20.4.3. Radiation of moving charges - Larmor’s formula

20.4.3.1. Green’s function solution of wave equations: Liénard-Wiechert potential

Wave equations in electromagmetism. See Classical Electromagnetism: Wave Equations in Electromagnetism. Using Lorentz gauge, \(\nabla \cdot \mathbf{a} + \frac{1}{c^2}\partial_t \varphi = 0\), the wave equations for the mathbftor potential \(\mathbf{a}\) and the scalar potential \(\varphi\) read

\[\begin{split}\begin{aligned} \frac{1}{c^2} \partial_{tt} \mathbf{a} - \Delta \mathbf{a} & = \mu \mathbf{j} \\ \frac{1}{c^2} \partial_{tt} \varphi - \Delta \varphi & = \frac{\rho}{\varepsilon} \\ \end{aligned}\end{split}\]

Solution of wave equation with Green’s function

The solution of a wave equation

\[\frac{1}{c^2} \partial_{tt} u - \Delta u = f \ ,\]

can be written using Math: Green’s function method (with assumptions of causality, and no contribution from i.c. and b.c.),

\[\begin{split}\begin{aligned} u(\mathbf{r},t) & = \frac{1}{4 \pi} \int_{t' \in T} \int_{\mathbf{r}' \in V} \dfrac{ \delta \left( t' - t + \frac{|\mathbf{r}' - \mathbf{r}|}{c} \right) }{| \mathbf{r}' - \mathbf{r} |} f (\mathbf{r}', t') d \mathbf{r}' d t' = \\ & = \frac{1}{4 \pi} \int_{\mathbf{r}' \in V} \frac{1}{| \mathbf{r}' - \mathbf{r} |} f\left( \mathbf{r}', t_{ret}(\mathbf{r}, t; \mathbf{r}') \right) d \mathbf{r}' \\ \end{aligned}\end{split}\]

with the retarded time \(t_{ret}(\mathbf{r}, t; \mathbf{r}') = t - \frac{| \mathbf{r}' - \mathbf{r} |}{c} \).

The solution of the wave equations reads

\[\begin{split}\begin{aligned} \mathbf{a}(\mathbf{r},t) & = \frac{\mu}{4 \pi} \int_{\mathbf{r}' \in V} \frac{1}{| \mathbf{r}' - \mathbf{r} |} \mathbf{j} \left( \mathbf{r}', t_{ret}(\mathbf{r}, t; \mathbf{r}') \right) d \mathbf{r}' \\ \varphi(\mathbf{r},t) & = \frac{1}{4 \pi \varepsilon} \int_{\mathbf{r}' \in V} \frac{1}{| \mathbf{r}' - \mathbf{r} |} \rho \left( \mathbf{r}', t_{ret}(\mathbf{r}, t; \mathbf{r}') \right) d \mathbf{r}' \\ \end{aligned}\end{split}\]

with the retarded time \(t_{ret}(\mathbf{r}, t; \mathbf{r}') = t - \frac{| \mathbf{r}' - \mathbf{r} |}{c} \).

Point charge. The charge density and the current density of a point charge with electrical charge \(q\), position \(\mathbf{r}_s(t)\) and velocity \(\mathbf{v}_s = \dot{\mathbf{r}}_s\) are respectively

\[\begin{split}\begin{aligned} \rho(\mathbf{r},t) & = q \, \delta ( \mathbf{r} - \mathbf{r}_s(t) ) \\ \mathbf{j}(\mathbf{r},t) & = q \mathbf{v}_s(t) \, \delta ( \mathbf{r} - \mathbf{r}_s(t) ) \\ \end{aligned}\end{split}\]

Thus, the electromagnetic potentials generated by a point charge are

\[\begin{split}\begin{aligned} \mathbf{a}(\mathbf{r},t) & = \frac{\mu}{4 \pi} \frac{1}{| \mathbf{r}_s(t_{ret}) - \mathbf{r} |} q \mathbf{v}_s(t_{ret}) \\ \varphi(\mathbf{r},t) & = \frac{1}{4 \pi \varepsilon} \frac{1}{| \mathbf{r}_s(t_{ret}) - \mathbf{r} |} q \\ \end{aligned}\end{split}\]

