📂Electrodynamics

자기장의 기호로 B를 사용하는 이유

Question

Electromagnetism is literally the study of electric fields $\mathbf{E}$ and magnetic fields $\mathbf{B}$. While studying electromagnetism, one might have wondered the following at least once.

Why is the symbol for magnetic fields $\mathbf{B}$ used?

It's understandable that the electric field is $\mathbf{E}$, derived from the Electric field, but why is the magnetic field $\mathbf{B}$ when it should be from Magnetic field? This notation might feel oddly placed, and it's because it was decided without any significant reason.

Answer

Maxwell’s Notation^{1 2}

Maxwell, often called the father of electromagnetism, completed classical electromagnetism³ with Maxwell’s equations. Like Newton, Leibniz, Euler, and other individuals who made significant achievements in mathematics/science, not only their names and accomplishments but also their notations have been passed down to future generations. This is true for Maxwell as well; the notation of using $\mathbf{B}$ for magnetic fields and $\mathbf{H}$ for auxiliary fields has naturally continued because Maxwell noted them this way.

The symbols for various vectors used by Maxwell in electromagnetism⁴ are as follows.⁵ Maxwell denoted these vectors with alphabets from A to J, and in cases where there were appropriate symbols like C, D, F, he assigned them, while the rest were seemingly chosen at his discretion.

Notation		Meaning
Maxwell	Today
$\frak{A}$	$\mathbf{A}$	The electromagnetic momentum at a point Currently known as vector potential.
$\frak{B}$	$\mathbf{B}$	The magnetic induction Nowadays referred to as the magnetic field.
$\frak{C}$	$I$	The (total) electric 'C'urrent, current
$\frak{D}$	$\mathbf{D}$	The electric 'D'isplacement, displacement field
$\frak{E}$	$\mathcal{E}$	The 'E'lectromotive intensity Today called electromotive force, emf.
$\frak{F}$	$\mathbf{F}$	The mechanical 'F'orce Currently known as Lorentz force.
$\frak{G}$		The velocity of a point
$\frak{H}$	$\mathbf{H}$	The magnetic force Today, this is referred to as the H-field, auxiliary field, magnetic field intensity, etc.
$\frak{I}$	$\mathbf{M}$	The 'I'ntensity of magnetization Seems to be what is called magnetization density today.
$\frak{J}$	$\mathbf{J}$	The current of conduction, conduction current

Most of these notations are still used today, with the current being denoted as $I$, following the initial of the intensity of current.

Furthermore, upon researching, one may find the argument that ’the symbol for magnetic fields comes from Biot in the Biot-Savart law’ but, in my opinion, that’s debatable. Firstly, the question ‘Why is the symbol for magnetic fields $\mathbf{B}$?’ is equivalent to asking ‘Why do we use $\mathbf{B}$ as the symbol for magnetic fields today?’, so the answer ‘Because Maxwell used it that way’ seems valid. Then that aside, one might wonder, ‘Did Maxwell choose $\mathbf{B}$ for the magnetic field symbol because it was named after Biot?’. There doesn’t seem to be a concrete basis for this answer either. Even if it were, it wouldn’t be because it was named after Biot’s name but rather ‘Among the vectors listed above, the one that best fits the symbol $\mathbf{B}$ is magnetic induction, related to the Biot-Savart law’ might be a more fitting explanation. (Honestly, the claims that it was derived from Biot, bi-polar field, boreal, etc., seem like a stretch to me)

Between B and H, What is the Magnetic Field?

Meanwhile, there’s also discussion on whether $\mathbf{B}$ or $\mathbf{H}$ should be called the magnetic field. $\mathbf{H}$ is a value that can be adjusted in experiments regardless of the medium. Hence, in engineering-related areas (for example, electrical engineering textbooks), $\mathbf{H}$ is commonly referred to as the magnetic field, whereas in physics-related areas, $\mathbf{B}$ is typically called the magnetic field. However, as explained in the Wikipedia article on magnetic fields, since $\mathbf{B}$ mediates the Lorentz force, it appears consistent and logical to call $\mathbf{E}$ the electric field, similarly $\mathbf{B}$ should be referred to as the magnetic field.

$$ \text{Lorentz force}: \mathbf{F}=Q\left[ \mathbf{E} + (\mathbf{v}\times\mathbf{B}) \right] $$