(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Griffith's problem 10.8

Show that retarded potentials satisfy the Lorentz condition. Hint proceed as follows

a) Show that

[tex] \nabla\cdot\left(\frac{J}{R}\right)=\frac{1}{R}\left(\nabla\cdot\vec{J}\right)+\frac{1}{R}\left(\nabla '\cdot\vec{J}\right)-\nabla '\cdot\left(\frac{J}{R}\right) [/tex]

b) Show that [tex] \nabla\cdot\vec{J}=-\frac{1}{c}\frac{\partial\vec{J}}{\partial t_{r}}\cdot(\nabla R)[/tex]

c) Note that [tex] \vec{J}=\vec{J}\left(\vec{r'},t_{r}\right)[/tex]

[tex] \nabla '\cdot J=-\frac{\partial \rho}{\partial t}-\frac{1}{c}\frac{\partial J}{\partial t_{r}}\cdot (\nabla ' R) [/tex]

where [tex] \vec{R}=\vec{r}-\vec{r'} [/tex]

2. The attempt at a solution

I managed to do the first and second parts but its the third part that i am unable to prove.

Ok so i know that [tex] \vec{J}=\vec{J}\left(\vec{r'},t_{r}\right)=\vec{J}\left(\vec{r'},t-\frac{\vec{r}-\vec{r'}}{c}\right)[/tex]

To make it simpler for me to understand lets do it for one dimension.

[tex] \frac{\partial J_{x}}{\partial x'} = \frac{\partial J_{x}}{\partial t_{r}}\frac{\partial t_{r}}{\partial x'}[/tex]

But [tex] \frac{\partial t_{r}}{\partial x'}=\frac{1}{c}\frac{\partial R}{\partial x'} [/tex]

so [tex] \frac{\partial J_{x}}{\partial x'} = \frac{1}{c}\frac{\partial J_{x}}{\partial t_{r}}\frac{\partial R}{\partial x'}[/tex]

THat explains the second term which i need to get in the proof. But how do i get the first term?

Also is it supposed to be [tex] \nabla\cdot J=-\frac{\partial \rho}{\partial t}[/tex]

or is it supposed to be [tex] \nabla'\cdot J=-\frac{\partial \rho}{\partial t}[/tex]

Thanks for your help!

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# Retarded potentials

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