Please explain Taylor expansion in radiation.

[yungman]

For retarded scalar potential of arbigtrary source around origin:

$$V(\vec r, t) = \frac 1 {4\pi\epsilon_0}\int \frac { \rho(\vec r\;',t-\frac {\eta}{c}) }{\eta} d\;\tau' \;\hbox { where }\;\eta =\sqrt{r^2 + r'^2 - 2 \vec r \cdot \vec r\;' }$$

Where $\;\vec r \;$ point to the field point where V is measured. And $\;\vec r\;' \;$ points to the source point.



For $\;\vec r\;' \;$ << $\;\vec r \;$:

$$\eta \approx \; r- \hat r \cdot \vec r\;' \Rightarrow \rho(\vec r\;',\;t-\frac {\eta}{c}) \approx \rho (\vec r\;',\;t-\frac {r}{c} + \frac {\vec r \cdot \vec r\;'}{c})$$

This next step is where I don't understand how the book do the Taylor expansion. I am going to type the exact word from the book:


Expanding $\rho \;$ as a Taylor series in t about the retarded time at the origin,

$$t_0=t-\frac r c$$

We have

$$\rho(\vec r\;',\;t-\frac {\eta}{c}) \approx \rho (\vec r\;',\; t_0) + \dot{\rho} (\vec r\;',\; t_0)\left ( \frac {\vec r \cdot \vec r\;'}{c}\right ) + \frac 1 {2!} \ddot{\rho} \left ( \frac {\vec r \cdot \vec r\;'}{c}\right )^2 + \frac 1 {3!} \rho^{...}_{ } \left ( \frac {\vec r \cdot \vec r\;'}{c}\right )^3 ...$$

Why are they use $\left ( \frac {\vec r \cdot \vec r\;'}{c}\right )\;$ as x for the expansion. I just don't follow this. Please help.

thanks

Alan

 

[Born2bwire]

What step don't you understand?

You have some function f(x) that you wish to do the Taylor's expansion about the point x_0, hence you wish to approximate f(x+x_0) for small values of x << 1.

So, if we choose r' << r, then:

$$x+x_0 \approx t-\frac{r}{c}+\frac{\mathbf{r}'\cdot\mathbf{r}}{c}$$
We choose x_0 to be t-r/c and x to be r' dot r/c and do our Taylor's expansion about our x.

 

[yungman]

What step don't you understand?

You have some function f(x) that you wish to do the Taylor's expansion about the point x_0, hence you wish to approximate f(x+x_0) for small values of x << 1.

So, if we choose r' << r, then:

$$x+x_0 \approx t-\frac{r}{c}+\frac{\mathbf{r}'\cdot\mathbf{r}}{c}$$
We choose x_0 to be t-r/c and x to be r' dot r/c and do our Taylor's expansion about our x.

I don't understand why it is in power of $\frac {\vec r-\vec r\;'}{c}$. How does this fit into x-x0?

 

[Born2bwire]

The Taylor's expansion of f(x) about zero is the Maclaurin Series.

$$f(x) \approx f(0) + f'(0)x + 0.5f''(0)x^2 + \dots$$

If we want to expand about some point x_0 then

$$f(x_0+x) \approx f(x_0) + f'(x_0)x + \dots$$

Or something like that. So it seems that the author desires to do a Taylor's expansion of the charge distribution about the retarded time at the origin, t_0. That is, he is expanding the charge about the charge picture as time progressed at the origin where the charges are clustered. So you can think of \rho(r', t_0) as the charge distribution that occurred if the charges about the origin instantly communicated their information to the origin.

So if t-r/c is our x_0, then our x is r' \cdot r/c. So we expand out in powers of r' \cdot r/c.

 

[yungman]

The Taylor's expansion of f(x) about zero is the Maclaurin Series.

$$f(x) \approx f(0) + f'(0)x + 0.5f''(0)x^2 + \dots$$

If we want to expand about some point x_0 then

$$f(x_0+x) \approx f(x_0) + f'(x_0)x + \dots$$

Or something like that. So it seems that the author desires to do a Taylor's expansion of the charge distribution about the retarded time at the origin, t_0. That is, he is expanding the charge about the charge picture as time progressed at the origin where the charges are clustered. So you can think of \rho(r', t_0) as the charge distribution that occurred if the charges about the origin instantly communicated their information to the origin.

So if t-r/c is our x_0, then our x is r' \cdot r/c. So we expand out in powers of r' \cdot r/c.

Again thanks for your time. I thought Taylor expansion is:

$$f(x)\approx f(x_0)+f'(a)(x-a) +\frac { f''(x_0)(x-x_0)^2 }{2!} +...\frac { fn(x_0)(x-x_0)^n }{n!}...$$

In this case, it should be x=t and $x_0=\frac r c -\frac {\vec r-\vec r\;'}{c}$ so the Taylor expansion should be:

$$f(t)\approx f(\frac r c -\frac {\vec r-\vec r\;' } c )+f' (\frac r c -\frac {\vec r-\vec r\;'} c)( t-\frac r c -\frac {\vec r-\vec r\;'} c) + \frac { f''(\frac r c -\frac {\vec r-\vec r\;'} c)( t-\frac r c + \frac {\vec r-\vec r\;'} c )^2 }{2!} +...\frac { f^n(\frac r c -\frac {\vec r-\vec r\;'} c)( t-\frac r c +\frac {\vec r-\vec r\;'} c )^n }{n!}...$$But the book use $t_0=t-\frac r c$ this mean the independent variable is $\frac {\vec r-\vec r\;'} c$

$$\Rightarrow f'(t)\;=\; \frac {d (f(\frac {\vec r-\vec r\;'} c)}{d(\frac {\vec r-\vec r\;'} c)} \;\hbox { also}\; \frac {\vec r-\vec r\;'} c \;\hbox { is a constant at the given source and field point location!}$$

I understand how the book come up with this, it is just:

$$x_0=t-\frac r c \;\;\hbox { and };\; x=\frac {\vec r-\vec r\;' } c$$

The question is why!

 

[yungman]

Anyone please?