Griffiths Electrodynamics gradient of charge distribution

  1. I do not understand the following from Griffiths’ Electrodynamics – page 424 Equation 10.21.

    [tex]
    \nabla p = \dot{p} \nabla {tr} = …
    [/tex]

    I’m not sure how much of this applies (I think my question is on the math) but p is the charge distribution, tr is the retarded time.

    Is this an application of the chain rule?

    With the gradient being a derivative with respect to spatial location (x,y,z), why is the time derivative showing up in the gradient? I initially want to say if something is dependent upon t but not on x, then its derivative with respect to x is zero.

    The result looks like the chain rule applied – I don’t see why the time dependent portion shows up.

    Can you help clear this up for me?

    Thanks
    Sparky_
     
  2. jcsd
  3. continuity equation?
     
  4. I do not see it yet.

    I see later on the same page

    [tex]
    \nabla \dot{p} = \ddot{p} \nabla {tr} = …
    [/tex]

    Can you explain further?

    Somehow the gradient is giving an additional time derivative.

    Thanks
    Sparky_
     
  5. [itex]\rho[/itex] has arguments like this:
    [itex]\rho (\vec{r}', t_r(\vec{r}, \vec{r}', t)) [/itex]

    The gradient is being applied w.r.t to the coordinates of [itex]\vec{r}[/itex] ( not [itex]\vec{r}'[/itex] which gets integrated away). The coordinates that we would be taking the derivative with respect to in order to obtain the gradient are only found in the parameters of [itex]t_r[/itex]. So this result is from the chain rule. Here is one component of the gradient, for example.
    [itex](\nabla \rho)_x = \frac{\partial \rho (\vec{r}', t_r(x, y, z, \vec{r}', t))}{\partial x} = \dot{\rho}\frac{\partial t_r}{\partial x} [/itex]
     
  6. Oh I see, did he specify that the dot derivative is with respective to retarded time?
     
  7. WannabeNewton

    WannabeNewton 5,767
    Science Advisor

    It won't matter. ##\partial _{t_{r}} = \frac{\partial t_{r}}{\partial t}\partial _{t} = \frac{\partial }{\partial t}(t - \frac{\mathfrak{r}}{c})\partial_{t} = \partial_{t}##.

    Anyways, as noted above ##\rho = \rho(r',t_{r}) ## and ##r'## is no longer a variable after integration but ##t_{r} = t_{r}(t,x,y,z,r')## so ##\nabla \rho = \partial _{t_{r}}\rho \nabla t_{r} = \partial_{t}\rho \nabla t_{r}##. Not sure what that has to do with the conservation of 4-current (continuity equation) ##\partial_{a}j^{a} = 0##.
     
    Last edited: Apr 8, 2013
  8. Thank you!!

    I went back in this section of the text and reread. I see that p (charge density) is specified p(r’, tr). That is actually the point of this topic (the nonstatic case).

    You confirmed that this is an application of the chain rule and p is a function of position and tr.

    Thank you for the help!
    Sparky_
     
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