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I have been going through Griffith's 'Introduction To Electrodynamics" 3rd Edition chapter 10 on potentials and fields and I am a little confused about the derivation of the Liénard Wiechert potentials, equations 10.39 and 10.40:

$V(\textbf{r},t)=\frac{1}{4\pi\epsilon_0}\frac{qc}{(rc-\textbf{r}\cdot\textbf{v})}$

$\textbf{A}(\textbf{r},t)=\frac{\textbf{v}}{c^2}V(\textbf{r},t)$

If my understanding proves correct, these equations are derived as such:

  1. Assume static, time-independent fields which leads to our familiar Poisson equation for the two potentials, $V(\textbf{r},t)$ and $\textbf{A}(\textbf{r},t)$, $\textbf{without}$ any gauge assumptions on the potentials; no choice of gauge was made.

  2. Extract the integral, static form of $V(\textbf{r},t)$ and $\textbf{A}(\textbf{r},t)$ (equation 10.17).

  3. Argue that charge and current densities are to be evaluated at the retarded time due to the finite speed of light and show, in doing so, the Lorenz gauge (10.12) with be satisfied along with its subsequent d'Alembertian-form inhomogeneous wave equations (equations 10.16), despite making no decision in gauge.

  4. Argue the geometrical doppler like effect for the charge and current densities and evaluate the integral forms of $V(\textbf{r},t)$ and $\textbf{A}(\textbf{r},t)$ (equation 10.17) to reach the final results of equations 10.39 and 10.40.

My confusion is that the Lorenz gauge did not seem to play a role in the above arguments, except maybe for point 3. But, the fact that point 3 was satisfied seemed like sheer coincidence and more so like the result of the solid argument that the information needs time to travel and be received; hence evaluation of the potentials at the retarded time.

So, are we forced to remain in the Lorenz gauge if we wish to use equations 10.39 and 10.40 for $V(\textbf{r},t)$ and $\textbf{A}(\textbf{r},t)$?

Was the fact that we must evaluate the potentials at the retarded time somehow automatically convoluted/correlated/ingrained with the Lorenz gauge?

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  • $\begingroup$ (a) Equations (10.39) and (10.40) come from equations (10.19). (b) Equations (10.19) are "The natural generalization of equations (10.17) for nonstatic sources". (c) Equations (10.17) are in turn the static case of equations (10.16). (d) Equations (10.16) come from equations (10.4) and (10.5) under the condition of the Lorentz gauge, equation (10.12) $$ \boldsymbol{\nabla}\boldsymbol{\cdot}\mathbf{A}=-\mu_{o}\epsilon_{o}\dfrac{\partial V}{\partial t} \tag{10.12} $$ So equations (10.39) and (10.40) are valid under the condition of the Lorentz gauge. $\endgroup$ – Frobenius Jan 8 '17 at 10:34
  • $\begingroup$ That's why the author comments two paragraphs under equations (10.19) about them : "To prove them, I must show that they satisfy the inhomogeneous wave equation (10.16) and meet the Lorentz condition (10.12)." $\endgroup$ – Frobenius Jan 8 '17 at 10:34
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From the OP's reference textbook(1)


(a) Equations (10.39) and (10.40) $$ \bbox[yellow] { V\left(\mathbf{r},t\right)=\dfrac{1}{4\pi \epsilon_{0}}\dfrac{qc}{\left(\mathfrak{r}c-{ \widehat{\boldsymbol{\mathfrak{r}}} }\boldsymbol{\cdot}\mathbf{v}\right)} } \tag{10.39} $$ $$ \bbox[yellow] { \mathbf{A}\left(\mathbf{r},t\right)=\dfrac{\mu_{0}}{4\pi}\dfrac{qc\mathbf{v}}{\left(\mathfrak{r}c-{ \widehat{\boldsymbol{\mathfrak{r}}} }\boldsymbol{\cdot}\mathbf{v}\right)}=\dfrac{\mathbf{v}}{c^{2}}V\left(\mathbf{r},t\right) } \tag{10.40} $$ come from equations (10.19).


