# Electromagnetic induction

Hi,
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 $\vec{B}$ 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 $\vec{j}(t)$ over time we have still a very weak field $\vec{B}$ outside the solenoid, say it is more or less 0.
The inner of the solenoid has a changing field $\vec{B}$. This means that the second equation is not zero. Which means we get an $\vec{E}$ 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 $\vec{B}(t)$ is weak but $\frac{\partial\vec{B}}{\partial t}$ is strong?
Why if the magnetic field outside is more or less zero the change of the flux $\vec{B}\cdot\vec{A}$ is detected strongly?

Thanks!