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## Homework Statement

- A current [itex]I(t)= (0,160 A s^{-3}) t^3[/itex] flows through an ideal solenoid with a turns density [itex]n = 9,00 \cdot 10^{-3} m^{-1}[/itex] and a cross sectional area [itex]A_s=2,00\cdot 10^{-4} m^2 [/itex]
- 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 [itex]t_1 [/itex] when the current in the solenoid is [itex]3.20 A[/itex].
- Calculate the amount of electric charge that flowed through the wire of the solenoid between instants [itex]t_0=0s [/itex] and [itex]t_1 [/itex]

## Homework Equations

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

## The Attempt at a Solution

- from the equation for the current I can find [itex]t_1= \sqrt[3]{2}\, s \approx 1.2599\, s[/itex]
- From Faraday's equation [itex]{\cal E}(t) = A_s \mu_0 n \dot I(t) [/itex]. The required EMF is obtained after plugging time [itex]t_1 [/itex] in this equation.
- 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.