Transformer equation: Using Faraday's law "the wrong way"?

[greypilgrim]

Hi.

All derivations of the (ideal) transformer equation $\frac{U_p}{U_s}=\frac{n_p}{n_s}$ use Faraday's law of induction
$$U=-n\cdot \frac{d\Phi}{dt}$$
for primary and secondary and equate the change of flux $\frac{d\Phi}{dt}$.

Until now, in my textbooks it was always like this: Electrical currents or permanent create magnetic fields, and change of magnetic flux creates voltage by Faraday's law.
Now here Faraday's law seems to be applied the other way around: The voltage on the primary creates magnetic flux change. Why is it suddenly possible to use this equation "the other way around"?

If I had been to derive the transformer equation without knowledge of above "traditional" derivation, I'd probably have started with the magnetic field created by the current in the primary coil (introducing its resistance $R_p$), then compute the induced voltage $U_s$ in the secondary coil and the induced current (introducing the total secondary resistance of secondary coil and load $R_s+R_{load}$). Then I'd probably have remembered that this current also generates a magnetic field that would influence the primary coil, which would lead to an iteration. Then I would have tried to find a fixed point of this iteration which would hopefully be the stable solution.

Obviously this derivation would be very complicated, and it also uses resistances that are not present in the transformer equation. Still, would this work as well?

 

[Dale]

Why is it suddenly possible to use this equation "the other way around"?

It is always possible to use any equation the other way around. If A=B then B=A.

 

[greypilgrim]

Sure. What I meant is: Until now, in Faraday's law
$$U=-n\cdot\frac{d\Phi}{dt}$$
the change of flux $\frac{d\Phi}{dt}$ was always cause and $U$ was effect. It was also proven this way in my textbooks.
But then suddenly $U_p$ is cause and $\frac{d\Phi}{dt}$ is effect in the primary coil. How is this justified?

 

[Dale]

It is justified because it never was a cause and effect relationship to begin with.

What was unjustified was the verbal description as cause and effect. The math never justified that wording in the first place. Causes always come before effects, so where does that equation identify which side comes first?

Consider 2+2=4. You can just as easily write it 4=2+2, and either way is valid. What is not valid is to verbally translate that equation as "2+2 causes 4". You would say "2+2 is 4" or you could even use the word "equivalent" or "implies" if it fits more naturally. Implication and equivalence go both directions, cause and effect do not

 

[greypilgrim]

So let's use this equation for a different system, for example a single solenoid attached to a DC voltage source $U$. The voltage leads to a constant current that leads to a constant magnetic field that leads to a constant magnetic flux $\Phi$ through the solenoid. So now we have $U\neq 0$ but $\frac{d\Phi}{dt}=0$. How can
$$U=-n\cdot\frac{d\Phi}{dt}$$
be true now?

 

[Dale]

According to that equation a constant $U$ implies that $\Phi$ increases (in the opposite direction) without bound.

Whenever you are using an equation in physics it is important to understand the assumptions that were made in deriving the equation. What are the assumptions for that equation, and are they met by the situation you described?

 

[greypilgrim]

Well you tell me. As I said, all derivations of this equation I know start from a changing flux and show that it induces a voltage in a solenoid. You said it's possible to use this equation (as any equation) the other way around, as it apparently happens in the primary coil of a transformer.

So what's the difference in the assumptions when we look at a single solenoid compared to the primary coil of a transformer?

 

[Dale]

all derivations of this equation I know start from a changing flux and show that it induces a voltage

Hmm, those derivations seem to have been unclear. It induces an EMF, not a voltage. In a typical solenoid the voltage is due not only to the EMF, but also due to Ohm's law.

The application to the solenoid assumes that the resistance is negligible, so that in the case of 0 resistance then the ever-increasing $\Phi$ would be correct. The reason that it seems wrong is because you are intuitively thinking about a normal coil with non-negligible resistance.

 

[lychette]

Hmm, those derivations seem to have been unclear. It induces an EMF, not a voltage. In a typical solenoid the voltage is due not only to the EMF, but also due to Ohm's law.

The application to the solenoid assumes that the resistance is negligible, so that in the case of 0 resistance then the ever-increasing $\Phi$ would be correct. The reason that it seems wrong is because you are intuitively thinking about a normal coil with non-negligible resistance.

What do you mean by the statement..."in a typical solenoid the voltage is due not only to the EMF, but also to Ohm's law"

 

[Dale]

What do you mean by the statement..."in a typical solenoid the voltage is due not only to the EMF, but also to Ohm's law"

I mean that the wire in a solenoid has resistance. If a nonzero current runs through the wire then some of the voltage at the terminal is due to Ohmic resistance.

 

[greypilgrim]

I mean that the wire in a solenoid has resistance. If a nonzero current runs through the wire then some of the voltage at the terminal is due to Ohmic resistance.

Ok that's probably what the derivations I know have omitted. As a very first example they look at a conducting rod moving in a magnetic field and derive the induced voltage between its ends by looking at the equilibrium case where Lorentz force and electrical force acting on the electrons cancel. Then they use this to derive Faraday's law for loops and solenoids.

So it seems to me that they actually only derive
$$U=-n\cdot \frac{d\Phi}{dt}$$
in the equilibrium case $\frac{dU}{dt}=\frac{d^2\Phi}{dt^2}=0$ and where $I=0$ in the solenoid.