(moving-charge-radiation)=
# Radiation of moving charges - Larmor's formula

(moving-charge-radiation:lienard-wiechert)=
## Green's function solution of wave equations: Liénard-Wiechert potential

**Wave equations in electromagmetism.** See [Classical Electromagnetism: Wave Equations in Electromagnetism](https://basics2022.github.io/bbooks-physics-electromagnetism/ch/waves-equation.html). 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

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

```{dropdown} 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](https://basics2022.github.io/bbooks-math-miscellanea/ch/pde/bem-poisson-helmholtz-waves.html#wave-equation) (with assumptions of causality, and no contribution from i.c. and b.c.),

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

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

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

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

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

Thus, the electromagnetic potentials generated by a point charge are

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


````{dropdown} Derivatives of the retarded time
:open:


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

```{dropdown} Details

Time derivative

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

and thus

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

Space derivatives

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

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,

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

```

````

````{dropdown} Derivatives of $\ |\mathbf{r} - \mathbf{r}_s(t_{ret})|^{-1}$
:open:

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

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


```{dropdown} Details

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

Now,

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

and thus

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

and

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


```

````

````{dropdown} From potentials to the electromagnetic field
:open:

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

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

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


```{dropdown} Details

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


$$\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}$$

```

````

(moving-charge-radiation:larmor)=
## 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

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

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

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

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

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

**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 $|\mathbf{a}(t)| = \omega^2 R$

```{dropdown} Systems with $\ V(\mathbf{r}) = \dfrac{1}{2} m \omega^2 |\mathbf{r}|^2$
:open:

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

and thus

$$\langle \Phi \rangle = \dfrac{q^2 \omega^2}{6 \pi \varepsilon m c^3} E =: \gamma E \ .$$ (eq:larmor:quadratic:power-energy)

...**todo** Proof using virial theorem...


```


```{dropdown} Vector identities

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


```