查看“︁學院物理/磁場”︁的源代码

==定義==
電量為 '''''q''''' 的電荷，以速度 '''''v''''' 射入，受到與運動方向垂直的作用力 '''''F'''''，則區域內有磁場。

<math>
\vec F=q\,\vec v\times\vec B=i\,\vec l\times\vec B
</math>
==應用==
===質譜儀===
<math>
\begin{align}&
F=qvB=\frac{mv^2}r
\\&
r=\frac{mv}{qB}
\end{align}
</math>
===速度選擇器===
<math>
\begin{align}&
qE=qvB
\\&
v=\frac E B
\end{align}
</math>
===電荷的螺線運動===
<math>
\begin{align}&
r=\frac{mv}{qB}
\\&
T=\frac{2\pi r}v=\frac{2\pi m}{qB}
\\&
\omega=\frac{qB}m
\\&
P=v_xT
\end{align}
</math>
==例題==
===三===
電子受電位差 '''''ΔV'''''' 從靜止加速進入均勻磁場，電子受垂直行進方向磁場作用，行進路徑彎曲為半徑 '''''r'''''，求磁場與角頻率？

<math>
\begin{align}
\frac{mv^2}2&-0=-q\Delta V
\\
v=&\sqrt{\frac{-2q\Delta V}m}
\\
&B=\frac{mv}{er}
\\
&\omega=\frac v r
\end{align}
</math>
===四===
<div style="float: right;position: relative;width: 200px;">
<div style="border-bottom: 4px solid;height: 50px;overflow: hidden;width: 108px;"><div style="border: 4px solid;border-radius: 50%;solid;height: 100px;width: 100px;"></div></div>
圖28.20
</div>
半徑 '''''R''''' 電流 '''''I''''' 的封閉半圓導線在 xy 平面上受 y 軸方向的均勻磁場，求導線受力？

由於直線導線和磁場垂直，

<math>
\begin{align}&
F_1=2iaB\hat z
\\&
F_2=-2iaB\hat z
\end{align}
</math>
===五===
Torque

<math>
\begin{align}&

\end{align}
</math>
==霍爾效應==
{{Wikipedia|霍爾效應}}
==例題==
===六===
<math>
\begin{align}&

\end{align}
</math>
==畢歐沙伐==
{{Wikipedia|必歐沙伐}}
<math>
d\vec B=\frac{\mu i}{4\pi}\frac{d\vec l\times\hat r}{r^2}
</math>
==例題==
===一===
<div style="float: right;position: relative;width: 200px;">
<div style="border: solid;border-width: 0 2px 8px 0;height: 60px;width: 60px;"></div>
<div style="border-bottom: 8px solid;height: 60px;left: 62px;position: absolute;top: 0;width: 40px;"></div>
<div style="border-top: 2px dashed;left: 0px;position: absolute;top: 60px;transform: rotate(-45deg);transform-origin: 0 0;width: 85px;"></div>
<div style="border-top: 2px dashed;left: 62px;position: absolute;top: 0px;transform: rotate(56deg);transform-origin: 0 0;width: 72px;"></div>
<div style="left: 44px;position: absolute;top: 14px;">''θ<sub>1</sub>''</div>
<div style="left: 65px;position: absolute;top: 24px;">''θ<sub>2</sub>''</div>
<div style="margin-left: 60px;"><math>\begin{matrix}\longrightarrow\\i\quad\vec l\quad x\end{matrix}</math></div>
<div style="left: 96px;position: absolute;top: 24px;"><span style="display: inline-block;font-size: 30px;transform: rotate(5deg);">↖</span> <math>\vec r</math></div>
圖29.4
</div>
<math>
\begin{align}&
d\vec l\times\hat r=\sin\left(\pi-\frac\pi2-\theta\right)\,dx\,\hat\odot=\cos\theta\,dx\,\hat\odot
\\&
\vec B=\frac{\mu i}{4\pi}\int\frac{d\vec l\times\hat r}{r^2}=\frac{\mu i}{4\pi}\int\frac{\cos\theta\,dx}{r^2}\,\hat\odot
\\&
B=\frac{\mu i}{4\pi}\int\frac{\left|d\vec l\times\hat r\right|}{r^2}=\frac{\mu i}{4\pi}\int\frac{\cos\theta\,dx}{r^2}=\frac{\mu i}{4\pi}\int\frac{y\,dx}{r^3}
\\&
\qquad x=y\tan\theta\qquad dx=y\sec^2\theta\,d\theta\qquad r=y\sec\theta
\\&
=\frac{\mu i}{4\pi}\int\frac{y^2\sec^2\theta\,d\theta}{y^3\sec^3\theta}=\frac{\mu i}{4\pi y}\int\cos\theta\,d\theta=\frac{\mu i}{4\pi y}\Big[\sin\theta\Big]_{\theta_2}^{\theta_1}
\\&
=\frac{\mu i}{4\pi y}\big(\sin\theta_1-\sin\theta_2\big)
\\&
\\&
\theta_1=\frac\pi2\qquad\theta_2=-\frac\pi2\qquad B=\frac{\mu i}{2\pi y}
\end{align}
</math>
===二===
<div style="float: right;position: relative;width: 200px;">
<div style="border: 4px solid transparent;border-radius: 50%;border-right: 4px solid;height: 100px;left: -50px;margin: 20px;position: relative;width: 100px;"></div>
<div style="border-top: 4px solid;left: 54px;position: absolute;top: 22px;transform: rotate(-45deg);width: 40px;"></div>
<div style="border-top: 4px solid;left: 54px;position: absolute;top: 122px;transform: rotate(45deg);width: 40px;"></div>
圖29.5


