Table of Contents
- 1. Electric Field
- 2. Electric Potential
- 3. Electric Dipole Moment
- 4. Polarization
- 5. Displacement Field
- 6. Current Density
- 7. Magnetic Field
- 8. Magnetic Vector Potential
- 9. Magnetization
- 10. Magnetizing Field
- 11. Bound Charge
- 12. Bound Current
- 13. Electromagnetic Tensor
- 14. Terminology
- 15. References
1. Electric Field
- 0\(\mathbf{E}\), 전기장, 전장, 전계
2. Electric Potential
\(\phi\)
3. Electric Dipole Moment
\(\mathbf{p}\)
3.1. Definition
3.1.1. Basic Definition
- The electric dipole memont when charge \(q\) is seperated with the
displacement \(\mathbf{d}\) is:
- \[ \mathbf{p} := q\mathbf{d}. \]
3.1.2. Properties
- The torque on the dipole \(\mathbf{p}\)
- \[ \bm{\tau} = \mathbf{p}\times \mathbf{E} \]
- Notice the similarity to the torque on the magnetic dipole made of
the circular circuit
- \[ \bm{\tau} = i\mathbf{A}\times \mathbf{B} = \mathbf{m}\times \mathbf{B} \]
- The force on the dipole
- \[ \mathbf{F} = \mathbf{p}\cdot \nabla \mathbf{E} \]
- The electric potential energy of the dipole
- \[ U = -\mathbf{p}\cdot \mathbf{E} \]
- The electric potential caused by the dipole
- \[ V = \frac{1}{4\pi\varepsilon_0}\frac{\mathbf{p}\cdot \mathbf{\hat{r}}}{r^2} \]
4. Polarization
- \(\mathbf{P}\), Electric Polarization, Polarization Density
- Electric dipole moment density
4.1. Definition
- Volume density of the 3.
- \[ \mathbf{P} = \operatorname*{equil\,lim}_{V\to 0} \frac{\mathbf{p}}{V} \]
- It is a statistical quantity.
4.2. Properties
- The electric potential due to the polarization field \(\mathbf{P}\)
- \[ V(\mathbf{r}) = \frac{1}{4\pi\varepsilon_0}\int_\Omega \frac{\mathbf{P}(\mathbf{r}')\cdot \widehat{\mathbf{r}-\mathbf{r}'}}{|\mathbf{r}-\mathbf{r}'|^2}\,\mathrm{d}^{\wedge 3}\mathbf{r}' = \frac{1}{4\pi\varepsilon_0}\int_\Omega \mathbf{P}(\mathbf{r}')\cdot \nabla\big|^{\mathbf{r}'}\left(\frac{1}{|\mathbf{r}-\mathbf{r}'|}\right)\,\mathrm{d}^{\wedge 3}\mathbf{r}' \]
\[
\begin{align*} V(\mathbf{r})\ \ =\quad\ & \frac{1}{4\pi\varepsilon_0}\left[\int_\Omega\nabla\big|^{\mathbf{r}'}\cdot\left(\frac{\mathbf{P}(\mathbf{r}')}{|\mathbf{r}-\mathbf{r}'|}\right)\,\mathrm{d}^{\wedge 3}\mathbf{r}' -\int_\Omega \frac{1}{|\mathbf{r}-\mathbf{r}'|}\nabla\big|^{\mathbf{r}'}\mathbf{P}(\mathbf{r}')\,\mathrm{d}^{\wedge 3}\mathbf{r}' \right] \\[1em] \stackrel{\text{div. thm}}{=} \ & \frac{1}{4\pi\varepsilon_0}\left[\oint_{\partial\Omega}\frac{\mathbf{P}(\mathbf{r}')}{|\mathbf{r}-\mathbf{r}'|}\cdot\mathrm{d}^{\wedge 3}\mathbf{r}' -\int_\Omega \frac{1}{|\mathbf{r}-\mathbf{r}'|}\nabla\big|^{\mathbf{r}'}\mathbf{P}(\mathbf{r}')\,\mathrm{d}^{\wedge 3}\mathbf{r}' \right] \\[1em] \ \ =\quad\ & \frac{1}{4\pi\varepsilon_0}\left[\oint_{\partial\Omega}\frac{\mathbf{P}(\mathbf{r}')}{|\mathbf{r}-\mathbf{r}'|}\,\cdot{\mathbf{\hat{n}}}\,\mathrm{d}^{\wedge 2}\mathbf{r}' -\int_\Omega \frac{1}{|\mathbf{r}-\mathbf{r}'|}\nabla\big|^{\mathbf{r}'}\mathbf{P}(\mathbf{r}')\,\mathrm{d}^{\wedge 3}\mathbf{r}' \right] \\ \end{align*}\]
- Using the bound charges and additionally letting \(\mathscr{r}'' := |\mathbf{r} - \mathbf{r}'|\),
- \[ V(\mathbf{r}) = \frac{1}{4\pi\varepsilon_0}\oint_{\partial \Omega} \frac{\sigma_b}{r''}\,\mathrm{d}^{\wedge 2}\mathbf{r}' +\frac{1}{4\pi\varepsilon_0}\int_\Omega \frac{\rho_b}{r''}\,\mathrm{d}^{\wedge 3}\mathbf{r}' \]
- Bound Charge
id:: 669acb4b-c304-482b-b516-99d6762775fa
- The bound surface charge density \(\sigma_b\) and the bound charge
density \(\rho_b\)
- \[ \sigma_b := \mathbf{P}\cdot \mathbf{\hat{n}},\quad \rho_b := -\nabla\cdot \mathbf{P} \]
- Reference
- Griffith
- https://en.wikipedia.org/wiki/Polarization_density
5. Displacement Field
- \(\mathbf{D}\)
- \(\mathbf{D} = \varepsilon_0\mathbf{E} + \mathbf{P}\)
- It is the \(\mathbf{E}\) field outside, but with the dipole field inside.
- See
6. Current Density
\(\mathbf{J}\)
7. Magnetic Field
- \(\mathbf{B}\), Magnetic Flux Density, Magnetic Induction
8. Magnetic Vector Potential
- \(\mathbf{A}\)
- \(V = -q\mathbf{v}\cdot\mathbf{A}\)
8.1. Gauge Invariance
- The choice of magnetic vector potential does not affect the laws of physics as long as it complies with the constraints imposed by context.
9. Magnetization
- \(\mathbf{M}\)
- Magnetic dipole moment density
- \[ \mathbf{M} = \operatorname*{equil\,lim}_{V\to 0}\frac{\mathbf{m}}{V} \] where \(\mathbf{m}\) is the magnetic moment.
10. Magnetizing Field
- \(\mathbf{H}\), Magnetic Field Strength, H-Field, Magnetic Field
\(\mathbf{H} = \mathbf{B}/\mu_0 - \mathbf{M}\)
- It is the \(\mathbf{B}\) field outside, but without the solenoid field inside.
11. Bound Charge
The bound surface charge density \( \sigma_b \) and the bound charge density \( \rho_b \) is defined as \[ \sigma_b := \mathbf{P}\cdot \mathbf{\hat{n}},\quad \rho_b := -\nabla\cdot \mathbf{P} \]
12. Bound Current
\[ \mathbf{J}_b = \nabla\times \mathbf{M} + \frac{\partial \mathbf{P}}{\partial t}. \]
13. Electromagnetic Tensor
14. Terminology
** Engineering
- 전계
- -> Electric Field
- 쇄교 자속
- 쇄교: 사슬저럼 교차하다
- Magnetic Flux Linkage, Weber Turns(Wb Turns)
- \[ \lambda = N\Phi \]
- -> Magnetic Flux