# How to compute the divergence of retarded scalar potential

Tags:
1. Dec 2, 2014

### genxium

I'm learning time-dependent Maxwell's Equations and having difficulty understanding the following derivative:

Given $f(\textbf{r}, \textbf{r}', t) = \frac{[\rho(\textbf{r}, t)]}{|\textbf{r} - \textbf{r}'|}$

where

$\textbf{r} = x \cdot \textbf{i} + y \cdot \textbf{j} + z \cdot \textbf{k}$, in Cartesian Coordinates

$\textbf{r}' = x' \cdot \textbf{i} + y' \cdot \textbf{j} + z' \cdot \textbf{k}$

$[\rho(\textbf{r}, t)] \stackrel{\Delta}{=} \rho(\textbf{r}, t_r)$ with $t_r = t-\frac{|\textbf{r} - \textbf{r}'|}{c}$ and $c$ is a non-zero constant(speed of EM wave indeed)

The tutorial I'm reading "infers" that

$\nabla f(\textbf{r}, \textbf{r}', t) = \nabla \frac{1}{|\textbf{r} - \textbf{r}'|} \cdot [\rho(\textbf{r}, t)] + \frac{1}{|\textbf{r} - \textbf{r}'|} \cdot [\frac{\partial \rho(\textbf{r}, t)}{\partial t}] \cdot \nabla t_r$ -- (a)

where $\nabla \stackrel{\Delta}{=} \frac{\partial}{\partial x} \cdot \textbf{i} + \frac{\partial}{\partial y} \cdot \textbf{j} + \frac{\partial}{\partial z} \cdot \textbf{k}$

I'm confused by the latter part of the equation above. By applying the identity $\nabla (g_1 \cdot g_2) = g_2 \cdot \nabla g_1 + g_1 \cdot \nabla g_2$ to $f(\textbf{r}, \textbf{r}', t)$ I get

$\nabla f(\textbf{r}, \textbf{r}', t) = \nabla \frac{1}{|\textbf{r} - \textbf{r}'|} \cdot [\rho(\textbf{r}, t)] + \frac{1}{|\textbf{r} - \textbf{r}'|} \cdot \nabla [\rho(\textbf{r}, t)]$ -- (b)

then if (a) is correct I'll have

$\nabla [\rho(\textbf{r}, t)] = [\frac{\partial \rho(\textbf{r}, t)}{\partial t}] \cdot \nabla t_r$ -- (c)

However, though trivial, $\textbf{r}(x, y, z) = x \cdot \textbf{i} + y \cdot \textbf{j} + z \cdot \textbf{k}$ is still a function of $x, y \, \text{and} \, z$, so in my calculation

$\nabla [\rho(\textbf{r}, t)] = \frac{\partial \rho(\textbf{r}, t_r)}{\partial \textbf{r}} \cdot \nabla \textbf{r} + \frac{\partial \rho(\textbf{r}, t_r)}{\partial t_r} \cdot \nabla t_r = \frac{\partial \rho(\textbf{r}, t_r)}{\partial \textbf{r}} \cdot \nabla \textbf{r} + [\frac{\partial \rho(\textbf{r}, t)}{\partial t}] \cdot \nabla t_r$ -- (d)

Obviously (d) contradicts (c) but unfortunately I can't figure out where I went wrong in my calculation.

Can someone help to point out my mistakes or guide me to some references? Any help is appreciated :)

2. Dec 6, 2014

### genxium

I have solved my problem. The mistake I made was that I took $[\rho] = [\rho(\textbf{r}, t)]$ while it should be $[\rho] = [\rho(\textbf{r}', t)]$.