experimental knowledge.

 

[SredniVashtar]

Maybe your confusion stems from the fact that you are trying to explain a transient effect (the establishment of an equilibrium condition between alternating voltage and current) with a theory that assumes said equilibrium has been reached an infinite time ago.

For example, if by alternating you mean sinusoidal steady state, then the easy relationship between voltage and current is the result of the stimulus being applied to your solenoid at time minus infinity.

 

[DaveE]

If we connect an inductor without ohmic resistance to the alternating voltage source

Maybe your confusion stems from the fact that you are trying to explain a transient effect (the establishment of an equilibrium condition between alternating voltage and current) with a theory that assumes said equilibrium has been reached an infinite time ago.

For example, if by alternating you mean sinusoidal steady state, then the easy relationship between voltage and current is the result of the stimulus being applied to your solenoid at time minus infinity.

OK, but this circuit has an infinite time constant (τ=L/R), since there are no losses. So the transient from the initial conditions never goes away. This, I think, is a common problem in circuit questions here, where idealized mathematical models are used and then referred to as if they were accurate models for real world things. @vanhees71 has showed us the answer, it is the simplest DE ever, it does not decay. Although he did make the normal assumption about zero ICs, an integration constant he implicitly set to zero.

 

[tech99]

If we connect an inductor without ohmic resistance to the alternating voltage source, voltage should induce in the inductor because of the Faraday's law. Voltage is induced by changing magnetic flux through the inductor which is accomplished by alternating current through it. Therefore, in order for us to find the induced voltage, we need to know what that current is, $I(t)$.

Which law defines this current? If we say, Faraday's law (or Kirchoff's voltage law in the context of the lumped model), we're already assuming that the voltage on the inductor is equal to the source voltage or that we know what the voltage is. But, we can't know this prior to knowing what the current, $I(t)$ is, as voltage is induced by changing the current through the inductor. If you ask me, there is a circularity in this reasoning which is often used in textbooks.

I'd highly appreciate your thoughts as I'm confused.

There is an analogue of this in the mechanical world when we accelerate a mass. If we consider mass analogous to inductance, then an accelerating force (applied voltage) will encounter an equal and opposite reaction (the induced voltage). We might suppose then that nothing happens, but the reaction at any instant arises because of the acceleration.

 

[Dario56]

The voltage on/in your inductor isn't induced, it's applied by your voltage source. You have defined it. That determines all of the other parameters (flux, current, etc). I guess you could argue that voltage is induced in each coil to force sharing in each, but they must sum to the applied source voltage.

If we say that the voltage on the inductor is equal to the source because it's connected to the source terminals than we don't need Faraday's law to explan it. We don't need time changing current to explain the induced voltage.

In that case, it seems that causal link is flipped. It's by connecting the inductor to the alternating voltage source that the time changing current is induced.

However, when learning about Faraday's law, causality is usually given in the another direction (changing current induces voltage). At least, that's how I learned it.

My point is, maybe causality goes in both directions.

 

[alan123hk]

I'd highly appreciate your thoughts as I'm confused.

My personal opinion is that Faraday's law defines the relationship between induced voltage and changes in magnetic flux. Since changes in magnetic flux are caused by changes in current, Faraday's law defines the relationship between the voltage and current of an inductor.

So you can calculate the current based on the voltage you define, or the voltage based on the current you define. However, you cannot define a voltage-current relationship that does not follow Faraday's law for an inductor , because that relationship doesn't exist in a real inductor.

 

[Dario56]

My personal opinion is that Faraday's law defines the relationship between induced voltage and changes in magnetic flux. Since changes in magnetic flux are caused by changes in current, Faraday's law defines the relationship between the voltage and current of an inductor.

So you can calculate the current based on the voltage you define, or the voltage based on the current you define. However, you cannot define a voltage-current relationship that does not follow Faraday's law for an inductor , because that relationship doesn't exist in a real inductor.

The thing is, it seems that connecting the inductor to the source defines it's voltage automatically (that makes sense to me). If that's the case, than voltage isn't really induced, it's just set to value of the source due to connection to the source terminals.

Causality direction seems to be flipped, it's by connecting the inductor to the alternating voltage source that induces the time changing current. Maybe causality can go in both directions.

 

[alan123hk]

The thing is, it seems that connecting the inductor to the source defines it's voltage automatically (that makes sense to me). If that's the case, than voltage isn't really induced, it's just set to value of the source due to connection to the source terminals.

