ED formulas (2)

Symbol Convention

Symbol Convention%

In accordance with ISO 80000-2, the following font conventions are employed:
- Scalars and components for vectors or tensors are represented by lightface italic type ( $a, ϕ, a_{i}, T_{i j}$ ).
- Vectors are represented by boldface italic type ( $a, ω$ ).
- Second-order tensors are represented by boldface sans-serif type ( $T, I$ ).
- Operators & Constants: Roman (upright) type is used for fixed mathematical constants (e.g., Pi $π$ , the imaginary unit $i$ ) and differential operators (e.g., the differential $d$ in $d x$ ).
- Calculus Notation: For integrals, a thin space (\,) is used to separate the integrand from the differential operator, e.g., $\int f (x) d x$ .
Minkowski Metric: The Minkowski metric tensor $η_{μ ν}$ is defined using the mostly-plus signature convention:

η_{μ ν} = diag (- 1, + 1, + 1, + 1)

Consequently, the invariant spacetime interval is given by $d s^{2} = - c^{2} d t^{2} + d x^{2} + d y^{2} + d z^{2} = - d τ^{2}$ .

Lorentz transformation

K^{'} \to K : x^{α} = Λ_{β}^{α} x^{' β}

For $x$ -axis boost (where $K^{'}$ moves with velocity $v {\hat{e}}_{x}$ relative to $K$ ), the transformation matrix $Λ$ is:

Λ_{β}^{α} = [\begin{matrix} γ & γ β \\ γ β & γ \\ 1 \\ 1 \end{matrix}]

1 Static Electric Field

1.1 Multipole expansion

Consider electrostatic potential expansion at an external observation point $x$ , with localized charge source $ρ (x^{'})$ confined within a volume $V^{'}$ , and $R = x - x^{'}, r = | x | (f i e l d), r^{'} = | x^{'} | (s o u r c e)$ , then

\begin{aligned} \frac{1}{R} = \frac{1}{| x - x^{'} |} = e^{- x^{'} \cdot \nabla} \frac{1}{r} & = [1 - x^{'} \cdot \nabla + \frac{1}{2!} (x^{'} x^{'} : \nabla \nabla) + \dots] \frac{1}{r} \\ I : \nabla \nabla \frac{1}{r} = 0 ⟹ & = \frac{1}{r} - x^{'} \cdot \nabla \frac{1}{r} + \frac{1}{6} (3 x^{'} x^{'} - r^{' 2} I) : \nabla \nabla \frac{1}{r} + \dots \end{aligned}

⇓

\begin{aligned} φ (x) = \frac{1}{4 π ϵ_{0}} \int \frac{d q}{R} & = \frac{1}{4 π ϵ_{0}} [Q - p \cdot \nabla + \frac{1}{6} D : \nabla \nabla + \dots] \frac{1}{r} \\ = \frac{Q}{4 π ϵ_{0} r} + \frac{p \cdot \hat{r}}{4 π ϵ_{0} r^{2}} + \frac{D : \hat{r} \hat{r}}{8 π ϵ_{0} r^{3}} + \dots \end{aligned}

Where $Q = \int d q = \int ρ (x^{'}) d V^{'}$ is total charge, $p = \int x^{'} d q$ is electric dipole moment, $D = \int (3 x^{'} x^{'} - r^{' 2} I) d q$ is electric quadrupole moment.

Electrostatic intensity expansion

E = - \nabla φ = \frac{Q \hat{r}}{4 π ϵ_{0} r^{2}} + \frac{3 (p \cdot \hat{r}) \hat{r} - p}{4 π ϵ_{0} r^{3}} + \frac{5 (D : \hat{r} \hat{r}) \hat{r} - 2 D \cdot \hat{r}}{8 π ϵ_{0} r^{4}} + \dots

1.2 Small charged body in electric field

Assuming the charge of charged bodies $x$ is so small compared with the source $x^{'}$ which generates the electric field that $\nabla^{2} φ (x) = 0$ . Let $ξ$ be the relative coordinate of a charge element $d q$ from the body's center $x$ . The potential energy $U = \int φ (x + ξ) d q$ of the small body is,

U \approx Q φ |_{x} + p \cdot \nabla φ |_{x} + \frac{1}{6} D : \nabla \nabla φ |_{x} = (Q φ) |_{x} - (p \cdot E) |_{x} - \frac{1}{6} (D : \nabla E) |_{x}

Note that $Q = \int d q = \int ρ (ξ) d V$ , $p = \int ξ d q$ , $D = \int (3 ξ ξ - ξ^{2} I) d q$ are intrinsic properties of the small charged body itself, not the external source.

