electromagnetism
A solenoid is connected to a source of constant E.M.F. for a long time. A soft iron piece is inserted into it.
$\phi=B.A=\dfrac{\mu_oNIA}{L}$
After insertion, $I$(current) remains the same. So should magnetic flux $\phi$ remain constant too ? Why or why not ?
I agree that this seems mysterious on first meeting.
Multiplication by $n$ is shown to be correct by something called Stokes's Theorem, which you won't have met yet, and which lets us translate the basic equations of electromagnetism (called their Maxwell Equations) between their local and "spread out" forms.
But at an easier level, think of a very long, time varying flux tube, like the one I've drawn below.
Now imagine separate loops around the flux tube. Each of these loops has flux $\phi$ through them, and each separately will have an EMF $-\frac{{\rm d}}{{\rm d}\,t}\phi$ between their ends.
You link them together in series, and it is just like linking voltage sources, or batteries together in series. The total EMF across the series cells is $-n\,\frac{{\rm d}}{{\rm d}\,t}\phi$. Now imagine bringing the loops together in a neatly wound solenoid. Electrically nothing changes: it doesn't matter if you put long, zero resistance wires between batteries in series, their EMFs still add in exactly the same way.
Since the whole purpose of an inductor is react to a change in current with a back EMF, the physics doesn't change if you state the equation $V = -n\,\frac{{\rm d}}{{\rm d}\,t}\phi$ as $L I = -n\,\phi$; you simply differentiate this one to get the form I just derived.
The third of Maxwell's equations (Faraday's law) says that a changing magnetic field has an E-field curling around it. The closed line integral of this electric field is the EMF that drives the induced current in the conducting wire. At a microscopic level, the curling electric field, which has a significant component parallel to the wire, exerts a force on the charges in the conductor.
If your question is "why are Maxwell's equations the way they are?", I'm afraid that isn't a good question for this site.
Best Answer
Your equation for $B$ is only valid when there is no material present in the loop.
When you add a magnetic material like soft iron, this material will be magnetized, which adds to the total magnetic flux. In fact, the equation you need is
$$\vec{B} = \mu_0(\vec{H} + \vec{M})$$
Where $\vec{H}$ is the magnetic field intensity, and $\vec{M}$ the magnetization. Now $H$ is related to $I$ only by $H=\frac{NIA}{L}$, but for $B$ you have to take account of the additional term.