When alternating current passes through an inductor, according to Faraday's law, unless the inductance of the inductor is zero, an induced voltage will inevitably be generated. For a zero-resistance inductor consisting of a coil, the induced voltage produced by its current is exactly equal to the source voltage. If there is no such induced voltage, how can we explain why the current does not become infinite?

Causality direction seems to be flipped, it's by connecting the inductor to the alternating voltage source that induces the time changing current. Maybe causality can go in both directions.

To put it simply, the causal relationship is that you first connect the voltage source to the inductor, and then the voltage source provides current to the inductor. Then this current will produce a change in magnetic flux inside the inductor, and this change in magnetic flux will produce an induced voltage. In this way, the induced voltage plus the voltage drop of the ohmic resistance of the coil is exactly equal to the source voltage, so that the inductor current is in a certain stable state.

 

[Dario56]

When alternating current passes through an inductor, according to Faraday's law, unless the inductance of the inductor is zero, an induced voltage will inevitably be generated. For a zero-resistance inductor consisting of a coil, the induced voltage produced by its current is exactly equal to the source voltage. If there is no such induced voltage, how can we explain why the current does not become infinite?To put it simply, the causal relationship is that you first connect the voltage source to the inductor, and then the voltage source provides current to the inductor. Then this current will produce a change in magnetic flux inside the inductor, and this change in magnetic flux will produce an induced voltage. In this way, the induced voltage plus the voltage drop of the ohmic resistance of the coil is exactly equal to the source voltage, so that the inductor current is in a certain stable state.

What is little bit confusing about the inductors is that you're dealing with the non-conservative electric fields. Even though electric field circulation is non-zero in these circuits, potential difference is nevertheless path-independent between the source terminals (if your path is outside the inductor coil) and therefore inductor voltage can be defined meaningfully.

This is because changes in magnetic flux are limited only to the inductor and not other parts of the circuit.

In another words, you can regard the inductor as a resistor (lumped model) and apply Kirchoff's voltage law which generally doesn't apply to non-conservative electric fields.

 

[Dale]

when learning about Faraday's law, causality is usually given in the another direction (changing current induces voltage). At least, that's how I learned it.

That is, unfortunately, a common misconception even by physics instructors. None of Newton’s laws, Maxwell’s equations, or Kirchoff’s laws are causal relationships. The left side does not cause the right side nor vice versa.

The issue is that they do not have the mathematical form for a causal relationship. Causes, by definition, occur at an earlier time than effects. So a correct causal equation will have a form like $$ e(t)=\int_{-\infty}^{t} c(t_r) dt_r$$ or just $$ e(t)=c(t_r)$$ where $t_r<t$, or essentially anything else where the time ordering is unambiguous.

Newton’s laws, Maxwell’s equations, and Kirchoff’s laws all have both sides of the equation occurring at the same time. Neither side comes before the other, so the relationship is not a causal relationship at all. It is simply incorrect to think of them in terms of cause and effect.

If you want to look at an actual causal relationship in EM then look at Jefimenko’s equations or the retarded potentials.

 

[Dario56]

That is, unfortunately, a common misconception even by physics instructors. None of Newton’s laws, Maxwell’s equations, or Kirchoff’s laws are causal relationships. The left side does not cause the right side nor vice versa.

The issue is that they do not have the mathematical form for a causal relationship. Causes, by definition, occur at an earlier time than effects. So a correct causal equation will have a form like $$ e(t)=\int_{-\infty}^{t} c(t_r) dt_r$$ or just $$ e(t)=c(t_r)$$ where $t_r<t$, or essentially anything else where the time ordering is unambiguous.

Newton’s laws, Maxwell’s equations, and Kirchoff’s laws all have both sides of the equation occurring at the same time. Neither side comes before the other, so the relationship is not a causal relationship at all. It is simply incorrect to think of them in terms of cause and effect.

If you want to look at an actual causal relationship in EM then look at Jefimenko’s equations or the retarded potentials.

If casual relationships aren't established then we shouldn't call Faraday's law, electromagnetic induction, but electromagnetic correlation. If you get my point. It's not really correct to say that alternating current from the source induces voltage as that implies casual link. They are correlated and Faraday's law gives the relationship for the inductor. As you said, experiments show the correlation and casual links don't really exist.

In physics, people often do think in terms of cause and effect relationships. We often say that force causes acceleration, but I agree that's actually wrong as force doesn't precede acceleration. Without acceleration, force wouldn't be a force.

Wow, from physics we got to the philosophy of physics.

 

[Dale]

we shouldn't call Faraday's law, electromagnetic induction, but electromagnetic correlation. If you get my point

We can call it a flubnubitz if we want. We just need to understand what the term means. We have agreed to call it induction, it is not a causal relationship, therefore induction is not causation.

