# 亥姆霍茲線圈

## 數學描述

${\displaystyle B={\left({\frac {4}{5}}\right)}^{3/2}{\frac {\mu _{0}nI}{R}}}$

### 推導

${\displaystyle B={\frac {\mu _{0}IR^{2}}{2(R^{2}+z^{2})^{3/2}}}}$

${\displaystyle B={\frac {\mu _{0}nIR^{2}}{2(R^{2}+z^{2})^{3/2}}}}$

${\displaystyle B={\frac {2\mu _{0}nIR^{2}}{2(R^{2}+(R/2)^{2})^{3/2}}}={\left({\frac {4}{5}}\right)}^{3/2}{\frac {\mu _{0}nI}{R}}}$

### 進階推導

${\displaystyle \mathbf {B} ={\frac {\mu _{0}IR^{2}}{2}}\left\{\left[R^{2}+(z-h/2)^{2}\right]^{-3/2}+\left[R^{2}+(z+h/2)^{2}\right]^{-3/2}\right\}{\hat {\mathbf {z} }}}$

${\displaystyle \mathbf {B} ={\frac {\mu _{0}IR^{2}}{d^{3}}}\left[1+{\frac {3(h^{2}-R^{2})z^{2}}{2d^{4}}}+{\frac {15(h^{4}-6h^{2}R^{2}+2R^{4})z^{4}}{16d^{8}}}+\dots \right]{\hat {\mathbf {z} }}}$

${\displaystyle \mathbf {B} ={\left({\frac {4}{5}}\right)}^{3/2}{\frac {\mu _{0}I}{R}}\left[1-{\frac {144}{125}}\ \left({\frac {z}{R}}\right)^{4}+\dots \right]{\hat {\mathbf {z} }}}$

${\displaystyle {\frac {\Delta B_{z}}{B_{z}}}\approx -{\frac {144}{125}}\ \left({\frac {z}{R}}\right)^{4}}$

${\displaystyle z=\pm R/2}$，線圈平面與z-軸相交處，磁場數值的差別為

${\displaystyle {\frac {\Delta B_{z}}{B_{z}}}\approx -{\frac {144}{125}}\ \left({\frac {1}{2}}\right)^{4}\approx -7\%}$

## 參考文獻

1. ^ 亥姆霍茲線圈內部磁場分析 互联网档案馆存檔，存档日期2012-03-24.
2. ^ 地磁場磁力儀：亥姆霍茲線圈" 互联网档案馆存檔，存档日期2008-11-01. by Richard Wotiz 2004
3. ^ 喬治亞州州立大學Georgia State University）線上物理網頁：Field on Axis of Current Loop.
4. ^ Jackson, John David, Classical Electrodynamic 3rd., USA: John Wiley & Sons, Inc.: pp. 226–227, 1999, ISBN 978-0-471-30932-1