The \(H\)-field inside a long solenoid is \(nI\). If there is a vacuum inside the solenoid, the B-field is \( \mu_o H = \mu_o nI\). If we now place an iron rod of permeability \(\mu\) inside the solen...The \(H\)-field inside a long solenoid is \(nI\). If there is a vacuum inside the solenoid, the B-field is \( \mu_o H = \mu_o nI\). If we now place an iron rod of permeability \(\mu\) inside the solenoid, this doesn't change \(H\), which remains \(nI\). The B-field, however, is now \(B=\mu H\). This is greater than \(\mu_oH\), and we can write \[B = \mu_o(H+M) \]