In physics, people often do think in terms of cause and effect relationships.

And that is fine. You can unambiguously and correctly say that charge and current densities cause electromagnetic fields. So the issue is not that it is wrong to think in terms of cause and effect relationships. It is just a misunderstanding about which relationships are causal and which are not.

 

[vanhees71]

That's a very important point, even some textbooks state sometimes in a sloppy way, which can easily be misunderstood. For simplicity let's discuss the local form of electromagnetic laws, i.e., Maxwell's equations in differential form, which anyway is the fundamental formulation of the theory.

Faraday's Law reads
$$\vec{\nabla} \times \vec{E}=-\partial_t \vec{B}.$$
It is sometimes state that "a time-varying magnetic field gives rise to (a vortex of) the electric field). This can be easily misunderstood as if it says that the time dependence of $\vec{B}$ "causes" an electric (vortex) field.

This is a historical misunderstanding, because the Maxwell equations were for quite some time not understood from the point of view of the (special) theory of relativity, which in fact turned out to be the way, one has to interpret this theory (Einstein 1905).

Then the issue becomes easily clear: first of all the split of the electromagnetic field in terms of electric and magnetic fields is dependent on the (inertial) reference frame, one describes the physics in. This already implies that the one field cannot be "the cause" of the other.

Analyzing the Maxwell equations in view of relativity it simply turns out that the electromagnetic field $(\vec{E},\vec{B})$ (in 4D notation the field-strength four-tensor $F_{\mu\nu}$) is "caused" by charge-current-density distributions in terms of the corresponding retarded solutions of the Maxwell equations (also known as Jefimenko equations).

 

[alan123hk]

I have a little question and I'm not sure what the answer is but it might be an easy question for you to answer

Take eqution $~\vec{\nabla} \times \vec{E}=-\partial_t \vec{B}~~$ as an example, Let $~~\vec{B}=\vec{sin~t} ~~~~~~$ then $~~~~\vec{\nabla} \times \vec{E}= \vec{-cos~t }$

Although $~~\vec{\nabla} \times \vec{E}~~~$ and $~~~\vec{B}~~~~$ are at the same point in space, they have exactly the same waveform and direction, $~~\vec{\nabla} \times \vec{E}~~$ lag 90 degree behind, the delay time of $~~\vec{\nabla} \times \vec{E}~~~$ relative to $\vec{B}~~~$ is 1.57seconds.

So can we say that these are two waveforms or two events that are causally related?

 

[vanhees71]

You math doesn't make sense. E.g., how do you define $\vec{sin} t$?

 

[alan123hk]

This is a vector with a fixed direction but the amplitude changes with time. The function of the amplitude changing with time is sin(t)

 

[vanhees71]

So you mean something like
$$\vec{E}=\vec{E}'(\vec{x}) \cos(\omega t)$$
and similar for $\vec{B}$?

That's a good ansatz. It's, however more convenient to use
$$\vec{E}=\vec{E}'(\vec{x}) \exp(-\mathrm{i} \omega t), \quad \vec{B}=\vec{B}'(\vec{x})\exp(-\mathrm{i} \omega t).$$
The physical fields are then the real parts of these complex fields. Going with this ansatz into the Maxwell equations you get equations for $\vec{E}'$ and $\vec{B}'$.

 

[Dario56]

We can call it a flubnubitz if we want. We just need to understand what the term means. We have agreed to call it induction, it is not a causal relationship, therefore induction is not causation.

Yes, okay, I've learned that the Faraday's law isn't describing the cause and effect relationship. Since that's the case, my question basically makes no sense as it implies that such a relationship exists, that there is a current that causes an induced voltage.

And that is fine. You can unambiguously and correctly say that charge and current densities cause electromagnetic fields. So the issue is not that it is wrong to think in terms of cause and effect relationships. It is just a misunderstanding about which relationships are causal and which are not.

Here, I do have some additional points to add.

As you said, cause needs to precede the effect and therefore if charge is the cause of the EM field it should somehow exist before it.

That doesn't make sense as charge would not be a charge without creating or influencing the field. They arise together, interdependently.

In another words, charge can't be defined independently of the field and therefore can't exist without (or prior) to the field.

 

[alan123hk]

First I understand that there is no causal relationship between the left and right sides of Maxwell's equations. The following are some of my personal additions.

This is how I personally understand it. Lumped components in a circuit, such as resistors, capacitors, and inductors, are simplified models in circuit analysis. They just have a simple mathematical equation that expresses the relationship between voltage and current. Because we assume that the delay time within them is negligible, these mathematical equations do not include the factor of time delay, so we do not have to study the cause and effect of their inner workings.

