Magnetic dipole moment derivation

[kidsasd987]

(2)

Hi. I am having a problem with understanding how to approximate 1/R in the forms of equations written above.

I took this equations from a blog, and it tells that I can use talyor polynomial. but I don't get there somehow.

 

[blue_leaf77]

$$
\sqrt{(x-x')^2+(y-y')^2+z^2} \approx \sqrt{x^2+y^2+z^2-2xx'-2yy'}
$$
where the terms square in $x'$ and $y'$ have been omitted. Then make substitution $r^2 = x^2+y^2+z^2$,
$$
r\sqrt{1-2\frac{x}{r^2}x'-2\frac{y}{r^2}y'}
$$
and apply Taylor expansion truncating the third term.

 

[kidsasd987]

$$
\sqrt{(x-x')^2+(y-y')^2+z^2} \approx \sqrt{x^2+y^2+z^2-2xx'-2yy'}
$$
where the terms square in $x'$ and $y'$ have been omitted. Then make substitution $r^2 = x^2+y^2+z^2$,
$$
r\sqrt{1-2\frac{x}{r^2}x'-2\frac{y}{r^2}y'}
$$
and apply Taylor expansion truncating the third term.

Hello, I am really sorry but could you provide me the taylor expansion of it?

 

[blue_leaf77]

To simplify the appearance, you can make the substitution $-2\frac{x}{r^2}x'-2\frac{y}{r^2}y' = u$ so that
$$
r\sqrt{1-2\frac{x}{r^2}x'-2\frac{y}{r^2}y'} = r(1+u)^p
$$
where $p=1/2$. Now look up online or in your textbook examples of Taylor series, especially the series which corresponds to a form $(1+u)^p$ with $|u|<1$ as is the case here.

 

[kidsasd987]

To simplify the appearance, you can make the substitution $-2\frac{x}{r^2}x'-2\frac{y}{r^2}y' = u$ so that
$$
r\sqrt{1-2\frac{x}{r^2}x'-2\frac{y}{r^2}y'} = r(1+u)^p
$$
where $p=1/2$. Now look up online or in your textbook examples of Taylor series, especially the series which corresponds to a form $(1+u)^p$ with $|u|<1$ as is the case here.

Thanks. I got it!