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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 ?

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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.

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