{"id":7655,"date":"2024-02-21T16:03:52","date_gmt":"2024-02-21T07:03:52","guid":{"rendered":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/?page_id=7655"},"modified":"2024-02-22T14:16:06","modified_gmt":"2024-02-22T05:16:06","slug":"%e4%b8%80%e6%a7%98%e3%81%aa%e9%9d%99%e7%a3%81%e5%a0%b4%e4%b8%ad%e3%81%ae%e8%8d%b7%e9%9b%bb%e7%b2%92%e5%ad%90%e3%81%ae%e9%81%8b%e5%8b%95","status":"publish","type":"page","link":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/%e9%9b%bb%e7%a3%81%e6%b0%97%e5%ad%a6-i\/%e3%83%ad%e3%83%bc%e3%83%ac%e3%83%b3%e3%83%84%e5%8a%9b%ef%bc%9a%e9%9b%bb%e7%a3%81%e5%a0%b4%e4%b8%ad%e3%82%92%e9%81%8b%e5%8b%95%e3%81%99%e3%82%8b%e8%8d%b7%e9%9b%bb%e7%b2%92%e5%ad%90%e3%81%8c%e5%8f%97\/%e4%b8%80%e6%a7%98%e3%81%aa%e9%9d%99%e7%a3%81%e5%a0%b4%e4%b8%ad%e3%81%ae%e8%8d%b7%e9%9b%bb%e7%b2%92%e5%ad%90%e3%81%ae%e9%81%8b%e5%8b%95\/","title":{"rendered":"\u4e00\u69d8\u306a\u9759\u78c1\u5834\u4e2d\u306e\u8377\u96fb\u7c92\u5b50\u306e\u904b\u52d5"},"content":{"rendered":"

<\/p>\n

\u96fb\u5834\u306f\u306a\u3044\u3068\u3057\u3066 $\\boldsymbol{E} = \\boldsymbol{0}$\uff0c\u78c1\u5834\u306b\u3088\u308b\u30ed\u30fc\u30ec\u30f3\u30c4\u529b\u3092\u53d7\u3051\u305f\u8377\u96fb\u7c92\u5b50\u306e\u904b\u52d5\u65b9\u7a0b\u5f0f\u306f\uff0c<\/p>\n

$$m \\frac{d\\boldsymbol{v}}{dt} = q\\,\\boldsymbol{v}\\times \\boldsymbol{B}$$<\/p>\n

\u4e21\u8fba\u306b $\\boldsymbol{v}$ \u306e\u5185\u7a4d\u3092\u304b\u3051\u3066<\/p>\n

\\begin{eqnarray}
\nm \\boldsymbol{v}\\cdot\\frac{d\\boldsymbol{v}}{dt}
\n&=& q\\,\\boldsymbol{v}\\cdot\\left(\\boldsymbol{v}\\times \\boldsymbol{B}\\right) \\\\
\n&=& q \\boldsymbol{B}\\cdot\\left(\\boldsymbol{v}\\times \\boldsymbol{v}\\right) \\\\
\n&=& 0 \\\\
\n\\therefore\\ \\ \\frac{d}{dt} \\left(\\frac{1}{2} m \\boldsymbol{v}\\cdot\\boldsymbol{v} \\right) &=& 0 \\\\
\n\\therefore\\ \\ \\frac{1}{2} m \\boldsymbol{v}\\cdot\\boldsymbol{v} &=& \\mbox{const.} \\equiv E
\n\\end{eqnarray}<\/p>\n

\u4e00\u69d8\u306a\u9759\u78c1\u5834\u306e\u5411\u304d\u3092\u8868\u3059\u5358\u4f4d\u30d9\u30af\u30c8\u30eb<\/p>\n

$$\\displaystyle \\boldsymbol{n} \\equiv \\frac{\\boldsymbol{B}}{\\sqrt{\\boldsymbol{B}\\cdot\\boldsymbol{B}}}=\\frac{\\boldsymbol{B}}{B}, \\quad \\frac{d\\boldsymbol{n} }{dt} = \\boldsymbol{0} $$<\/p>\n

\u3092\u4f7f\u3063\u3066\u901f\u5ea6\u30d9\u30af\u30c8\u30eb $\\boldsymbol{v}$ \u3092 $\\boldsymbol{n}$ \u306b\u5e73\u884c\u306a\u30d1\u30fc\u30c8 $\\boldsymbol{v}_{\\scriptscriptstyle\/\/}$ \u3068\u5782\u76f4\u306a\u30d1\u30fc\u30c8 $\\boldsymbol{v}_{\\perp }$ \u306b\u5206\u89e3\u3059\u308b\u3002<\/p>\n

\\begin{eqnarray}
\n\\boldsymbol{v}_{\\scriptscriptstyle\/\/} &\\equiv& \\left(\\boldsymbol{v}\\cdot\\boldsymbol{n} \\right)\\,\\boldsymbol{n} \\\\
\n\\boldsymbol{v}_{\\perp }&\\equiv& \\boldsymbol{v} -\\boldsymbol{v}_{\\scriptscriptstyle\/\/}
\n\\end{eqnarray}<\/p>\n

\u904b\u52d5\u65b9\u7a0b\u5f0f\u306b $\\boldsymbol{n}$ \u306e\u5185\u7a4d\u3092\u304b\u3051\u3066<\/p>\n

\\begin{eqnarray}
\n\\boldsymbol{n}\\cdot \\frac{d\\boldsymbol{v}}{dt}
\n&=& \\frac{q}{m} \\boldsymbol{n}\\cdot \\left( \\boldsymbol{v}\\times \\boldsymbol{B}\\right) \\\\
\n&=& \\frac{q}{m B} \\boldsymbol{v}\\cdot \\left( \\boldsymbol{B}\\times \\boldsymbol{B}\\right) \\\\
\n&=& 0 \\\\
\n\\therefore\\ \\ \\frac{d}{dt} \\left(\\boldsymbol{v}\\cdot\\boldsymbol{n} \\right) &=& 0 \\\\
\n\\therefore\\ \\ \\boldsymbol{v}_{\\scriptscriptstyle\/\/} &=& \\mbox{const.} \\equiv \\boldsymbol{V}_{\\scriptscriptstyle\/\/} \\\\
\n\\therefore\\ \\ \\boldsymbol{r}_{\\scriptscriptstyle\/\/} &=& \\int^t \\boldsymbol{v}_{\\scriptscriptstyle\/\/}\\, dt =\u00a0 \\boldsymbol{V}_{\\scriptscriptstyle\/\/} \\,t
\n\\end{eqnarray}<\/p>\n

\u3053\u3053\u3067\uff0c\u7c21\u5358\u306e\u305f\u3081\u306b\u521d\u671f\u6761\u4ef6\u3092 $t=0$ \u3067 $\\boldsymbol{r}_{\\scriptscriptstyle\/\/} = \\boldsymbol{0}$ \u3068\u3057\u305f\u3002<\/p>\n

\u901f\u5ea6\u30d9\u30af\u30c8\u30eb $\\boldsymbol{v}$ \u306e $\\boldsymbol{n}$ \u306b\u5e73\u884c\u306a\u30d1\u30fc\u30c8 $\\boldsymbol{v}_{\\scriptscriptstyle\/\/}$ \u306f\u5b9a\u30d9\u30af\u30c8\u30eb\u3067\u3042\u308b\u3053\u3068\u304c\u308f\u304b\u3063\u305f\u3002<\/p>\n

\u6b21\u306b\uff0c$\\displaystyle \\frac{d\\boldsymbol{v}_{\\scriptscriptstyle\/\/}}{dt}\u00a0 = \\boldsymbol{0}$ \u3092\u4f7f\u3046\u3068\uff0c\u904b\u52d5\u65b9\u7a0b\u5f0f\u306f<\/p>\n

\\begin{eqnarray}
\n\\frac{d\\boldsymbol{v}_{\\perp}}{dt} &=& \\frac{q}{m}\\,\\boldsymbol{v}_{\\perp}\\times \\boldsymbol{B} \\\\
\n\\frac{d^2\\boldsymbol{v}_{\\perp}}{dt^2}
\n&=&\\frac{q}{m}\\,\\frac{d\\boldsymbol{v}_{\\perp}}{dt}\\times \\boldsymbol{B} \\\\
\n&=& \\left(\\frac{q}{m} \\right)^2 \\left( \\boldsymbol{v}_{\\perp}\\times\\boldsymbol{B}\\right)\\times\\boldsymbol{B} \\\\
\n&=& -\\left(\\frac{q}{m} \\right)^2 \\left\\{\\left(\\boldsymbol{B}\\cdot\\boldsymbol{B} \\right)\u00a0 \\boldsymbol{v}_{\\perp} – \\left(\\boldsymbol{B}\\cdot\\boldsymbol{v}_{\\perp}\\right)\u00a0 \\boldsymbol{B} \\right\\} \\\\
\n&=& -\\left(\\frac{q B}{m} \\right)^2 \\boldsymbol{v}_{\\perp} \\\\
\n\\therefore\\ \\ \\boldsymbol{v}_{\\perp}\u00a0 &=& \\boldsymbol{A} \\cos \\omega t + \\boldsymbol{B} \\sin \\omega t, \\quad \\omega \\equiv \\frac{q B}{m} \\\\
\n\\therefore\\ \\ \\boldsymbol{r}_{\\perp}\u00a0 &=& \\int^t \\boldsymbol{v}_{\\perp}\u00a0 \\, dt = \\frac{ \\boldsymbol{A}}{\\omega} \\sin \\omega t -\\frac{ \\boldsymbol{B} }{\\omega}\\cos \\omega t
\n\\end{eqnarray}<\/p>\n

\u521d\u671f\u6761\u4ef6\u3092 $t=0$ \u3067 $\\boldsymbol{r}_{\\perp} = \\boldsymbol{R}_{\\perp}, \\boldsymbol{v}_{\\perp} = \\boldsymbol{V}_{\\perp}$ \u3068\u3059\u308b\u3068\uff0c<\/p>\n

\\begin{eqnarray}
\n\\boldsymbol{v}_{\\perp}\u00a0 &=& \\boldsymbol{V}_{\\perp}\\cos \\omega t -\\omega\\boldsymbol{R}_{\\perp}\\sin \\omega t\\\\
\n\\boldsymbol{r}_{\\perp}\u00a0 &=&\u00a0 \\frac{ \\boldsymbol{V}_{\\perp}}{\\omega} \\sin \\omega t +\\boldsymbol{R}_{\\perp}\\cos \\omega t
\n\\end{eqnarray}<\/p>\n

\u3068\u306a\u308b\u304c\uff0c\u904b\u52d5\u30a8\u30cd\u30eb\u30ae\u30fc\u304c\u4e00\u5b9a\u3067\u3042\u308b\u3053\u3068\u304b\u3089 $\\boldsymbol{v}_{\\perp}\\cdot\\boldsymbol{v}_{\\perp}$ \u3082\u4e00\u5b9a\u3067\u3042\u308b\u3053\u3068\u306b\u306a\u308a\uff0c<\/p>\n

$$ |\\boldsymbol{V}_{\\perp}| =\\omega |\\boldsymbol{R}_{\\perp}|,\u00a0 \\quad\\boldsymbol{V}_{\\perp}\\cdot\\boldsymbol{R}_{\\perp} =0$$<\/p>\n

\u3067\u3042\u308b\u3053\u3068\u3082\u308f\u304b\u308b\u3002\u306a\u305c\u304b\u3068\u3044\u3046\u3068\uff0c<\/p>\n

\\begin{eqnarray}
\n\\boldsymbol{v}_{\\perp}\\cdot\\boldsymbol{v}_{\\perp} &=&
\n\\boldsymbol{V}_{\\perp}\\cdot \\boldsymbol{V}_{\\perp} \\cos^2 \\omega\\,t \\\\
\n&&+
\n\\omega^2 \\boldsymbol{R}_{\\perp}\\cdot\\boldsymbol{R}_{\\perp} \\sin^2 \\omega\\,t \\\\
\n&& -2 \\omega \\boldsymbol{V}_{\\perp}\\cdot\\boldsymbol{R}_{\\perp} \\sin\\omega\\,t \\ \\cos\\omega \\, t
\n\\end{eqnarray}<\/p>\n

\u306f\u3044\u3064\u3067\u3082\u4e00\u5b9a\u3067\u3042\u308b\u306e\u3067\uff0c<\/p>\n

\\begin{eqnarray}
\n\\boldsymbol{v}_{\\perp}\\cdot\\boldsymbol{v}_{\\perp} &=&
\n\\boldsymbol{V}_{\\perp}\\cdot \\boldsymbol{V}_{\\perp}\\quad\\mbox{for} \\ \\ t=0 \\\\
\n&=& \\omega^2 \\boldsymbol{R}_{\\perp}\\cdot\\boldsymbol{R}_{\\perp} \\quad\\mbox{for} \\ \\ \\omega\\,t=\\frac{\\pi}{2} \\\\
\n&=&\\frac{1}{2} \\boldsymbol{V}_{\\perp}\\cdot \\boldsymbol{V}_{\\perp}\u00a0 +
\n\\frac{1}{2}\\omega^2 \\boldsymbol{R}_{\\perp}\\cdot\\boldsymbol{R}_{\\perp}\u00a0\u00a0 – \\omega \\boldsymbol{V}_{\\perp}\\cdot\\boldsymbol{R}_{\\perp}\\quad\\mbox{for} \\ \\ \\omega\\,t=\\frac{\\pi}{4}
\n\\end{eqnarray}<\/p>\n

\u3068\u306a\u308b\u304b\u3089\u3002<\/p>\n

\u6700\u7d42\u7684\u306b $\\boldsymbol{B}$ \u306e\u5411\u304d\u3092 $z$ \u8ef8\u306b\u3068\u308a\uff0c<\/p>\n

\\begin{eqnarray}
\n\\boldsymbol{V}_{\\scriptscriptstyle\/\/} &=& (0, 0, V_{\\scriptscriptstyle\/\/}) \\\\
\n\\boldsymbol{V}_{\\perp} &=& (0, V_{\\perp}, 0) \\\\
\n\\boldsymbol{R}_{\\perp} &=&\u00a0 \\left(\\frac{V_{\\perp}}{\\omega}, 0, 0\\right)\\\\
\n\\boldsymbol{r}_{\\scriptscriptstyle\/\/} &=& (0, 0, z) \\\\
\n\\boldsymbol{r}_{\\perp} &=& (x, y, 0)
\n\\end{eqnarray}<\/p>\n

\u3068\u3059\u308c\u3070\uff0c<\/p>\n

\\begin{eqnarray}
\nx &=& \\frac{V_{\\perp}}{\\omega} \\cos \\omega\\, t \\\\
\ny &=& \\frac{V_{\\perp}}{\\omega} \\sin \\omega\\, t \\\\
\nz &=& V_{\\scriptscriptstyle\/\/} \\, t
\n\\end{eqnarray}<\/p>\n

\u4e00\u69d8\u306a\u9759\u78c1\u5834\u4e2d\u3067\u306f\uff0c\u8377\u96fb\u7c92\u5b50\u306f\u3089\u305b\u3093\u8ecc\u9053\u3092\u63cf\u3044\u3066\u904b\u52d5\u3059\u308b\u3053\u3068\u304c\u308f\u304b\u308b\u3002<\/p>\n

\"\"<\/p>\n","protected":false},"excerpt":{"rendered":"","protected":false},"author":33,"featured_media":0,"parent":2637,"menu_order":10,"comment_status":"closed","ping_status":"closed","template":"","meta":{"inline_featured_image":false,"footnotes":""},"class_list":["post-7655","page","type-page","status-publish","hentry","nodate","item-wrap"],"aioseo_notices":[],"_links":{"self":[{"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/pages\/7655","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/users\/33"}],"replies":[{"embeddable":true,"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/comments?post=7655"}],"version-history":[{"count":44,"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/pages\/7655\/revisions"}],"predecessor-version":[{"id":7721,"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/pages\/7655\/revisions\/7721"}],"up":[{"embeddable":true,"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/pages\/2637"}],"wp:attachment":[{"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/media?parent=7655"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}