The relationship between the magnetic susceptibility ($$\chi$$) and the magnetic permeability ($$\mu$$) is given by: ($$\mu_0$$ is the permeability of free space and $$\mu_r$$ is relative permeability)

The magnetisation $$M$$ produced in a material kept in an external magnetic field $$H$$ is written as
$$M = \chi\,H$$, where $$\chi$$ is called the magnetic susceptibility.

The magnetic induction $$B$$ inside the same material is given by
$$B = \mu\,H$$, where $$\mu$$ is the magnetic permeability of the material.

For vacuum (or free space) the induction is $$B_0 = \mu_0\,H$$, so the relative permeability of the material is defined as
$$\mu_r = \frac{\mu}{\mu_0}$$.

By definition we can also write
$$B = \mu_0\,(H + M)$$. Substituting $$M = \chi\,H$$ gives
$$B = \mu_0\,(H + \chi H) = \mu_0\,(1 + \chi)\,H$$.

Comparing the two expressions for $$B$$:
$$\mu\,H = \mu_0\,(1 + \chi)\,H$$.
Cancelling the common factor $$H$$, we get
$$\mu = \mu_0\,(1 + \chi)$$.

Divide both sides by $$\mu_0$$:
$$\frac{\mu}{\mu_0} = 1 + \chi$$.

The left side is just $$\mu_r$$, so
$$\mu_r = 1 + \chi \quad$$(relation between $$\mu_r$$ and $$\chi$$).

Rearranging for susceptibility:
$$\chi = \mu_r - 1 = \frac{\mu}{\mu_0} - 1$$.

Thus the correct option is
$$\chi = \frac{\mu}{\mu_0} - 1$$ → Option A.