Furthermore, The total electrostatic force $F$ acting on the small charged body is,

F = - \nabla U = Q E_{ext} |_{x} + (p \cdot \nabla) E_{ext} |_{x} + \frac{1}{6} (D : \nabla \nabla) E_{ext} |_{x} + \dots

The total force moment $N$ acting on the small charged body about its center $x$ is,

N = \int ξ \times E_{ext} (x + ξ) d q = p \times E_{ext} + \frac{1}{3} \nabla \cdot (D \times E_{ext}) + \dots

1.3 Spherical harmonic expansion

1.3.1 Orthonormal Complete Set of Functions

Definition

Orthonormality

\int_{a}^{b} ψ_{n}^{*} (x) ψ_{m} (x) d x = δ_{n m}, with Integration Limits [a, b]

Completeness: Any square-integrable function $f (x)$ can be expanded as

f (x) = \sum_{n} C_{n} ψ_{n} (x), where C_{n} = \int_{a}^{b} ψ_{n}^{*} (x) f (x) d x

which implies

\sum_{n} ψ_{n}^{*} (x^{'}) ψ_{n} (x) = δ (x - x^{'})

Trigonometric & Complex Exponential Functions

ψ_{n} (x) = \frac{1}{\sqrt{a}} \sin \frac{n π x}{a}, \frac{1}{\sqrt{a}} \cos \frac{n π x}{a}, \frac{1}{\sqrt{2 a}} \exp (i \frac{n π x}{a}), with Integration Limits [- a, a]

Note: $n \in N_{+}$ for sine set; $n \in N$ for cosine set; $n \in Z$ for complex set.

Legendre Polynomials

ψ_{l} (x) = \sqrt{\frac{2 l + 1}{2}} P_{l} (x), l \in N, with Integration Limits [- 1, 1]

P_{l} (x) = \frac{1}{2^{l} l!} \frac{d^{l}}{d x^{l}} (x^{2} - 1)^{l} = {\begin{cases} 1, l = 0 \\ x, l = 1 \\ \frac{1}{2} (3 x^{2} - 1), l = 2 \\ \frac{1}{2} (5 x^{3} - 3 x), l = 3 \\ \dots \end{cases}

Spherical Harmonics

ψ_{l m} (θ, φ) = Y_{l}^{m} (Ω) = \sqrt{\frac{1}{2 π} \frac{2 l + 1}{2} \frac{(l - m)!}{(l + m)!}} P_{l}^{m} (\cos θ) e^{i m φ},

l \in N, m \in Z & | m | \leq l, with θ \in [0, π], φ \in [0, 2 π]

P_{l}^{m} (x) = (- 1)^{m} (1 - x^{2})^{\frac{m}{2}} \frac{d^{m}}{d x^{m}} P_{l} (x), for m \geq 0

For negative $m$ , the relations are given by:

P_{l}^{- m} (x) = (- 1)^{m} \frac{(l - m)!}{(l + m)!} P_{l}^{m} (x) or Y_{l}^{- m} (θ, φ) = (- 1)^{m} Y_{l}^{m *} (θ, φ)

The first few associated Legendre functions are,

\begin{aligned} P_{0}^{0} (x) & = 1, \\ P_{1}^{0} (x) & = x, P_{1}^{1} (x) = - \sqrt{1 - x^{2}}, \\ P_{2}^{0} (x) & = \frac{1}{2} (3 x^{2} - 1), P_{2}^{1} (x) = - 3 x \sqrt{1 - x^{2}}, P_{2}^{2} (x) = 3 (1 - x^{2}), \\ P_{3}^{0} (x) & = \frac{1}{2} (5 x^{3} - 3 x), P_{3}^{1} (x) = - \frac{3}{2} (5 x^{2} - 1) \sqrt{1 - x^{2}}, P_{3}^{2} (x) = 15 x (1 - x^{2}), P_{3}^{3} (x) = - 15 (1 - x^{2})^{3 / 2} \\ \dots \end{aligned}