(b) Equations (10.19) $$ \boxed { V\left(\mathbf{r},t\right)=\dfrac{1}{4\pi\epsilon_{0}}\int\dfrac{\rho\left(\mathbf{r'},t_{r}\right)}{\mathfrak{r}}\mathrm{d}\tau'\,, \quad \mathbf{A}\left(\mathbf{r},t\right)=\dfrac{\mu_{0}}{4\pi}\int\dfrac{\mathbf{J}\left(\mathbf{r'},t_{r}\right)}{\mathfrak{r}}\mathrm{d}\tau' } \tag{10.19} $$ are "The natural generalization of equations (10.17) for nonstatic sources".(1)


(c) Equations (10.17) $$ V\left(\mathbf{r}\right)=\dfrac{1}{4\pi\epsilon_{0}}\int\dfrac{\rho\left(\mathbf{r'}\right)}{\mathfrak{r}}\mathrm{d}\tau'\,, \quad \mathbf{A}\left(\mathbf{r}\right)=\dfrac{\mu_{0}}{4\pi}\int\dfrac{\mathbf{J}\left(\mathbf{r'}\right)}{\mathfrak{r}}\mathrm{d}\tau' \tag{10.17} $$ are in turn the static case of equations (10.16).


(d) Equations (10.16) \begin{align} (i)\quad \Box^{2}V&=\boldsymbol{-}\dfrac{1}{\epsilon_{0}}\rho\, ,\\ \tag{10.16}\\ (ii)\quad \Box^{2}\mathbf{A}&=\boldsymbol{-}\mu_{0}\mathbf{J}\, . \end{align} come from equations (10.4) and (10.5) $$ \boldsymbol{\nabla}^{2}V+\dfrac{\partial}{\partial t}\left(\boldsymbol{\nabla}\boldsymbol{\cdot} \boldsymbol{A} \right) =\boldsymbol{-}\dfrac{1}{\epsilon_{0}}\rho \tag{10.4} $$ $$ \biggl(\boldsymbol{\nabla}^{2} \mathbf{A}-\mu_{0}\epsilon_{0}\dfrac{\partial^{2} \mathbf{A}}{\partial t^{2}}\biggr)-\boldsymbol{\nabla}\biggl(\boldsymbol{\nabla}\boldsymbol{\cdot} \mathbf{A}+\mu_{0}\epsilon_{0}\dfrac{\partial V}{\partial t}\biggr) =\boldsymbol{-}\mu_{0} \mathbf{J}\, . \tag{10.5} $$ under the condition of the Loren(t)z gauge, equation (10.12) $$ \boldsymbol{\nabla}\boldsymbol{\cdot}\mathbf{A}=-\mu_{0}\epsilon_{0}\dfrac{\partial V}{\partial t} \tag{10.12} $$


As we see from this backwards procedure the equations (10.39) and (10.40) are created with initial condition the Loren(t)z gauge. On the other hand the author comments two paragraphs under equations (10.19) about these last equations(1): "To prove them, I must show that they satisfy the inhomogeneous wave equation (10.16) and meet the Lorentz condition (10.12)."

(1) David J. Griffiths "Introduction to Electrodynamics " 3rd Edition 1999.

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  • $\begingroup$ Thanks for the answer! But I am still a little fuzzy. To assume/argue equations 10.19 (the potentials evaluated at the retarded time), must we be in the Lorenz gauge? In any other gauge is this assumption incorrect? $\endgroup$ – Donkey Kong Jan 8 '17 at 20:43
  • $\begingroup$ @Donkey Kong : I think that (10.19) are not valid generally for other gauges. For other gauge we'll have different "(10.19)-type" equations and different solutions for the potentials $\:V,\mathbf{A}\:$, but the field vectors $\:\mathbf{E},\mathbf{B}\:$ would be the same. An example is with the Coulomb gauge $\:\boldsymbol{\nabla}\boldsymbol{\cdot} \mathbf{A}=0\:$. With this gauge the first of equations (10.19) for the scalar potential is the same, see equation (10.10) in Griffiths, but for the vector potential the equation is very complicated and difficult to solve, see equation (10.11). $\endgroup$ – Frobenius Jan 8 '17 at 21:20
  • $\begingroup$ @Donkey Kong : I suggest to read the paragraphs "10.1.2 Gauge Transformations" and "10.1.3 Coulomb Gauge and Lorentz* Gauge" in Griffiths. $\endgroup$ – Frobenius Jan 8 '17 at 21:21

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