Derivatives of the retarded time

\[\begin{split}\begin{aligned} \partial_t t_{ret} & = \frac{1}{1 - \hat{\mathbf{r}} \cdot \boldsymbol\beta} \\ \nabla t_{ret} & = - \frac{1}{c} \frac{\hat{\mathbf{r}}}{1 - \hat{\mathbf{r}} \cdot \boldsymbol\beta} \ . \end{aligned}\end{split}\]

Details

Time derivative

\[\begin{split}\begin{aligned} \partial_t t_{ret}(\mathbf{r}, t, \mathbf{r}_s(t)) & = \partial_t \left( t - \frac{|\mathbf{r}_s(t_{ret}(\mathbf{r},t, \mathbf{r}_s(t_{ret})) - \mathbf{r}|}{c} \right) = \\ & = 1 - \frac{1}{c} \frac{d |\mathbf{r}_s - \mathbf{r}|}{d t_{ret}} \partial_t t_{ret} = \\ & = 1 + \frac{x_{k} - x_{s,k}}{|\mathbf{r}_s - \mathbf{r}|} \frac{v_{s,k}}{c} \partial_t t_{ret} = \\ & = 1 + \hat{\mathbf{r}} \cdot \boldsymbol\beta \, \partial_t t_{ret} \ , \end{aligned}\end{split}\]

and thus

\[\partial_t t_{ret} = \frac{1}{1 - \hat{\mathbf{r}} \cdot \boldsymbol\beta} \ .\]

Space derivatives

\[\begin{split}\begin{aligned} \partial_k t_{ret}(\mathbf{r}, t, \mathbf{r}_s(t)) & = \partial_k \left( t - \frac{|\mathbf{r}_s(t_{ret}) - \mathbf{r}|}{c} \right) = \\ & = - \frac{1}{c} \partial_k |\mathbf{r}_s(t_{ret}) - \mathbf{r}| = \\ & = \frac{1}{c} \frac{r_i - r_{s,i}(t_{ret})}{|\mathbf{r}_s(t_{ret}) - \mathbf{r}|} \partial_k ( r_{s,i}(t_{ret}) - r_i )= \\ & = \frac{1}{c} \frac{r_i - r_{s,i}(t_{ret})}{|\mathbf{r}_s(t_{ret}) - \mathbf{r}|} \left( v_{s,i}(t_{ret}) \partial_k t_{ret} - \delta_{ik} \right) = \\ & = \hat{\mathbf{r}} \cdot \boldsymbol\beta \ , \partial_k t_{ret} - \frac{1}{c}\frac{r_k - r_{s,k}(t_{ret})}{|\mathbf{r}_s(t_{ret}) - \mathbf{r}|} \ , \end{aligned}\end{split}\]

so that

\[\partial_k t_{ret} = - \frac{1}{c} \frac{1}{1 - \hat{\mathbf{r}} \cdot \boldsymbol\beta} \frac{r_k - r_{s,k}(t_{ret})}{|\mathbf{r}_s(t_{ret}) - \mathbf{r}|} \]

or, using vector notation,

\[\nabla t_{ret} = - \frac{1}{c} \frac{\hat{\mathbf{r}}}{1 - \hat{\mathbf{r}} \cdot \boldsymbol\beta} \ .\]

Derivatives of \(\ |\mathbf{r} - \mathbf{r}_s(t_{ret})|^{-1}\)

\[\begin{aligned} \partial_t | \mathbf{r} - \mathbf{r}_s(t_{ret}) |^n & = - n | \mathbf{r} - \mathbf{r}_s(t_{ret}) |^{n-1} \frac{ \hat{\mathbf{r}} \cdot \boldsymbol\beta }{1 - \hat{\mathbf{r}} \cdot \boldsymbol\beta}c \end{aligned}\]

\[\begin{aligned} \partial_k | \mathbf{r} - \mathbf{r}_s(t_{ret}) |^n & = n | \mathbf{r}_s(t_{ret}) - \mathbf{r} |^{n-1} \left[ \frac{ \hat{\mathbf{r}} \cdot \boldsymbol\beta }{1 - \hat{\mathbf{r}} \cdot \boldsymbol\beta} + 1 \right] \hat{r}_k \end{aligned}\]