Orthonormality

\oint Y_{l^{'}}^{m^{'} *} (Ω) Y_{l}^{m} (Ω) d Ω = δ_{l l^{'}} δ_{m m^{'}}

Completeness

\sum_{l = 0}^{+ \infty} \sum_{m = - l}^{l} Y_{l}^{m} (Ω^{'}) Y_{l}^{m} (Ω) = δ (Ω - Ω^{'}) = δ (\cos θ - \cos θ^{'}) δ (φ - φ^{'})

1.3.2 Multipole expansion *

By using the Legendre generating function, one finds

\frac{1}{R} = \frac{1}{| x - x^{'} |} = \sum_{l = 0}^{\infty} \frac{r_{<}^{l}}{r_{>}^{l + 1}} P_{l} (\cos γ), γ = \arccos \frac{x \cdot x^{'}}{| x \cdot x^{'} |}, r_{<} = min (x, x^{'}), r_{>} = max (x, x^{'})

P_{l} (\cos γ) = 2 π \cdot \frac{2}{2 l + 1} \cdot \sum_{m = - l}^{l} Y_{l}^{m *} (θ^{'}, φ^{'}) Y_{l}^{m} (θ, φ)

Consider a localized charge distribution $ρ (x^{'})$ confined within a volume $V^{'}$ . The exact electrostatic potential $Φ (x)$ at an external observation point $x$ (where $r = | x | > r^{'} = | x^{'} |$ , hence $r_{<} = r^{'}$ and $r_{>} = r$ ) is given by,

Φ (x) = \frac{1}{4 π ε_{0}} \int_{V^{'}} \frac{ρ (x^{'})}{| x - x^{'} |} d^{3} x^{'} = \frac{1}{4 π ε_{0}} \sum_{l = 0}^{\infty} \frac{1}{r^{l + 1}} \int_{V^{'}} ρ (x^{'}) (r^{'})^{l} P_{l} (\cos γ) d^{3} x^{'}

Monopole Term (Point Charge). For $l = 0$ , since $P_{0} (\cos γ) = 1$ , the first term simplifies to:

Φ^{(0)} (x) = \frac{1}{4 π ε_{0} r} \int_{V^{'}} ρ (x^{'}) d^{3} x^{'} = \frac{1}{4 π ε_{0}} \frac{q}{r}, with q = \int_{V^{'}} ρ (x^{'}) d^{3} x^{'}

Dipole Term. For $l = 1$ , since $P_{1} (\cos γ) = \cos γ = \frac{x \cdot x^{'}}{r r^{'}}$ , the second term becomes:

Φ^{(1)} (x) = \frac{1}{4 π ε_{0} r^{2}} \int_{V^{'}} ρ (x^{'}) r^{'} (\frac{x \cdot x^{'}}{r r^{'}}) d^{3} x^{'} = \frac{1}{4 π ε_{0}} \frac{p \cdot \hat{r}}{r^{2}}

with p = \int_{V^{'}} ρ (x^{'}) x^{'} d^{3} x^{'}

Quadrupole Term. For $l = 2$ , since $P_{2} (\cos γ) = \frac{1}{2} (3 \cos^{2} γ - 1)$ , substituting $\cos γ$ into the expression yields:

\begin{aligned} Φ^{(2)} (x) & = \frac{1}{4 π ε_{0} r^{3}} \int_{V^{'}} ρ (x^{'}) (r^{'})^{2} \cdot \frac{1}{2} [3 \frac{(x \cdot x^{'})^{2}}{r^{2} (r^{'})^{2}} - 1] d^{3} x^{'} \\ = \frac{1}{8 π ε_{0} r^{5}} x_{i} x_{j} \int_{V^{'}} ρ (x^{'}) [3 x_{i}^{'} x_{j}^{'} - δ_{i j} (r^{'})^{2}] d^{3} x^{'} \\ = \frac{1}{4 π ε_{0}} \frac{1}{2 r^{5}} \sum_{i, j} D_{i j} x_{i} x_{j} \\ = \frac{D : \hat{r} \hat{r}}{8 π ϵ_{0} r^{3}} \end{aligned}

with D_{i j} = \int_{V^{'}} ρ (x^{'}) [3 x_{i}^{'} x_{j}^{'} - δ_{i j} (r^{'})^{2}] d^{3} x^{'}

