## Magnetic Susceptibility

\[\vec{M}=\frac{d \vec{\mu}}{dV}\]

Long before we had any understanding of atomic or molecular structure, Ampere proposed a model of magnetism in which the magnetisation of materials Is due to microscopic current loops inside the the material. We now know that these current loops are a classical model for the orbital motion and spin of the electrons in atoms. Consider a cylinder of magnetised material in which atomic current loops in the cylinder aligned with their magnetic moments along the axis of the cylinder. Because of cancellation of neighbouring current loops, the net current at any point inside the material is zero, leaving a net current on the surface of the material. This surface current, called an amperian current, is similar to the real current in the windings of the solenoid.

Consider a short cylinder of cross-sectional area

\[A\]

, length \[dl\]

and volume \[V=Adl\]

.Let

\[dl\]

be the amperian current on the curved surface of the cylinder. The magnitude of the magnetic dipole moment of the cylinder is the same as that of a current loop that has an area \[A\]

and carries a current \[di\]

\[d \mu =Adi\]

.The magnitude of the magnetisation of the cylinder is the magnetic moment per unit volume:

\[M= \frac{ d \mu}{dV}= \frac{A d i}{A dl} = \frac{d i}{dl}\]

.Thus, the magnitude of the magnetisation vector is the amperian current per unit length along the surface of the magnetised material. We see from this result that the SI units for magnetisation M are amperes per metre.

Consider a cylinder that has a uniform magnetisation

\[\vec{M}\]

parallel to its axis. The effect of the magnetisation is the same as if the cylinder carried a surface current per unit length of magnitude M. This current is similar to the current carried by a tightly wound solenoid. For a solenoid, the current per unit length is \[nI\]

, where \[n\]

is the number of turns per unit length and \[I\]

is the current in each turn. The magnitude of the magnetic field \[B_m\]

inside the cylinder and far from its ends is thus given by \[B_m= \mu_0 nI\]

for a solenoid with {jatex options: inline}I}{/jatex} replaced by \[M\]

.Suppose we place a cylinder of magnetic material inside a long solenoid that has

\[n\]

turns per unit length and carries a current \[I\]

. The applied field of the solenoid \[(B_{app}=\mu_0 nI\]

inside the cylinder and far from its ends is The resultant magnetic field at a point inside the solenoid and far from its ends due to the current inside the solenoid ) magnetises the material so that it has a magnetisation M. The the current in the solenoid plus the magnetised material is \[\vec{B}= \vec{B}_{app}+\mu_0 n \vec{I}\]

.For paramagnetic and ferromagnetic materials,

\[\vec{M}\]

is in the same direction as \[\vec{B}\]

, and for diamagnetic materials, \[\vec{M}\]

is opposite to \[\vec{B}\]

. For paramagnetic materials, the magnetisation is found to be proportional to the applied magnetic field that produces the alignment of the magnetic dipoles in the material. We can thus write \[\vec{M}=\chi_m \frac{\vec{B}}{\mu_0}\]

where \[\chi_m\]

is a dimensionless constant called the magnetic susceptibility.We can write

\[\vec{B}=\vec{B}_{app}+ \mu_0 \vec{M}= \vec{B}_{app}(1+ \chi_m)=K_m \vec{B}_m\]

where \[K_m\]

is called the relative permeability of the material. For paramagnetic materials \[\chi_m\]

is small and positive, depending on temperature. For diamagnetic materials other than superconductors, \[\chi_m\]

is small, constant and independent of temperature. For ferromagnetic materials \[K_m\]

ranges from 50,00 to 100,000 and is not defined for permanent magnets which may retain some magnetism in the absence of a magnetic field.