💼 Case: Potential for a point charge $q$ moving with a constant velocity $\vec \beta =\beta \hat x'$
The observer is at $P(\vec r,t)$
Light cone is defined as follows:
$$ \Delta x'=c\Delta t =\Delta x^0 $$
ie all points that can be connected with $P(\vec r,t)$ by a signal travelling at the $c$
🗒️ Note: In the past, world lines can cross the past light of $P(\vec r,t)$ at one and only one point in space time
where $\vec A'=0$ because it is at rest
💼 Case: charge $q$ at rest at origin of frame $S'$
⚽ Goal: evaluate the potential $A^\mu$ in frame $S$ in which the charge is moving $\vec \beta=\beta \hat x'$
🗒️ Note: We need to transform $A'^\mu$ but also $x'^1=\gamma [x^1-\beta x^0]$
In rest frame $S'$ we have
$$ A'^0 =\frac{V'}{c}=\frac{q}{4\pi\epsilon_0 c}\frac{1}{R'} \quad \vec A'=0 $$
where
$$ (R')^2=(x'^1)^2+(x'^2)^2+(x'^3)^2=(\gamma[x^1-\beta x^0])^2+(x^2)^2+(x^3)^3 $$
We apply the inverse Lorentz transformations as we are going from primed to unprime
$$ \begin{aligned} \qquad \qquad \frac{V}{c}=A^0&=\gamma (A'^0+\beta \overbrace{A'^1}^{=0})=\gamma A'^0 \qquad \qquad\\ A^0&=\frac{q}{4\pi \epsilon_0 c} \gamma \frac{1}{[(\gamma [x^1-\beta x^0])^2+(x^2)^2+(x^3)^2]^\frac 12} \\ A^1&=\gamma (\beta A'^0+A'^1)=\gamma\beta A'^0=\beta A^0 \\ &=\frac{q}{4\pi \epsilon_0 c} \gamma \beta \frac{1}{[(\gamma [x^1-\beta x^0])^2+(x^2)^2+(x^3)^2]^\frac 12} \\ A^2&=A^3=0
\end{aligned} $$
🗒️ Note: we can show that the $A^2$ is invariant ie $A'^\mu A'\mu=A^\mu A\mu$
To visualize the what these potentials we will consider 2 case
Along the $x^1$-axis ie $x^2=x^3=0$ then we have
$$ A^0=\frac{V}{c}=\frac{q}{4\pi \epsilon_0 c} \gamma \frac{1}{[(\gamma [x^1-\beta x^0])^2]^\frac 12} =\frac{q}{4\pi \epsilon_0 c}\frac{1}{|x^1-\beta x^0|} $$
🗒️ Note: this is the same as for a stationary charge $q$ positioned at $x^1=\beta x^0$, $x^2=x^3=0$
Along the transverse of the $x^1$-axis at position $x^1=\beta x^0$ (the “current” position of the charge)
$$ A^0=\frac{V}{c}=\gamma \left [\frac{q}{4\pi \epsilon_0 c}\frac{1}{[(x^2)^2+(x^3)^2]^\frac 12} \right ] $$
🗒️ Note: this is the $\gamma$ times the potential for a stationary charge at $x^1=\beta x^0,x^2=x^3=0$
Stationary charge $x'$ coordinates
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moving charge with $\gamma =2$, $x$ coordinates
🗒️ Note: The equipotential are centred around the “current” position $x'=\beta x^0$ of the charge not at the position of the charge at the retard time (which is where the electromagnetic signal is sent)
⚽ Goal: analyse the $\vec E$ and $\vec B$ fields produced by a moving point charge