Details

\[\begin{split}\begin{aligned} \partial_k | \mathbf{r} - \mathbf{r}_s(t_{ret}) |^n & = \partial_k \left[ ( \mathbf{r} - \mathbf{r}_s(t_{ret}) ) \cdot ( \mathbf{r} - \mathbf{r}_s(t_{ret}) ) \right]^{\frac{n}{2}} = \\ & = n | \mathbf{r} - \mathbf{r}_s(t_{ret}) |^{n-2} ( \mathbf{r} - \mathbf{r}_s(t_{ret}) ) \cdot \partial_k ( \mathbf{r} - \mathbf{r}_s(t_{ret}) ) \end{aligned}\end{split}\]

Now,

\[\begin{split}\begin{aligned} \partial_t ( r_{i} - r_{s,i}(t_{ret})) & = - \partial_t t_{ret} v_{s,i}(t_{ret}) = \\ & = - \frac{v_{s,i}(t_{ret})}{1 - \hat{\mathbf{r}} \cdot \boldsymbol{\beta}} \\ \partial_k (r_{i} - r_{s,i}(t_{ret})) & = \delta_{ik} - \partial_k t_{ret} v_{s,i} (t_{ret}) = \\ & = \delta_{ik} + \frac{1}{c} \frac{\hat{r}_k}{1-\hat{\mathbf{r}} \cdot \boldsymbol\beta} v_{s,i} (t_{ret}) \ , \end{aligned}\end{split}\]

and thus

\[\begin{split}\begin{aligned} \partial_t | \mathbf{r} - \mathbf{r}_s(t_{ret}) |^n & = \partial_t \left[ ( \mathbf{r} - \mathbf{r}_s(t_{ret}) ) \cdot ( \mathbf{r} - \mathbf{r}_s(t_{ret}) ) \right]^{\frac{n}{2}} = \\ & = n | \mathbf{r} - \mathbf{r}_s(t_{ret}) |^{n-2} ( \mathbf{r} - \mathbf{r}_s(t_{ret}) ) \cdot \partial_t ( \mathbf{r} - \mathbf{r}_s(t_{ret}) ) = \\ & = - n | \mathbf{r} - \mathbf{r}_s(t_{ret}) |^{n-1}\frac{ \hat{\mathbf{r}} \cdot \mathbf{v}_s(t_{ret}) }{1 - \hat{\mathbf{r}} \cdot \boldsymbol\beta} \end{aligned}\end{split}\]

and

\[\begin{split}\begin{aligned} \partial_k | \mathbf{r} - \mathbf{r}_s(t_{ret}) |^n & = \partial_k \left[ ( \mathbf{r} - \mathbf{r}_s(t_{ret}) ) \cdot ( \mathbf{r} - \mathbf{r}_s(t_{ret}) ) \right]^{\frac{n}{2}} = \\ & = n | \mathbf{r} - \mathbf{r}_s(t_{ret}) |^{n-2} ( \mathbf{r} - \mathbf{r}_s(t_{ret}) ) \cdot \partial_k ( \mathbf{r} - \mathbf{r}_s(t_{ret}) ) = \\ & = n | \mathbf{r} - \mathbf{r}_s(t_{ret}) |^{n-1} \left[ \frac{1}{c} \frac{ \hat{\mathbf{r}} \cdot \mathbf{v}_s(t_{ret}) }{1 - \hat{\mathbf{r}} \cdot \boldsymbol\beta} + 1 \right] \hat{r}_k \end{aligned}\end{split}\]

From potentials to the electromagnetic field

\[\begin{aligned} \mathbf{e} & = - \frac{q}{4 \pi \varepsilon} \frac{\mathbf{a}_s(t_{ret})}{|\mathbf{r}| c^2} \frac{1}{1 - \hat{\mathbf{r}} \cdot \boldsymbol\beta} + \frac{q}{4 \pi \varepsilon} \frac{1}{|\mathbf{r}|^2} \left( \frac{ \hat{\mathbf{r}} - \boldsymbol\beta (\hat{\mathbf{r}} \cdot \boldsymbol\beta) }{1 - \hat{\mathbf{r}} \cdot \boldsymbol\beta} \right) \ . \end{aligned}\]

\[\begin{aligned} \mathbf{b} & = \frac{q}{4 \pi \varepsilon} \frac{1}{c} \left[ -\frac{\hat{\mathbf{r}} \times \mathbf{a}_s(t_{ret})}{|\mathbf{r}| c^2} \frac{1}{1-\hat{\mathbf{r}} \cdot \boldsymbol\beta} - \frac{\hat{\mathbf{r}}}{|\mathbf{r}|^2} \times \boldsymbol\beta \left( \frac{1}{1 - \hat{\mathbf{r}} \cdot \boldsymbol\beta} \right) \right] \ . \end{aligned}\]

It’s easy to realize that \(\mathbf{b} = \frac{1}{c} \hat{\mathbf{r}} \times \mathbf{e}\).

