Gradient equation with retarded time

[ucclarke]

For a radiation problem,
i am desperate about the expansion of the following equation:
\nabla ( \hat{r} /r^2 \cdot \vec{p}(t_o))

where t_o is the retarded time at the center
t_o=t-r/c

and \vec{p}(t_o) is the electric dipole moment at t_o

actually, it expands to 4 main parts and i am unable to figure out the last one, namely:
\hat{r} /r^2 \times (\nabla \times \vec{p}(t_o))

it would be magnifique if anyone can figure out the expansion of that term

 

[dextercioby]

\nabla\left[\frac{\vec{r}}{r^{3}}\cdot\vec{p}\left(t_{o}\right)\right]

=\left(\frac{\vec{r}}{r^{3}}\cdot\nabla\right)\vec{p}\left(t_{o}\right)+\left[\vec{p}\left(t_{o}\right)\cdot\nabla\right]\frac{\vec{r}}{r^{3}}\ +\frac{\vec{r}}{r^{3}}\times\left[\nabla\times\vec{p}\left(t_{o}\right)\right]

+\vec{p}\left(t_{o}\right)\times\left(\nabla\times\frac{\vec{r}}{r^{3}}\right)

All 4 of them,okay?

Daniel.

 

[ucclarke]

ok, it's actually \nabla\left[\frac{\vec{r}}{r^{2}}\cdot\vec{p}\left(t_{o}\right )\right], but really fine, thanks a lot.

so may i ask which one of these terms give out (\hat{r}\cdot\dot{p}(t_o))\hat{r} and how?

 

[dextercioby]

Nope,u said

\frac{\hat{r}}{r^{2}}\equiv\frac{\vec{r}}{r^{3}}

The first.

Daniel.

 

[ucclarke]

yes, :)
you caught me, I've misread yours

but what about (\hat{r}\cdot\dot{p}(t_o))\hat{r} ? do you happen to figure out which of four gives it and how?

 

[dextercioby]

I've told you,use the chain rule for the first of the 4.

Daniel.

 

[ucclarke]

i find this out of the first one
(p(t_o)-3(\hat{r}\cdot p(t_o))\hat{r})/r^3

approved?

 

[ucclarke]

no, okay this's from the second one i fnd

 

[ucclarke]

i can't expand =\left(\frac{\vec{r}}{r^{3}}\cdot\nabla\right)\vec {p}\left(t_{o}\right) as it should be. mine doesn't satisfy the given answer

can anyone help?

i am going NUTS here

 

[dextercioby]

I=\left(\frac{\vec{r}}{r^{3}}\cdot\nabla\right)\vec{p}\left(t_{o}\right) (1)

Use cartesian tensors

I=\frac{x_{i}}{r^{3}}\partial_{i}p_{j}\left(t_{o}\right)\vec{e}_{j} (2)

\partial_{i}p_{j}=-\frac{1}{c}\frac{dp_{j}}{dt_{o}}\frac{\partial r}{\partial x_{j}}=-\frac{\dot{p}_{j}\left(t_{o}\right)}{c} \frac{x_{i}}{r} (3)

I=-\frac{\dot{p}_{j}\left(t_{o}\right)}{c} \frac{x_{i}}{r^{3}}\frac{x_{i}}{r} \vec{e}_{j} =-\frac{\vec{p}\left(t_{o}\right)}{c r^{2}}

Daniel.

 

[ucclarke]

thanx a lot, that's exactly what i found too

then my mistake is certainly in the expansion of \left[\vec{p}\left(t_{o}\right)\cdot\nabla\right]\frac{\vec{r}}{r^{3}} term

 

[dextercioby]

That cannot give a derivative of the moment of dipole vector.It's the one with a curl acting on the vector.

The one u mentioned is quite easy to compute,just use the Leibniz rule carefully.So what is

\partial_{i}\left(\frac{x_{j}}{r^{3}}\right)

equal to...?


Daniel.