For more complex cases we can use transmission line theory. Voltage and current on a transmission line occur simultaneously, there is no sequential difference between them. However, due to delays in transmitting current and voltage through transmission lines, transmission line theory also provides causal relationships for voltage/current at different locations.

For the most complex general case, the transmission and variation of the electromagnetic field throughout the system needs to be considered. Since the transmission of electromagnetic fields takes time, it contains very complex causal relationships.

Therefore, the internal working of resistors, inductors, and capacitors are not without causal relationships, we just don't need to deal with it in simplified models.

For example, if we say that connecting a voltage source to an inductor is the cause and then the inductor produces an back emf which is the effect. This is no problem, as this can refer to the steady-state back EMF that takes a delay time to reach after connecting the voltage source in AC circuit analysis.

 

[Dale]

That doesn't make sense as charge would not be a charge without creating or influencing the field. They arise together, interdependently.

Remember that, due to conservation of charge, charges are always created in pairs. They form a dipole and the fields radiate out from this dipole at $c$ like the wave from a standard dipole antenna.

Yes, you can say in some sense that the field and the charges exist at the same time but the field that “exists” at the moment of creation of the dipole is a degenerate case with 0 volume and 0 amplitude. The relationship is still causal, even with this degenerate case.

 

[Dario56]

First I understand that there is no causal relationship between the left and right sides of Maxwell's equations. The following are some of my personal additions.

This is how I personally understand it. Lumped components in a circuit, such as resistors, capacitors, and inductors, are simplified models in circuit analysis. They just have a simple mathematical equation that expresses the relationship between voltage and current. Because we assume that the delay time within them is negligible, these mathematical equations do not include the factor of time delay, so we do not have to study the cause and effect of their inner workings.

For more complex cases we can use transmission line theory. Voltage and current on a transmission line occur simultaneously, there is no sequential difference between them. However, due to delays in transmitting current and voltage through transmission lines, transmission line theory also provides causal relationships for voltage/current at different locations.

For the most complex general case, the transmission and variation of the electromagnetic field throughout the system needs to be considered. Since the transmission of electromagnetic fields takes time, it contains very complex causal relationships.

Therefore, the internal working of resistors, inductors, and capacitors are not without causal relationships, we just don't need to deal with it in simplified models.

For example, if we say that connecting a voltage source to an inductor is the cause and then the inductor produces an back emf which is the effect. This is no problem, as this can refer to the steady-state back EMF that takes a delay time to reach after connecting the voltage source in AC circuit analysis.

Hmm, since the Faraday's law isn't really cause and effect relationship, how can we explain the time changing current in the inductor in terms of casual relationship? In another words, what causes time changing current in inductors? This isn't something that is commonly discussed in physics textbooks.

 

[DaveE]

Hmm, since the Faraday's law isn't really cause and effect relationship, how can we explain the time changing current in the inductor in terms of casual relationship? In another words, what causes time changing current in inductors? This isn't something that is commonly discussed in physics textbooks.

$i(t) = \int v(t)\, dt + I_o$ explains it pretty well. The present value is about the history of the applied voltage. If you're not sure look at the step response, not steady state solutions.

OK, while this isn't the classic form of Faraday's law, it's a simple result from $v(t) = -\frac{d \Phi}{dt}$ which is a common form. This describes the rate of change at time $t$; the slope of $i(t)$, or $\Phi(t)$, now, in the present.

 

[alan123hk]

since the Faraday's law isn't really cause and effect relationship, how can we explain the time changing current in the inductor in terms of casual relationship? In another words, what causes time changing current in inductors? This isn't something that is commonly discussed in physics textbooks.

Since there is no causal relationship between the left and right sides of the Maxwell's Equations, we cannot simply say that changes in current cause magnetic fields, and changes in magnetic fields cause induced electric fields, etc., because this may cause misunderstanding.

So if you want to understand all the real details of the cause and effect of time-varying currents in an inductor, then you can try to consider the most complex case, i.e. how these currents vary with time and space in the inductor. This is similar to the case of voltage and current wave propagation, reflection, and ohmic losses in lossy transmission line models.

Of course, in practical applications it may be difficult to perform such an overly complex analysis, and analysis based on quasi-static fields and circuit theory is usually sufficient.

However, the simplest relationship can also be said with certainty that the voltage and current waves from the external source are the cause, while the voltage and current waves in the inductor are the effect, since these waves require a certain time delay to propagate into the inductor.