§１４．一般角柱磁石の磁界５）まとめ

§１４．一般角柱磁石により作られる磁界

５）まとめ

　ここでは、１４-４　の1)　～ 6)　までの計算結果をまとめる。

即ち　Fig.14-7　で示されている一般角柱磁石の点 $P_{0} (X_{0}, Y_{0}, Z_{0})$ に作られる磁界を、一般角柱磁石（左手系）のローカル座標を用いて

${\overset{⃑}{H}}_{L} (X_{0}, Y_{0}, Z_{0})$ 式(14-43)

と表す。すると１４-４　の　1）上面からの磁場への寄与　から　6）底面からの寄与　までの総和として ${\overset{⃑}{H}}_{L} (X_{0}, Y_{0}, Z_{0})$ のX成分は

${({\overset{⃑}{H}}_{L} (X_{0}, Y_{0}, Z_{0}))}_{X}$

$= \frac{M \cos θ}{4 π μ_{0}} \log (\frac{(Y_{0} + b) + \sqrt{{(X_{0} - a)}^{2} + {(Y_{0} + b)}^{2} + {Z_{0}}^{2}}}{(Y_{0} + b) + \sqrt{{{(X_{0} + a)}^{2} + (Y_{0} + b)}^{2} + {Z_{0}}^{2}}} \frac{(Y_{0} - b) + \sqrt{{(X_{0} + a)}^{2} + {(Y_{0} - b)}^{2} + {Z_{0}}^{2}}}{(Y_{0} - b) + \sqrt{{(X_{0} - a)}^{2} + {(Y_{0} - b)}^{2} + {Z_{0}}^{2}}})$

$+ \frac{M sin θ sin φ}{4π μ_{0}} \log (\frac{(Z_{0} + 2 c) + \sqrt{{(Z_{0} + 2 c)}^{2} + {(X_{0} - a)}^{2} + {(Y_{0} - b)}^{2}}}{(Z_{0} + 2 c) + \sqrt{{{(Z_{0} + 2 c)}^{2} + (X_{0} + a)}^{2} + {(Y_{0} - b)}^{2}}} \frac{Z_{0} + \sqrt{{{Z_{0}}^{2} + (X_{0} + a)}^{2} + {(Y_{0} - b)}^{2}}}{Z_{0} + \sqrt{{{Z_{0}}^{2} + (X_{0} - a)}^{2} + {(Y_{0} - b)}^{2}}})$

$+ \frac{M sin θ cos φ}{4π μ_{0}} [\{\tan^{- 1} (\frac{1}{(X_{0} - a)} \frac{(Y_{0} + b) (Z_{0} + 2 c)}{\sqrt{{(Y_{0} + b)}^{2} + {(Z_{0} + 2 c)}^{2} + {(X_{0} - a)}^{2}}})$

$- \tan^{- 1} (\frac{1}{(X_{0} - a)} \frac{(Y_{0} + b) Z_{0}}{\sqrt{{(Y_{0} + b)}^{2} + {Z_{0}}^{2} + {(X_{0} - a)}^{2}}})\}$

$- \{\tan^{- 1} (\frac{1}{(X_{0} - a)} \frac{(Y_{0} - b) (Z_{0} + 2 c)}{\sqrt{{(Y_{0} - b)}^{2} + {(Z_{0} + 2 c)}^{2} + {(X_{0} - a)}^{2}}})$

$- \tan^{- 1} (\frac{1}{(X_{0} - a)} \frac{(Y_{0} - b) Z_{0}}{\sqrt{{(Y_{0} - b)}^{2} + {Z_{0}}^{2} + {(X_{0} - a)}^{2}}})\}]$

$- \frac{M sin θ cos φ}{4π μ_{0}} [\{\tan^{- 1} (\frac{1}{(X_{0} + a)} \frac{(Y_{0} + b) (Z_{0} + 2 c)}{\sqrt{{(Y_{0} + b)}^{2} + {(Z_{0} + 2 c)}^{2} + {(X_{0} + a)}^{2}}})$

$- \tan^{- 1} (\frac{1}{(X_{0} + a)} \frac{(Y_{0} + b) Z_{0}}{\sqrt{{(Y_{0} + b)}^{2} + {Z_{0}}^{2} + {(X_{0} + a)}^{2}}})\}$

$- \{\tan^{- 1} (\frac{1}{(X_{0} + a)} \frac{(Y_{0} - b) (Z_{0} + 2 c)}{\sqrt{{(Y_{0} - b)}^{2} + {(Z_{0} + 2 c)}^{2} + {(X_{0} + a)}^{2}}})$