</div>
<math>
\begin{align}&
B=\frac{\mu i}{4\pi r^2}\int dl
\end{align}
</math>
===三===
<div style="clear: right;float: right;width: 200px;">
{{see also|學院物理/連續電荷分布與高斯定律#二|l1=圖23.3}}
<div style="margin-top: 8px;position: relative;">
<div style="border: 4px solid;border-radius: 50%;height: 100px;width: 100px;"></div>
<div style="border: 2px dashed;border-right: 0;border-top: 0;height: 50px;left: 53px;position: absolute;top: 3px;width: 100px;"></div>
<div style="font-family: Times;font-size: 18px;left: 114px;position: absolute;top: 29px;"><i><b>x</b></i></div>
<div style="font-family: Times;font-size: 18px;left: 38px;position: absolute;top: 18px;"><i><b>y</b></i></div>
<div style="margin-left: 40px;"><math>\begin{align}&\to i\\&\;d\vec l\end{align}</math></div>
<div style="left: 80px;position: absolute;top: 76px;"><span style="display: inline-block;font-size: 30px;transform: rotate(20deg);">↗</span> <math>\vec r</math></div>
<div style="border-top: 2px dashed;height: 20px;left: 54px;position: absolute;top: 106px;transform: rotate(-28deg);transform-origin: 0 0;width: 112px;"></div>
<div style="font-size: 30px;left: 150px;position: absolute;top: 40px;transform: rotate(8deg);">↘</div>
<div style="left: 130px;position: absolute;top: 76px;"><math>\begin{matrix}d\vec B\\\propto\\d\vec l\times\hat r\end{matrix}</math></div>
</div>
圖29.6
</div>
<math>
\begin{align}&
d\vec l\times\hat r=dl\,\sin{90}^\circ\;\hat x=dl\,\hat x
\\&
B=\frac{\mu i}{4\pi}\int\frac{\left|d\vec l\times\hat r\right|}{r^2}=\frac{\mu i}{4\pi}\int\frac{dl}{r^2}=\frac{\mu iy}{2r^2}=\frac{\mu iy}{2\left(x^2+y^2\right)}
\\&
\qquad \int dl=\int y\,d\theta=y\int d\theta=y2\pi\qquad r^2=x^2+y^2
\\&
dB_x=\frac{\mu iy\,dl}{4\pi r^3}\qquad\quad\;\;r^3=\left(x^2+y^2\right)^\frac32
\\&
B_x=\frac{Ba}r=\frac{\mu iy^2}{2r^3}=\frac{\mu iy^2}{2\left(x^2+y^2\right)^\frac32}
\\&
\mu_A=i\vec A\qquad\mu_A=iy2\pi
\\&
x\to0\qquad B_x=\frac{\mu i}{2y}=\frac{\mu\mu_A}{4\pi y^2}
\end{align}
</math>

