# Magnetic dipole moment derivation

1. Feb 1, 2016

### 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.

2. Feb 1, 2016

### 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.

3. Feb 1, 2016

### kidsasd987

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

4. Feb 2, 2016

### 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.

5. Feb 3, 2016

### kidsasd987

Thanks. I got it!