EMF induced in a coil encircling an ideal solenoid

[FranzDiCoccio]

Homework Statement

• A current $I(t)= (0,160 A s^{-3}) t^3$ flows through an ideal solenoid with a turns density $n = 9,00 \cdot 10^{-3} m^{-1}$ and a cross sectional area $A_s=2,00\cdot 10^{-4} m^2$
• A single loop of wire has the same axis as the solenoid, but its radius is larger. That is: the loop is outside the solenoid and "encircles" it.
• Calculate the EMF induced in the loop at the instant in time $t_1$ when the current in the solenoid is $3.20 A$.
• Calculate the amount of electric charge that flowed through the wire of the solenoid between instants $t_0=0s$ and $t_1$

Homework Equations

1. $B(t) = \mu_0 n I(t)$ instantaneous magnetic field inside the solenoid
2. ${\cal E} = {\dot \Phi}_\ell$ Faraday's law involving the flux of the magnetic field through the loop.
3. $\Phi_\ell(t) = \Phi_s(t) = B(t) A_s$ the flux through the loop is the same as the flux through a cross section of the solenoid

The Attempt at a Solution

1. from the equation for the current I can find $t_1= \sqrt[3]{2}\, s \approx 1.2599\, s$
2. From Faraday's equation ${\cal E}(t) = A_s \mu_0 n \dot I(t)$. The required EMF is obtained after plugging time $t_1$ in this equation.
3. The charge that flowed through the solenoid is a simple integral in the relevant time interval.

I had one small doubt about this, and I looked through PF for some inspiration. I'm afraid I did not solve my doubt but another one came about.

First doubt.
The current through the solenoid is not constant. Should I worry about the self induced current in the solenoid? Also, the induced current in the loop is going to generate a magnetic field... How should I go about that?
My first instinct is: ignore self induction and only worry about the "first order" process.
What would one be supposed to do in this case? And does the problem give all of the data needed to do that?

Second doubt.
The magnetic field outside the solenoid is zero. My understanding of electromagnetism is that this does not matter. According to Faraday's law the only thing that matters for the EMF in the loop is the flux of the magnetic field through any surface enclosed by the loop (I mean, the variation thereof).
However, the discussion in this old post has somehow shaken my beliefs. I'm not a native English speaker, so my understanding of some claims in the discussion might be wrong. However, as far as I understand, the science advisor phyzguy is claiming that for the flux through a coil to be non-zero "some magnetic field lines should cross the wires".

If this was true (which goes against my understanding of how electromagnetism works) there would be no EMF in the loop of my problem, because the magnetic field is confined inside the solenoid.
I think that even for an ideal solenoid (absolutely no magnetic field outside it) there would be an EMF in the loop.

 

[Charles Link]

The changing magnetic field just needs to be inside the loop to induce an EMF in the loop. The number you have for the number of turns per meter $n$ appears to have a typo. Normally that number is around 500 or 1000 or higher, and not $9.0 \cdot 10^{-3 }$. $\\$ In the old post which you supplied a "link", I took part in that discussion, and I believe the matter was resolved. It was an interesting discussion, but phyzguy had the concept incorrect. $\\$ The homework problem you have is really quite straightforward. You have the correct formulas. And you do not need to worry about any EMF created in the solenoid. The current is given, and you can assume that the current source that is driving the solenoid is not affected by any EMF's in the solenoid.

 

[FranzDiCoccio]

Hi Charles,

thanks for your help. Of course you're right about the number of turns, I do not know what I was thinking. It cannot be right. I'm going to correct it.

I also thought that the exercise was pretty straightforward, but it was given to me suggesting that it was really hard, and I feared that there was something I was missing.
I thought of what you suggest, i.e. that the current (rather than the EMF) in the solenoid is given. Also, I expected that, in order to carry out a more complete analysis, I had to know other parameters, such as the resistance of the circuit, which is not given. It's a weak argument, I know...

As to the old post, it seems to me that it ends abruptly, and that phyzguy maintains his version throughout it. So I thought either there is a misunderstanding (possibly from my part) or the matter is not resolved (at least in that post).

Thanks again!
Franz

EDIT: it seems that I cannot edit my OP any more. I'm going to write it here:

• the corrrect number of turns is $n = 9.00\cdot 10^3 \, {\rm m}^{-1}$