Details

\[\begin{split}\begin{aligned} \mathbf{e} & = - \partial_t \mathbf{a} - \nabla \varphi = \\ & = - \partial_t \left( \frac{\mu}{4 \pi} \frac{q \mathbf{v}_s(t_{ret})}{|\mathbf{r}_s(t_{ret}) - \mathbf{r}|} \right) - \nabla \left( \frac{1}{4 \pi \varepsilon} \frac{q}{|\mathbf{r}_s(t_{ret}) - \mathbf{r}|} \right) = \\ & = - \frac{q \mu}{4 \pi} \left[ \frac{1}{|\mathbf{r}|} \mathbf{a}_s(t_{ret}) \, \partial_t t_{ret} + \mathbf{v}_s |\mathbf{r}|^{-2} \frac{\hat{\mathbf{r}} \cdot \boldsymbol\beta}{1 - \hat{\mathbf{r}} \cdot \boldsymbol\beta} c \right] - \frac{q}{4 \pi \varepsilon} |\mathbf{r}|^{-2} \left( - 1 - \frac{\hat{\mathbf{r}} \cdot \boldsymbol\beta}{1 - \hat{\mathbf{r}} \cdot \boldsymbol\beta} \right) \hat{\mathbf{r}} = \\ & = - \frac{q \mu c^2}{4 \pi} \left[ \frac{\mathbf{a}_s(t_{ret})}{|\mathbf{r}| c^2} \frac{1}{1 - \hat{\mathbf{r}} \cdot \boldsymbol\beta} + \frac{\boldsymbol\beta}{|\mathbf{r}|^2} \frac{\hat{\mathbf{r}} \cdot \boldsymbol\beta}{1 - \hat{\mathbf{r}} \cdot \boldsymbol\beta} \right] + \frac{q}{4 \pi \varepsilon} \frac{1}{|\mathbf{r}|^{2}} \left( 1 + \frac{\hat{\mathbf{r}} \cdot \boldsymbol\beta}{1 - \hat{\mathbf{r}} \cdot \boldsymbol\beta} \right) \hat{\mathbf{r}}= \\ & = - \frac{q}{4 \pi \varepsilon} \frac{\mathbf{a}_s(t_{ret})}{|\mathbf{r}| c^2} \frac{1}{1 - \hat{\mathbf{r}} \cdot \boldsymbol\beta} + \frac{q}{4 \pi \varepsilon} \frac{1}{|\mathbf{r}|^2} \left( \frac{ -\boldsymbol\beta (\hat{\mathbf{r}} \cdot \boldsymbol\beta) + \hat{\mathbf{r}} (\hat{\mathbf{r}} \cdot \boldsymbol\beta) }{1 - \hat{\mathbf{r}} \cdot \boldsymbol\beta} + \hat{\mathbf{r}} \right) = \\ & = - \frac{q}{4 \pi \varepsilon} \frac{\mathbf{a}_s(t_{ret})}{|\mathbf{r}| c^2} \frac{1}{1 - \hat{\mathbf{r}} \cdot \boldsymbol\beta} + \frac{q}{4 \pi \varepsilon} \frac{1}{|\mathbf{r}|^2} \left( \frac{ \hat{\mathbf{r}} - \boldsymbol\beta (\hat{\mathbf{r}} \cdot \boldsymbol\beta) }{1 - \hat{\mathbf{r}} \cdot \boldsymbol\beta} \right) \ . \end{aligned}\end{split}\]

