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Notes on magnetism III - Various physical quantities describing magnetic field

Talking about magnetic field, we have three relevant physical quantities - magnetic field \(\vec{H}\), magnetic induction field \(\vec{B}\) and magnetization field \(\vec{M}\). It is straightforward to understand \(\vec{M}\) - when applying magnetic field (here, by 'magnetic field', we mean something in general, but not specifically refer to \(\vec{H}\)), we use \(\vec{M}\) to characterize how much the matter in question is magnetized. However, concerning \(\vec{H}\) and \(\vec{B}\), it seems that both are describing some sort of 'strength' of magnetic field (again, we mean 'magnetic field' in general. The same applies below until we become specific about what we mean by 'magnetic field'). But why do we have two quantities here to describe the 'same' thing? The answer is - they are not the same thing, as described by \(\vec{H}\) and \(\vec{B}\), respectively. Fundamentally, this goes back to the foundation of the electromagnetic theory - specific

Why do we have spin half for electrons?

Illustration for Stern-Glach experiment [1] Angular moment means rotation - this applies to electron as well. For electrons, any movement is associated with current and therefore angular moment naturally is tightly linked to current as well. Furthermore, current means magnetic field, as originated by Orsted. Mathematically, starting from expressing the current with the angular moment of rotating electrons, followed by detailed mathematics based on Biot-Savart law (induction \(\vec{B}\) as the function of current), one can arrive at the expression of magnetic induction in terms of magnetic (dipole) moment \(\vec{m}_l\). In another word, it can be shown that the magnetic field generated by electron rotation current is equivalent to that generated by magnetic dipole, with the equivalent dipole moment defined as (see Ref. [2] for detailed derivation), \[\vec{m}_l = -\frac{1}{2}eR^2\vec{\omega}\] Given that the angular moment can be written as \(\vec{l} = m_eR^2\vec{\omega}\), we can obtain