$- \tan^{- 1} (\frac{1}{(X_{0} + a)} \frac{(Y_{0} - b) Z_{0}}{\sqrt{{(Y_{0} - b)}^{2} + {Z_{0}}^{2} + {(X_{0} + a)}^{2}}})\}]$

$- \frac{M sin θ sin φ}{4π μ_{0}} l o g (\frac{(Z_{0} + 2 c) + \sqrt{{(Z_{0} + 2 c)}^{2} + {(X_{0} - a)}^{2} + {(Y_{0} + b)}^{2}}}{(Z_{0} + 2 c) + \sqrt{{{(Z_{0} + 2 c)}^{2} + (X_{0} + a)}^{2} + {(Y_{0} + b)}^{2}}} \frac{Z_{0} + \sqrt{{{Z_{0}}^{2} + (X_{0} + a)}^{2} + {(Y_{0} + b)}^{2}}}{Z_{0} + \sqrt{{{Z_{0}}^{2} + (X_{0} - a)}^{2} + {(Y_{0} + b)}^{2}}})$

$- \frac{M \cos θ}{4 π μ_{0}} \log (\frac{(Y_{0} + b) + \sqrt{{(X_{0} - a)}^{2} + {(Y_{0} + b)}^{2} + {(Z_{0} + 2 c)}^{2}}}{(Y_{0} + b) + \sqrt{{{(X_{0} + a)}^{2} + (Y_{0} + b)}^{2} + {(Z_{0} + 2 c)}^{2}}} \frac{(Y_{0} - b) + \sqrt{{(X_{0} + a)}^{2} + {(Y_{0} - b)}^{2} + {(Z_{0} + 2 c)}^{2}}}{(Y_{0} - b) + \sqrt{{(X_{0} - a)}^{2} + {(Y_{0} - b)}^{2} + {(Z_{0} + 2 c)}^{2}}})$ 式(14-44)

と表すことができる。

同様に ${\overset{⃑}{H}}_{L} (X_{0}, Y_{0}, Z_{0})$ のY成分は

${({\overset{⃑}{H}}_{L} (X_{0}, Y_{0}, Z_{0}))}_{Y}$

$= \frac{M \cos θ}{4 π μ_{0}} \log (\frac{(X_{0} + a) + \sqrt{{(X_{0} + a)}^{2} + {(Y_{0} - b)}^{2} + {Z_{0}}^{2}}}{(X_{0} + a) + \sqrt{{{(X_{0} + a)}^{2} + (Y_{0} + b)}^{2} + {Z_{0}}^{2}}} \frac{(X_{0} - a) + \sqrt{{{(X_{0} - a)}^{2} + (Y_{0} + b)}^{2} + {Z_{0}}^{2}}}{(X_{0} - a) + \sqrt{{{(X_{0} - a)}^{2} + (Y_{0} - b)}^{2} + {Z_{0}}^{2}}})$

$+ \frac{M sin θ cos φ}{4π μ_{0}} \log (\frac{Z_{0} +2c + \sqrt{{(Y_{0} -b)}^{2} + {(Z_{0} +2c)}^{2} + {(X_{0} -a)}^{2}}}{Z_{0} +2c + \sqrt{{{(Y_{0} +b)}^{2} + (Z_{0} +2c)}^{2} + {(X_{0} -a)}^{2}}} \frac{Z_{0} + \sqrt{{(Y_{0} +b)}^{2} + {Z_{0}}^{2} + {(X_{0} -a)}^{2}}}{Z_{0} + \sqrt{{(Y_{0} -b)}^{2} + {Z_{0}}^{2} + {(X_{0} -a)}^{2}}})$

$+ \frac{M sin θ sin φ}{4π μ_{0}} [\{\tan^{- 1} (\frac{1}{(Y_{0} - b)} \frac{(Z_{0} + 2 c) (X_{0} + a)}{\sqrt{{(Z_{0} + 2 c)}^{2} + {(X_{0} + a)}^{2} + {(Y_{0} - b)}^{2}}})$

$- \tan^{- 1} (\frac{1}{(Y_{0} - b)} \frac{(Z_{0} + 2 c) (X_{0} - a)}{\sqrt{{(Z_{0} + 2 c)}^{2} + {(X_{0} - a)}^{2} + {(Y_{0} - b)}^{2}}})\}$

$- \{\tan^{- 1} (\frac{1}{(Y_{0} - b)} \frac{Z_{0} (X_{0} + a)}{\sqrt{{Z_{0}}^{2} + {(X_{0} + a)}^{2} + {(Y_{0} - b)}^{2}}})$