\[\begin{split}\begin{aligned} \mathbf{b} & = \nabla \times \mathbf{a} = \\ & = \nabla \times \left( \frac{\mu}{4 \pi} \frac{q \mathbf{v}_s(t_{ret})}{|\mathbf{r} - \mathbf{r}_s(t_{ret})|} \right) = \\ & = \hat{\mathbf{e}}_k \varepsilon_{klm} \partial_l \left( \frac{\mu}{4 \pi} \frac{q v_{s,m}(t_{ret})}{|\mathbf{r} - \mathbf{r}_s(t_{ret})|} \right) = \\ & = \hat{\mathbf{e}}_k \varepsilon_{klm} \frac{\mu q}{4 \pi} \partial_l \left( \frac{v_{s,m}(t_{ret})}{|\mathbf{r} - \mathbf{r}_s(t_{ret})|} \right) = \\ & = \hat{\mathbf{e}}_k \varepsilon_{klm} \frac{\mu q}{4 \pi} \left[ \frac{a_{s,m}(t_{ret}) \partial_l t_{ret}}{|\mathbf{r}-\mathbf{r}_s(t_{ret})|} - v_{s,m}(t_{ret})\frac{\hat{r}_l}{|\mathbf{r}|^2} \left( 1 + \frac{\hat{\mathbf{r}} \cdot \boldsymbol\beta}{1 - \hat{\mathbf{r}} \cdot \boldsymbol\beta} \right) \right] = \\ & = \hat{\mathbf{e}}_k \varepsilon_{klm} \frac{\mu q}{4 \pi} \left[ - \frac{a_{s,m}(t_{ret}) }{|\mathbf{r}-\mathbf{r}_s(t_{ret})|} \frac{1}{c} \frac{\hat{r}_l}{1 - \hat{\mathbf{r}} \cdot \boldsymbol\beta} - v_{s,m}(t_{ret})\frac{\hat{r}_l}{|\mathbf{r}|^2} \left( 1 + \frac{\hat{\mathbf{r}} \cdot \boldsymbol\beta}{1 - \hat{\mathbf{r}} \cdot \boldsymbol\beta} \right) \right] = \\ & = \frac{q \mu c^2}{4 \pi} \frac{1}{c^2} \left[ -\frac{\hat{\mathbf{r}} \times \mathbf{a}_s(t_{ret})}{|\mathbf{r}| c} \frac{1}{1-\hat{\mathbf{r}} \cdot \boldsymbol\beta} - \frac{\hat{\mathbf{r}}}{|\mathbf{r}|^2} \times \mathbf{v}_s(t_{ret}) \left( 1 + \frac{\hat{\mathbf{r}} \cdot \boldsymbol\beta}{1 - \hat{\mathbf{r}} \cdot \boldsymbol\beta} \right) \right] = \\ & = \frac{q}{4 \pi \varepsilon} \frac{1}{c} \left[ -\frac{\hat{\mathbf{r}} \times \mathbf{a}_s(t_{ret})}{|\mathbf{r}| c^2} \frac{1}{1-\hat{\mathbf{r}} \cdot \boldsymbol\beta} - \frac{\hat{\mathbf{r}}}{|\mathbf{r}|^2} \times \boldsymbol\beta \left( 1 + \frac{\hat{\mathbf{r}} \cdot \boldsymbol\beta}{1 - \hat{\mathbf{r}} \cdot \boldsymbol\beta} \right) \right] = \\ & = \frac{q}{4 \pi \varepsilon} \frac{1}{c} \left[ -\frac{\hat{\mathbf{r}} \times \mathbf{a}_s(t_{ret})}{|\mathbf{r}| c^2} \frac{1}{1-\hat{\mathbf{r}} \cdot \boldsymbol\beta} - \frac{\hat{\mathbf{r}}}{|\mathbf{r}|^2} \times \boldsymbol\beta \left( \frac{1}{1 - \hat{\mathbf{r}} \cdot \boldsymbol\beta} \right) \right] \ . \end{aligned}\end{split}\]

20.4.3.2. Larmor formula

Poynting vector reads

\[\mathbf{s}(\mathbf{r},t) := \frac{ \mathbf{e}(\mathbf{r},t) \times \mathbf{b}(\mathbf{r},t) }{\mu} \ ,\]

and thus, using the electromagnetic field of a moving charge,

\[\mathbf{s} = \frac{1}{\mu} \mathbf{e} \times \mathbf{b} = \frac{1}{c \mu} \mathbf{e} \times \left( \hat{\mathbf{r}} \times \mathbf{e} \right) = \frac{1}{\mu c} \left[ |\mathbf{e}|^2 \hat{\mathbf{r}} - ( \mathbf{e} \cdot \hat{\mathbf{r}} ) \mathbf{e} \right] \ .\]

