Week 7 Lex 3: Magnetostatics in a Material Media I

Having completed Magnetostatics in free space ( vaccuum) we now turn to study the Magnetostatics in Magnetic Materials. This is the magnetostatic analog of Electrostatics in Dielectric Material.


Recall that the source of Magnetism are currents. In magnetic materials the magnetism is due to atomic currents. These currents essentially arise due to two causes. (i) The orbital motion of the electron around the nucleus (ii) The intrinsic ( quantum mechanical) spin of the electron. These small atomic current loops are equivalent to tiny atomic dipoles from a macroscopic ( large scale ) point of view. Normally their effects cancel out due to (i) random orientation (ii) thermal motion of the atoms in a material. However in an applied external magnetic field B these magnetic dipoles tend to either align parallel or anti-parallel to the applied field giving rise to weak magnetic effects called (i) paramagnetism (ii) diamagnetism respectively. For permanent magnetic materials there is a FROZEN-IN or PERMANENT magnetization. Apart from this we also have Ferromagnetic Materials about which we will discuss later.




The torque on a magnetic dipole in an uniform field B is given as N=m X B whereas in a non uniform field the dipole experiences a force ∇ ( m.B) where the dipole moment m=I A where A is the vector area of the current loop. ( For flat current loops this is the usual area vector). This torque tends to orient the dipoles along the applied field direction resulting in a Magnetization. Due to the Pauli exclusion principle paired electrons in atoms with up and down spins cancel each others torques. So Paramagnetism is most readily observed in atoms with unpaired electrons. As paired electrons do not contribute to the magnetic effects Paramegnatism is a weak magnetic phenomena. Diamagnetism on the other hand arises due to the orbital motion of the electrons. The electron due to the magnetic forces speeds up/slows down in an external magnetic fields B. This causes a change in the dipole moment associated with the orbital motion anti-parallel to the field. ( Check Griffiths). The weak diamagnetic effects arise from this incremental dipole moment.


Hence at a macroscopic ( large scale) level a magnetic material has magnetic polariaztion and can be described by a magnetic dipole moment density which is called Magnetization M which is the magnetic dipole moment/unit volume. Just as in dielectrics it is now possible to describe the macrsocopic field due to a magnetized material by a distribution of surface and volume bound current densities. However the equations for the magntetic field being different the bound currents are now given as a surface bound current density K_b=M X η and a volume bound current desnity J_b=∇ X M ( Check Grifiths for the proof which is just like that in Dielectrics but now involving the Stokes Theorem.)


The bound current densities arise just as in dielectrics from now the cancellation between adjacent atomic current loops leving only the boundary surface contributions for uniform Magnetization M when every loop carries the same current. For non-uniform M the currents are different for different loops and only a partial cancellation occurs giving rise to also a volume bound current density inside the material apart from a surface bound current density.


The differential form of Ampere's Law can now be written as 
∇ X B= μ₀ J_tot where J_tot= J_free + J_b is the total current density consisting of the free current density if any in the material and the bound current density due to the Magnetization M. ( We are neglecting the surface bound current density for the same reason as in Dielcetrics. For real magnetic materials the Magnetization goes to ZERO rapidly within a small surface thickness and surface bound currents do not develop in real life.) But for our problems we will consider such IDEALIZED surface bound currents. ( DIPA/IPSA)


It is now easy to define the magnetic analog of the electric displacement D as the H field which satisfies ∇ X H= J_free. So the source for the H field is the free current density only and  
H= (B/μ₀ - M). We can easily define the corresponding integral form of the Amperes law for H as ∫_C H.dl=I^enc_free. When symmetry allows us we can calculate H from knowing the free currents ( see Eg 6.2 Griffths ).


Similar warning as in Dielcetrics for the displacement vector D is also in force with H. H is not quite like B as ∇.B=0 but ∇.H=-∇.M. Only if M is constant or uniform is B like H. In particular do not assume that H is ZERO because there are no free currents. The H field satisfies similar boundary conditions with both normal and tangential components discontinuous at a surface current density. This is obvious from a look at the equations for H namely ∇.H=-∇.M and  ∇ X H=-K_free for a surface current density.