Dimensions and Units

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Dimensions and Units
관련코스 전기역학
소분류 물리
선행 키워드
연관 키워드

Electromagnetism is a field of science where various unit systems are being used and hence very confusing.

In this wiki, we will stick to SI unit (as opposed to cgs unit being used in Jackson, for example), but still there are many confusing issues even within SI unit.

Dimensional analysis in EM

On top of those fundamental dimensions covered in classical mechanics (mass, length, and time), electromagnetism requires another dimension which is current. There can be doubts on why current, not charge, has to be the fundamental dimension, but we just follow what's been conventionally fixed for now.

Individual physical quantities

let's add another column for scalar/vector/pseudovector

Variable Dimension (MLTA)
acceleration [math]\mathbf{a}[/math] [math]0 1 \bar2 0[/math]
force [math]\mathbf{F}[/math] [math]1 1 \bar2 0[/math]
energy [math]E[/math] [math]1 2 \bar2 0[/math]
energy density [math]u[/math] [math]1 \bar1 \bar2 0[/math]
potential [math]V[/math] [math]1 2 \bar3 \bar1[/math]
electric field [math]\mathbf{E}[/math] [math]1 1 \bar3 \bar1[/math]
magnetic field [math]\mathbf{B}[/math] [math]1 0 \bar2 \bar1[/math]
permittivity [math]\epsilon[/math] [math]\bar1 \bar3 4 2[/math]
permeability [math]\mu[/math] [math]1 1 \bar2 \bar2[/math]
polarization [math]\mathbf{P}[/math] [math]0 \bar2 1 1[/math]
magnetization [math]\mathbf{M}[/math] [math]0 \bar1 0 1[/math]
displacement field [math]\mathbf{D}[/math] [math]0 \bar2 1 1[/math]
auxiliary field [math]\mathbf{H}[/math] [math]0 \bar1 0 1[/math]
charge density [math]\rho[/math] [math]0 \bar3 1 1[/math]
current density [math]\mathbf{J}[/math] [math]0 \bar2 0 1[/math]
resistance [math]R[/math] [math]1 2 \bar3 \bar2[/math]
conductivity [math]\sigma[/math] [math]\bar1 \bar3 3 2[/math]

Composite physical quantities

composite quantity dimension (MLTA) meaning
[math]E/B[/math] [math]0 1 \bar1 0[/math] c, speed of light
[math]{1 \over \sqrt{\epsilon_0\mu_0}}[/math] [math]0 1 \bar1 0[/math] [math]c[/math]
[math]\sqrt{\frac{\mu_0}{\epsilon_0}}[/math] [math]1 2 \bar3 \bar2[/math] Z, vacuum impedence (same dimension as Resistance)
[math]\mathbf{E}\times\mathbf{H}[/math] [math]1 0 \bar3 0[/math] S, Poynting vector (same dimension as [math]u\cdot c[/math])
[math]\frac{1}{2} \mathbf{E}\cdot\mathbf{D}[/math] [math]1 \bar1 \bar2 0[/math] Electrical energy density
[math]\frac{1}{2} \mathbf{B}\cdot\mathbf{H}[/math] [math]1 \bar1 \bar2 0[/math] Magnetic energy density
[math]\frac{\omega\epsilon}{\sigma}[/math] [math]0 0 0 0[/math] Q, Quality factor for EM wave in conductor

Dimensions in equations

[math]\mathbf{E}=\mathbf{E}_0 \sin(kx-\omega t)[/math]
Wave equation
[math]\frac{\partial^2 \mathbf{E}}{\partial x^2}-\frac1{c^2}\frac{\partial^2\mathbf{E}}{\partial t^2} = 0[/math]
[math]E=mc^2[/math] vs. [math]E=h\nu[/math] vs. [math]E=\mathbf{F}\cdot\mathbf{r}[/math]
Centripetal force
[math]F_c = m\frac{v^2}{r}[/math]