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I have a trivial question about electrodynamics.

If you have a very long coil, a long solenoid. Keep the current constant and you will have no [itex]\vec{B}[/itex] outside (magnetostatics).

Let's write down the Maxwell equations:

\begin{matrix}

\nabla\cdot\vec{B} &= &0 \\

\nabla\times\vec{E} &= &-\frac{\partial\vec{B}}{\partial t} \\

\nabla\times\vec{B} &= &\frac{\vec{j}}{\epsilon_0 c^2}

\end{matrix}

For the stationary case the second equation equals to zero.

If we slowly vary [itex]\vec{j}(t)[/itex] over time we have still a very weak field [itex]\vec{B}[/itex] outside the solenoid, say it is more or less 0.

The inner of the solenoid has a changing field [itex]\vec{B}[/itex]. This means that the second equation is not zero. Which means we get an [itex]\vec{E}[/itex] which works against the change - self induction, so we get a reactance from the basic solenoid.

If now another solenoid is wrapped around the basic solenoid, why does it feel a pretty strong induction?

Is it because [itex]\vec{B}(t)[/itex] is weak but [itex]\frac{\partial\vec{B}}{\partial t}[/itex] is strong?

Why if the magnetic field outside is more or less zero the change of the flux [itex]\vec{B}\cdot\vec{A}[/itex] is detected strongly?

Thanks!