🧠 Remember: If a point is travelling at constant speed the field lines of the electric field will meet at the point where it will be when we measure, not at the retarded time

Intuitive approach

💼 Case: lets consider a burst with the following setup

1. initially : Point charge $q$ travelling with a constant speed $v'\ll c$ in frame $S'$

2. At time $t'_\text{ret}$: $q$ rapidly decelerates to be at rest over an interval $\Delta t'$

   $$ \dot \beta'=\frac{v'}{c\Delta t} $$

3. At time $t'>t'_\text{ret}:$

   • Region within $R'=c(t'-t'\text{ret})$ sees field of stationary $q$ at $x'=v't'\text{ret}$
   • Region outside $R'+c\Delta t'$ still sees field of moving charge at $x'=v't'$

Initial state is as we expect and final state creates a discontinuity which is not allowed in gausses law thus we introduce red lines 🗒️ Note: this is a way of “cheating” relativity as only general relativity consider acceleration

💼 Case: lets look at one ray specifically

1. Consider the radial feel going to $A$ (usual case)

   $$ E'_R=\frac{1}{4\pi\epsilon_0}\frac{q}{(R')^2} $$

2. Within a boundary the tangential component $E'_T$ has to stay continuous, we can write it as

   $$ \begin{aligned} \frac{E_T'}{E'R}&=\frac{v'(t'-t'\text{ret}\sin\theta}{c\Delta t'} \\ \Rightarrow \qquad E'T&=\frac{q}{4\pi \epsilon_0} \frac{c(t'-t'\text{ret})}{(R')^2 c} \frac{v'}{c\Delta t'}\sin\theta'=\frac{q}{4\pi \epsilon_0 c} \frac{\dot \beta '}{R'}\sin \theta' \end{aligned} $$

💎 Conclusion:

• Acceleration has produced a tangential component of $\vec E$
• Associated with this will be a tangential component to the $\vec B$ field
• Both $\vec E$ and $\vec B$ fall off with $\frac 1 {R'}$ compared to constant velocity $\propto \frac 1 {(R')^2}$
• $\vec E \propto \sin \theta'$ is visible on the diagram, perpendicular to the acceleration the effect is larger

Vector calculus approach

In the previous section we showed that for acceleration we need to consider the electric field at the retarded time $t_\text{ret}=t-R/c$

⚽ Goal: recover our previous result in a formal way using vector calculus