Electromagnetism

Table of Contents

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

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    • The bound surface charge density \(\sigma_b\) and the bound charge density \(\rho_b\)
      • \[ \sigma_b := \mathbf{P}\cdot \mathbf{\hat{n}},\quad \rho_b := -\nabla\cdot \mathbf{P} \]

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. Bloch Equation

  • Introduced by Felix Bloch in 1946.

Macroscopic equations that describe the change in magnetization as a magnetic field is imposed.

\[ \frac{d\mathbf{M}}{dt}=\gamma\mathbf{M}\times\mathbf{B}-\mathbf{T}(\mathbf{M}-M_0\hat{\mathbf{z}}) \] where \(\mathbf{T}\) is the relaxation time matrix with \(T_{11}=T_{22}=T_2\) and \(T_{33}=T_1\).

14. Gyromagnetic Ratio

14.1. Classical

\[ \gamma := \frac{q}{2m} \]

14.2. Isolated Electron

\[ \gamma_{\rm e} := \frac{-e}{2m_{\rm e}}|g_{\rm e}| = \frac{g_{\rm e}\mu_{\rm B}}{\hbar} \] It deviate from the classical result by the g-factor.

14.3. Nucleus

\[ \gamma = \frac{e}{2m_{\rm p}}g_{\rm n} = \frac{g_{\rm n}\mu_{\rm N}}{\hbar} \]

15. Electromagnetic Tensor

16. References

Author: Jeemin Kim

Created: 2026-07-12 Sun 14:28