And extending to $l = 3, 4, \dots$ yields the octupole ( $l = 3$ ), hexadecapole ( $l = 4$ ), and more high $2^{l}$ -pole moments $\dots$

1.3.3 Wigner $D$ -Matrices *

Spherical harmonics $Y_{l}^{m} (θ, φ)$ form a $(2 l + 1)$ -dimensional irreducible representation of $S O (3)$ . Under a coordinate rotation $R$ parameterized by the Euler angles $(α, β, γ)$ , the total angular momentum quantum number $l$ remains invariant because the rotation operator commutes with the total angular momentum squared: $[R, L^{2}] = 0$ . However, the azimuthal projection $m$ undergoes a linear transformation.

The transformed spherical harmonic in the rotated frame can be expressed as a linear combination of the original basis functions,

Y_{l}^{m} (θ^{'}, φ^{'}) = \sum_{m^{'} = - l}^{l} D_{m^{'} m}^{l} (α, β, γ) Y_{l}^{m^{'}} (θ, φ)

where $D_{m^{'} m}^{l} (α, β, γ)$ are the elements of the Wigner $D$ -matrix. We find that the entire rotation operator can be directly written as $R (α, β, γ) = e^{- i α L_{z}} e^{- i β L_{y}} e^{- i γ L_{z}}$ . Thus, Wigner $D$ -matrix can be factorized into a product of exponentials and Wigner's small $d$ -matrix,

D_{m^{'} m}^{l} (α, β, γ) = e^{- i m^{'} α} d_{m^{'} m}^{l} (β) e^{- i m γ}

Here, Wigner's small $d$ -matrix $d_{m^{'} m}^{l} (β)$ describes the rotation around the $y$ -axis,

d_{m^{'} m}^{l} (β) = ⟨ l, m^{'} | e^{- i β L_{y}} | l, m ⟩

Since spatial rotations preserve the norm of the state space, the Wigner $D$ -matrix is strictly unitary, satisfying,

\sum_{m = - l}^{l} D_{m k}^{l *} (α, β, γ) D_{m n}^{l} (α, β, γ) = δ_{k n}

Specially, when the second projection index is set to zero ( $m = 0$ ), the Wigner $D$ -matrix simplifies directly to a spherical harmonic,

D_{m^{'} 0}^{l} (α, β, 0) = \sqrt{\frac{4 π}{2 l + 1}} Y_{l}^{m^{'} *} (β, α)

You can computes the Wigner $D$ -matrix and small $d$ -matrix elements analytically by using the built-in WignerD function with the following syntax in Wolfram Mathematica (MMA),

\begin{aligned} D_{m^{'} m}^{l} (α, β, γ) & ⟹ WignerD[{l, m’, m}, α, β, γ] \\ d_{m^{'} m}^{l} (β) & ⟹ WignerD[{l, m’, m}, β] \end{aligned}

1.4 Method: separation of variables

1.4.1 Cartesian coordinate system

1.4.2 Cylindrical coordinate system

1.4.3 Spherical coordinate system

1.4.4 Green function *

1.5 Cases

If all charges are contained within the sphere $V$ (see Appendix for ED formulas)

∭_{V} E (x) d^{3} x = - \frac{p}{3 ϵ_{0}}

An ellipsoid uniformly charged with Q (in the principal inertia axis frame)

D_{11} = \frac{Q}{5} (2 a_{1}^{2} - a_{2}^{2} - a_{3}^{2}), D_{22} = \dots, D_{i \neq j} = 0

2 Static Magnetic Field

3 EM Wave

4 Electromagnetic Radiation