The percentage change in the magnetic field when the space within a current-carrying solenoid is filled with magnesium is calculated as follows:

We start by using the formula for percentage change in the magnetic field:
change in
Substituting the expressions for the magnetic field, we have:

Where:
is the permeability of the material inserted (here, magnesium).
is the permeability of free space.

is the number of turns per unit length of the solenoid.
is the current through the solenoid.

Given that the relationship between the permeability of the medium ( ) and relative permeability ( ) is:

We can simplify this to:

Which reduces to:

The relative permeability is equal to , where is the magnetic susceptibility. Thus, we have:

When magnesium is used, with a magnetic susceptibility , the calculation becomes: