导航：题目详情 | 答案解析 | 相关问题 | 最新问题

将一磁导率为μ 半径为R0的球体放人均匀磁场H0内求总磁感应强度B和诱导磁矩m。

大学 2020-10-14 22:06:11 磁导率,球体,半径已帮助人

将一磁导率为μ，半径为R₀的球体，放人均匀磁场H₀内，求总磁感应强度B和诱导磁矩m。

热搜：圆锥摆的摆球质量为m,速率为v,圆半径为r

答案解析

难度★★☆ 搜索

根据题意，以球心为原点建立球坐标系，取H0的方向为ez向，此球体在磁场中被磁化，且具有球对称性。又因为无传导电流，即 $latex.action?latex=XGJpZ3RyaWFuZ2xlZG93bg%3D%3D$ ×H=0，所以我们可以用磁标势法来解答，设球外区域为1区域，球内区域为2区域。
本题所满足的边界条件为
$latex.action?latex=XGxlZnRce1xiZWdpbnthcnJheX17Y2N9XHBoaSBfezF9XG1pZCBfe1IgXHRvIFxpbmZ0eSB9PS1I%0D%0AX3swfVJjb3NcdGhldGEgICAgJiDikaBcXCBccGhpIF97Mn1cbWlkIF97UiBcdG8gXGluZnR5IH3k%0D%0AuLrmnInpmZDlgLwgICAmIOKRoVxcXHBoaV97MX1cbWlkIF97UiBcdG8gXGluZnR5IH09XHBoaSBf%0D%0AezJ9XG1pZCBfe1I9Ul97MH19ICAgICbikaIgXFxcbXVfezB9XGZyYWN7XHBhcnRpYWwgXHBoaSBf%0D%0AezF9fXtccGFydGlhbCBSfVxtaWQgX3tSPVJfezB9fT1cbXVcZnJhY3tccGFydGlhbCBccGhpIF97%0D%0AMn19e1xwYXJ0aWFsIFJ9XG1pZCBfe1I9Ul97MH19ICAgICbikaMgXGVuZHthcnJheX0%3D$
ψ1，ψ2仇满足
$latex.action?latex=XGJpZ3RyaWFuZ2xlZG93bl57Mn0%3D$ ψ1=0 (R＞R0)
$latex.action?latex=XGJpZ3RyaWFuZ2xlZG93bl57Mn0%3D$ ψ2=0 (R＜R0)
由于ψ1，ψ2仇有轴对称性，所以有
$latex.action?latex=XHBoaSBfezF9PVxzdW1fe259KGFfe259Ul57bn0rXGZyYWN7Yl97bn19e1Jee24rMX19KVBfe259%0D%0AKGNvc1x0aGV0YSAp$
$latex.action?latex=XHBoaSBfezJ9PVxzdW1fe259KGNfe259Ul57bn0rXGZyYWN7ZF97bn19e1Jee24rMX19KVBfe259%0D%0AKGNvc1x0aGV0YSAp$
由边界条件确定ψ1，ψ2表达式中的系数。
(1) 由①有
$latex.action?latex=XHN1bV97bn1hX3tufVJee259UF97bn0oY29zXHRoZXRhICk9LUhfezB9UmNvc1x0aGV0YSA%3D$
∴a1=-H0，an=0(n≠1)
$latex.action?latex=XHBoaSBfezF9PS1IX3swfVJjb3NcdGhldGEgK1xzdW1fe259XGZyYWN7Yl97bn19e3Jee24rMX19%0D%0AUF97bn0oY29zXHRoZXRhICk%3D$
(2) 由②有dn=0， $latex.action?latex=XHBoaSBfezJ9PVxzdW1fe259Y197bn1SXntufVBfe259KGNvc1x0aGV0YSAp$
(3) 由③有 $latex.action?latex=LUhfezB9Ul97MH1jb3NcdGhldGEgK1xzdW1fe259XGZyYWN7Yl97bn19e1JfezB9XntuKzF9fVBf%0D%0Ae259KGNvc1x0aGV0YSApPVxzdW1fe259Y197bn1SX3swfV57bn1QX3tufShjb3NcdGhldGEgKSDi%0D%0AkaQ%3D$
(4) 由④有
$latex.action?latex=XG11X3swfVxsZWZ0WyAtSF97MH1jb3NcdGhldGEgK1xzdW1fe259XGZyYWN7KG4rMSliX3tufX17%0D%0AUl97MH1ee24rMn19UF97bn0oY29zXHRoZXRhICkgXHJpZ2h0XT1cbXVcc3VtX3tufW5jX3tufVJf%0D%0AezB9XntuLTF9UF97bn0oY29zXHRoZXRhICnikaU%3D$
由⑤，⑥式可得到(比较对应系数)
$latex.action?latex=XGxlZnRce1xiZWdpbnthcnJheX17Y30tUl97MH1IX3swfStcZnJhY3tiX3sxfX17Ul97MH1eezJ9%0D%0AfT1jX3sxfVJfezB9XFwtSF97MH0rXGZyYWN7MmJfezF9fXtSX3swfV57M319PVxmcmFje1xtdX17%0D%0AXG11X3swfX1jX3sxfVxcY197bn1iX3tufT0wIChuXG5lcSAxKSAgXGVuZHthcnJheX0%3D$
可解得
$latex.action?latex=Y197MX09LVxmcmFjezNcbXVfezB9fXtcbXUrMlxtdV97MH19SF97MH0%3D$
$latex.action?latex=Yl97MX09LVxmcmFje1xtdS1cbXVfezB9fXtcbXUrMlxtdV97MH19SF97MH1SX3swfV57M30%3D$
故，有 $latex.action?latex=XHBoaSBfezF9PS1IX3swfVJjb3NcdGhldGEgK1xmcmFje1xtdS1cbXVfezB9fXtcbXUrMlxtdV97%0D%0AMH19SF97MH1SX3swfV57M31cZnJhY3tjb3NcdGhldGEgfXtSXnszfX09LUhfezB9XGNkb3QgUitc%0D%0AZnJhY3tcbXUtXG11X3swfX17XG11KzJcbXVfezB9fVJfezB9XnszfVxmcmFje0hfezB9XGNkb3Qg%0D%0AUn17Ul57M319$
$latex.action?latex=XHBoaV97Mn09LVxmcmFjezNcbXVfezB9fXtcbXUrMlxtdV97MH19SF97MH1SY29zXHRoZXRhID0t%0D%0AXGZyYWN7M1xtdV97MH19e1xtdSsyXG11X3swfX1IX3swfVxjZG90IFI%3D$
而
$latex.action?latex=XGxlZnRce1xiZWdpbnthcnJheX17Y31IX3sxfT0tXGJpZ3RyaWFuZ2xlZG93biBccGhpIF97MX09%0D%0ASF97MH0rXGZyYWN7XG11LVxtdV97MH19e1xtdSsyXG11X3swfX1SX3swfV57M31cbGVmdFsgXGZy%0D%0AYWN7MyhIX3swfVxjZG90IFIpUn17Ul57NX19XGZyYWN7SF97MH19e1JeezN9fSBccmlnaHRdXFxC%0D%0AX3sxfT1cbXVfezB9SF97MX09XG11X3swfUhfezB9K1xmcmFje1xtdS1cbXVfezB9fXtcbXUrMlxt%0D%0AdV97MH19Ul97MH1eezN9XGxlZnRbIFxmcmFjezMoSF97MH1cY2RvdCBSKVJ9e1JeezV9fVxmcmFj%0D%0Ae0hfezB9fXtSXnszfX0gXHJpZ2h0XX0gIFxlbmR7YXJyYXl9$
$latex.action?latex=XGxlZnRce1xiZWdpbnthcnJheX17Y31IX3syfT0tXGJpZ3RyaWFuZ2xlZG93biBccGhpIF97Mn09%0D%0AXGZyYWN7M1xtdV97MH19e1xtdSsyXG11X3swfX1IX3swfVxcQl97Mn09XG11SF97Mn09XGZyYWN7%0D%0AM1xtdVxtdV97MH19e1xtdSszXG11X3swfX1IX3swfSAgXGVuZHthcnJheX0%3D$
即 $latex.action?latex=Qj1cbGVmdFx7XGJlZ2lue2FycmF5fXtjY31cZnJhY3szXG11XG11X3swfX17XG11KzJcbXVfezB9%0D%0AfUhfezB9ICYgKFLvvJxSX3swfSlcXFxtdV97MH1IX3swfStcZnJhY3tcbXUtXG11X3swfX17XG11%0D%0AKzJcbXVfezB9fVxtdV97MH1SX3swfV57M31cbGVmdFsgXGZyYWN7MyhIX3swXGNkb3QgUn0pUn17%0D%0AUl57NX19LVxmcmFje0hfezB9fXtSXnszfX0gXHJpZ2h0XSAgICYoUu%2B8nlJfezB9KSBcZW5ke2Fy%0D%0AcmF5fQ%3D%3D$
球内磁化强度为
$latex.action?latex=TV97Mn09XGZyYWN7Ql97MH19e1xtdV97MH19LUhfezJ9PVxmcmFjezMoXG11LVxtdV97MH0pfXtc%0D%0AbXUrMlxtdV97MH19SF97MH0%3D$
诱导磁矩为
$latex.action?latex=bT1cZnJhY3s0fXszfVxwaVJfezB9XnszfU09XGZyYWN7NFxwaShcbXUtXG11X3swfSl9e1xtdSsy%0D%0AXG11X3swfX1SX3swfV57M31IX3swfQ%3D%3D$