$- \tan^{- 1} (\frac{1}{(Y_{0} - b)} \frac{Z_{0} (X_{0} - a)}{\sqrt{{Z_{0}}^{2} + {(X_{0} - a)}^{2} + {(Y_{0} - b)}^{2}}})\}]$

$- \frac{M sin θ cos φ}{4π μ_{0}} \log (\frac{(Z_{0} + 2 c) + \sqrt{{(Y_{0} - b)}^{2} + {(Z_{0} + 2 c)}^{2} + {(X_{0} + a)}^{2}}}{(Z_{0} + 2 c) + \sqrt{{{(Y_{0} + b)}^{2} + (Z_{0} + 2 c)}^{2} + {(X_{0} + a)}^{2}}} \frac{Z_{0} + \sqrt{{(Y_{0} + b)}^{2} + {Z_{0}}^{2} + {(X_{0} + a)}^{2}}}{Z_{0} + \sqrt{{(Y_{0} - b)}^{2} + {Z_{0}}^{2} + {(X_{0} + a)}^{2}}})$

$- \frac{M sin θ sin φ}{4π μ_{0}} [\{\tan^{- 1} (\frac{1}{(Y_{0} + b)} \frac{(Z_{0} + 2 c) (X_{0} + a)}{\sqrt{{(Z_{0} + 2 c)}^{2} + {(X_{0} + a)}^{2} + {(Y_{0} + b)}^{2}}})$

$- \tan^{- 1} (\frac{1}{(Y_{0} + b)} \frac{(Z_{0} + 2 c) (X_{0} - a)}{\sqrt{{(Z_{0} + 2 c)}^{2} + {(X_{0} - a)}^{2} + {(Y_{0} + b)}^{2}}})\}$

$- \{\tan^{- 1} (\frac{1}{(Y_{0} + b)} \frac{Z_{0} (X_{0} + a)}{\sqrt{{Z_{0}}^{2} + {(X_{0} + a)}^{2} + {(Y_{0} + b)}^{2}}})$

$- \tan^{- 1} (\frac{1}{(Y_{0} + b)} \frac{Z_{0} (X_{0} - a)}{\sqrt{{Z_{0}}^{2} + {(X_{0} - a)}^{2} + {(Y_{0} + b)}^{2}}})\}]$

$- \frac{M \cos θ}{4 π μ_{0}} \log (\frac{(X_{0} + a) + \sqrt{{(X_{0} + a)}^{2} + {(Y_{0} - b)}^{2} + {(Z_{0} + 2 c)}^{2}}}{(X_{0} + a) + \sqrt{{{(X_{0} + a)}^{2} + (Y_{0} + b)}^{2} + {(Z_{0} + 2 c)}^{2}}} \frac{(X_{0} - a) + \sqrt{{{(X_{0} - a)}^{2} + (Y_{0} + b)}^{2} + {(Z_{0} + 2 c)}^{2}}}{(X_{0} - a) + \sqrt{{{(X_{0} - a)}^{2} + (Y_{0} - b)}^{2} + {(Z_{0} + 2 c)}^{2}}})$ 式(14-45)

と表すことができる。
同様に ${\overset{⃑}{H}}_{L} (X_{0}, Y_{0}, Z_{0})$ のZ成分は

${({\overset{⃑}{H}}_{L} (X_{0}, Y_{0}, Z_{0}))}_{Z}$

$= \frac{M \cos θ}{4 π μ_{0}} [\{\tan^{- 1} (\frac{1}{Z_{0}} \frac{(X_{0} + a) (Y_{0} + b)}{\sqrt{{(X_{0} + a)}^{2} + {(Y_{0} + b)}^{2} + {Z_{0}}^{2}}})$

$- \tan^{- 1} (\frac{1}{Z_{0}} \frac{(X_{0} + a) (Y_{0} - b)}{\sqrt{{(X_{0} + a)}^{2} + {(Y_{0} - b)}^{2} + {Z_{0}}^{2}}})\}$

$- \{\tan^{- 1} (\frac{1}{Z_{0}} \frac{(X_{0} - a) (Y_{0} + b)}{\sqrt{{(X_{0} - a)}^{2} + {(Y_{0} + b)}^{2} + {Z_{0}}^{2}}})$

$- \tan^{- 1} (\frac{1}{Z_{0}} \frac{(X_{0} - a) (Y_{0} - b)}{\sqrt{{(X_{0} - a)}^{2} + {(Y_{0} - b)}^{2} + {Z_{0}}^{2}}})\}]$