Far-field approximation (also assuming that \(\boldsymbol\beta \sim \mathbf{0}\). todo slow charge? Non-relativistic limit?),

\[\mathbf{e} \sim - \frac{q}{4 \pi \varepsilon} \frac{\mathbf{a}_{s,ret}}{|\mathbf{r}| c^2}\]

and thus

\[\begin{split}\begin{aligned} \mathbf{s} & = \frac{1}{\mu c} \left( \frac{q}{4 \pi \varepsilon |\mathbf{r}| c^2} \right)^2 \, \mathbf{a} \times \left( \hat{\mathbf{r}} \times \mathbf{a} \right) = \\ & = \frac{1}{\mu c} \left( \frac{q}{4 \pi \varepsilon |\mathbf{r}| c^2} \right)^2 \left( |\mathbf{a}|^2 \hat{\mathbf{r}} - ( \mathbf{a} \cdot \hat{\mathbf{r}} ) \mathbf{a} \right) = \\ \end{aligned}\end{split}\]

The elementary power flux at \(\mathbf{r} = R \hat{\mathbf{r}}\), assuming that the motion is ina region much smaller than \(R\), so that \(t_{ret} = t - \frac{R}{c}\), is

\[\begin{split}\begin{aligned} \hat{\mathbf{r}} \cdot \mathbf{s} & = \frac{1}{\mu c} \left( \frac{q}{4 \pi \varepsilon R c^2} \right)^2 \left( |\mathbf{a}|^2 - ( \mathbf{a} \cdot \hat{\mathbf{r}} )^2 \right) = \\ \end{aligned}\end{split}\]

Let’s define the direction of \(\mathbf{a}_{ret} = a \hat{\mathbf{t}}\), and sperical coordinates so that \(\hat{\mathbf{r}} \cdot \mathbf{\mathbf{t}} = \cos \theta\). Integration over the sphere of radius \(R\) gives

\[\begin{split}\begin{aligned} \Phi & := \int_{S} \hat{\mathbf{r}} \cdot \mathbf{s} = \\ & = \int_{\theta=0}^{\pi} \int_{\phi = 0}^{2 \pi} \frac{1}{\mu c} \left( \frac{q}{4 \pi \varepsilon R c^2} \right)^2 \left( 1 - \cos^2 \theta \right)^2 |\mathbf{a}|^2 R^2 \sin \theta \, d \theta \, d \phi = \\ & = \frac{1}{\mu c} \left( \frac{q |\mathbf{a}|}{4 \pi \varepsilon c^2} \right)^2 \int_{\theta=0}^{\pi} \int_{\phi = 0}^{2 \pi} ( 1 - \cos^2 \theta) \sin \theta \, d \theta \, d \phi = \\ & = \frac{1}{\mu c} \left( \frac{q |\mathbf{a}|}{4 \pi \varepsilon c^2} \right)^2 \cdot \frac{4}{3} \cdot 2 \pi = \\ & = \frac{q^2 |\mathbf{a}_{s,ret}|^2}{6 \pi \varepsilon c^3} \ . \end{aligned}\end{split}\]

Average power.

\[\langle \Phi \rangle = \frac{q^2}{6 \pi \varepsilon c^3} \langle |\mathbf{a}|^2 \rangle\]

Examples:

• harmonic oscillator \(\mathbf{a}(t) = - A \omega^2 \cos(\omega t) \hat{\mathbf{x}}\), \(\langle |\mathbf{a}(t)|^2 \rangle = \frac{1}{2} A^2 \omega^4 \)

• circular orbits, with constant speed. Acceleration is \(\mathbf{a}(t) = - R \omega^2 \hat{\mathbf{r}}(t)\), and has constan magnitude \(|\mathbf{a}(t)| = \omega^2 R\)

Systems with \(\ V(\mathbf{r}) = \dfrac{1}{2} m \omega^2 |\mathbf{r}|^2\)

\[\langle | \mathbf{a} |^2 \rangle = \dfrac{\omega^2}{m} E \ ,\]

and thus

(20.4)\[\langle \Phi \rangle = \dfrac{q^2 \omega^2}{6 \pi \varepsilon m c^3} E =: \gamma E \ .\]

…todo Proof using virial theorem…

Vector identities

\[\varepsilon_{abc} A_b \varepsilon_{cde} B_d C_e = ( \delta_{ad} \delta_{be} - \delta_{ae} \delta_{bd} ) A_b B_d C_e = A_b B_a C_b - A_b B_b C_a = \mathbf{A} \cdot \mathbf{C} \, \mathbf{B} - \mathbf{A} \cdot \mathbf{B} \, \mathbf{C}\]