$+ \frac{M sin θ cos φ}{4π μ_{0}} \log (\frac{(Y_{0} + b) + \sqrt{{(Y_{0} + b)}^{2} + {Z_{0}}^{2} + {(X_{0} - a)}^{2}}}{(Y_{0} + b) + \sqrt{{{(Y_{0} + b)}^{2} + (Z_{0} + 2 c)}^{2} + {(X_{0} - a)}^{2}}} \frac{(Y_{0} - b) + \sqrt{{{(Y_{0} - b)}^{2} + (Z_{0} + 2 c)}^{2} + {(X_{0} - a)}^{2}}}{(Y_{0} - b) + \sqrt{{(Y_{0} - b)}^{2} + Z_{0}^{2} + {(X_{0} - a)}^{2}}})$

$+ \frac{M sin θ sin φ}{4π μ_{0}} \log (\frac{(X_{0} + a) + \sqrt{{Z_{0}}^{2} + {(X_{0} + a)}^{2} + {(Y_{0} - b)}^{2}}}{(X_{0} + a) + \sqrt{{{(Z_{0} + 2 c)}^{2} + (X_{0} + a)}^{2} + {(Y_{0} - b)}^{2}}} \frac{(X_{0} - a) + \sqrt{{(Z_{0} + 2 c)}^{2} + {(X_{0} - a)}^{2} + {(Y_{0} - b)}^{2}}}{(X_{0} - a) + \sqrt{{Z_{0}}^{2} + {(X_{0} - a)}^{2} + {(Y_{0} - b)}^{2}}})$

$- \frac{M sin θ cos φ}{4π μ_{0}} \log (\frac{(Y_{0} + b) + \sqrt{{(Y_{0} + b)}^{2} + {Z_{0}}^{2} + {(X_{0} + a)}^{2}}}{(Y_{0} + b) + \sqrt{{{(Y_{0} + b)}^{2} + (Z_{0} + 2 c)}^{2} + {(X_{0} + a)}^{2}}} \frac{(Y_{0} - b) + \sqrt{{{(Y_{0} - b)}^{2} + (Z_{0} + 2 c)}^{2} + {(X_{0} + a)}^{2}}}{(Y_{0} - b) + \sqrt{{(Y_{0} - b)}^{2} + Z_{0}^{2} + {(X_{0} + a)}^{2}}})$

$- \frac{M sin θ sin φ}{4π μ_{0}} \log (\frac{(X_{0} + a) + \sqrt{{Z_{0}}^{2} + {(X_{0} + a)}^{2} + {(Y_{0} + b)}^{2}}}{(X_{0} + a) + \sqrt{{{(Z_{0} + 2 c)}^{2} + (X_{0} + a)}^{2} + {(Y_{0} + b)}^{2}}} \frac{(X_{0} - a) + \sqrt{{(Z_{0} + 2 c)}^{2} + {(X_{0} - a)}^{2} + {(Y_{0} + b)}^{2}}}{(X_{0} - a) + \sqrt{{Z_{0}}^{2} + {(X_{0} - a)}^{2} + {(Y_{0} + b)}^{2}}})$

$- \frac{M \cos θ}{4 π μ_{0}} [\{\tan^{- 1} (\frac{1}{(Z_{0} + 2 c)} \frac{(X_{0} + a) (Y_{0} + b)}{\sqrt{{(X_{0} + a)}^{2} + {(Y_{0} + b)}^{2} + {(Z_{0} + 2 c)}^{2}}})$

$- \tan^{- 1} (\frac{1}{(Z_{0} + 2 c)} \frac{(X_{0} + a) (Y_{0} - b)}{\sqrt{{(X_{0} + a)}^{2} + {(Y_{0} - b)}^{2} + {(Z_{0} + 2 c)}^{2}}})\}$

$- \{\tan^{- 1} (\frac{1}{(Z_{0} + 2 c)} \frac{(X_{0} - a) (Y_{0} + b)}{\sqrt{{(X_{0} - a)}^{2} + {(Y_{0} + b)}^{2} + {(Z_{0} + 2 c)}^{2}}})$

$- \tan^{- 1} (\frac{1}{(Z_{0} + 2 c)} \frac{(X_{0} - a) (Y_{0} - b)}{\sqrt{{(X_{0} - a)}^{2} + {(Y_{0} - b)}^{2} + {(Z_{0} + 2 c)}^{2}}})\}]$ 式(14-46)

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公開日：2015年3月9日　　　　　　更新日：2023年10月16日　　　　　　
作成者：児島　伸生
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