&nbsp;'''''μ<sub>A</sub>''''' 磁矩，'''''μ''''' = '''''μ<sub>0</sub>''''' = 磁導係數
==電流磁力==
<math>F=\frac{\mu ii_0l}{y2\pi}</math>
==例題==
===四===
電流磁力
==安培定律==
{{Wikipedia|安培定律}}
<math>\int \vec B\,d\vec l=\mu i</math>
==例題==
===五===
<div style="float: right;position: relative;width: 200px;">
<div style="background-color: lightgrey;border: 2px solid;border-radius: 50%;height: 50px;margin: 10px;width: 50px;"></div>
<div style="border-top: 2px dashed;left: 12px;position: absolute;top: 36px;width: 25px;"></div>
<div style="font-family: Times;font-size: 18px;left: 22px;position: absolute;top: 30px;"><i><b>a</b></i></div>
圖29.13
</div>
<math>
\begin{align}&
B(r\ge a)=\frac{\mu i}{r2\pi}
\\&
B(r<a)=\frac{\mu i_0}{r2\pi}=\frac{\mu ir^2}{r2\pi a^2}=\frac{\mu ir}{2\pi a^2}
\\&
\qquad i_0=\frac{ir^2}{a^2}
\end{align}
</math>
===六===
線圈磁場
==高斯定律==
{{Wikipedia|高斯定律}}
==例題==
===七===
<div style="float: right;position: relative;width: 200px;">
<div style="border-left: 2px solid;height: 90px;letter-spacing: 10px;overflow: hidden;width: 100px;word-break: break-all;">xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx</div>
<div style="border: 2px solid;height: 60px;left: 20px;position: absolute;top: 15px;width: 40px;"></div>
圖29.21
</div>
<math>
\begin{align}&
\Phi=\int B\,dA
\\&
B=\frac{\mu i}{4\pi r^2}\int\frac{\left|dl\times r\right|}{r^2}=\frac{\mu i}{2\pi y}
\end{align}
</math>
==法拉第定律==
{{Wikipedia|法拉第定律}}
<math>\epsilon=-\frac{d\Phi}{dt}=-\frac d{dt}\int B\,dA</math>
==例題==
===一===
線圈

<math>
\begin{align}&
\epsilon=-N\frac{d\Phi}{dt}
\\&
i=\frac{\left|\epsilon\right|}R=\frac{Blv}R
\end{align}
</math>
===二===
<div style="float: right;position: relative;width: 200px;">
<div style="border-bottom: 2px solid;border-top: 2px solid;color: lightgrey;height: 60px;letter-spacing: 10px;margin: 10px;overflow: hidden;width: 100px;word-break: break-all;">xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx</div>
<div style="border-left: 2px solid;height: 80px;left: 80px;position: absolute;top: 2px;width: 100px;"></div>
<div style="left: 24px;position: absolute;top: 10px;transform: rotate(90deg);transform-origin: 0 0;">----WW----</div>
<div style="font-family: Times;font-size: 18px;left: 24px;position: absolute;top: 30px;">'''''R'''''</div>
圖30.9
</div>
<math>
\begin{align}&
i=\frac{Blv}R
\\&
mv\frac{dv}{dt}=-\left(\frac{Blv}R^2\right)R
\\&
v=v_0e^{-t\,/\tfrac{mR}{B^2l^2}}=v_0e^\tfrac{-B^2l^2t}{mR}=\frac{dx}{dt}
\\&
x=v_0\frac{mR}{B^2l^2}
\end{align}
</math>
===三===
<div style="float: right;position: relative;width: 200px;">
<div style="height: 90px;letter-spacing: 10px;overflow: hidden;width: 100px;word-break: break-all;">xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx</div>
<div style="border-left: 4px solid;height: 60px;left: 24px;position: absolute;top: 14px;width: 60px;"></div>
圖30.10
</div>
<math>
\begin{align}&
\epsilon=\frac{-d\Phi}dt=\frac{Bl^2\,d\theta}{2\,dt}=\frac{Bl^2\omega}2
\\&
d\Phi=B\,dA=\frac{Bl^2\pi\,d\theta}{2\pi}=\frac{Bl^2\,d\theta}2
\end{align}
</math>
===四===
冷次
===六===
General
螺線管半徑 {{mvar|a}} 求 r 處 {{mvar|E}}
<div style="float: right;position: relative;width: 200px;">
<div style="background: repeating-linear-gradient(to right, transparent 1px,  black 4px);height: 40px;width: 100px;"></div>
<div style="border-left: 2px solid;border-right: 2px solid;height: 20px;width: 96px;"></div>
圖30.16
</div>
<math>
\begin{align}&
B=\mu ni
\\&
Ea2\pi=\frac{-d\Phi}{dt}=\frac{-\pi a^2dB}{dt}=\frac{-\pi a^2\mu ndi}{dt}
\end{align}
</math>
===七===
發電

<math>
\begin{align}&
\epsilon=NBA\omega
\\&
\omega=2\pi f
i=\frac\epsilon R
\end{align}
</math>
===八===
馬達的感應電流

<math>
\begin{align}&
i=\frac\epsilon R
\end{align}
</math>
==電感==
{{Wikipedia|電感}}
<math>\epsilon=\frac{-L\,di}{dt}</math>
==例題==
===一===
螺線管

<math>
\begin{align}&
L=\frac{N\Phi}i=\frac{\mu N^2A}l
\end{align}
</math>
===二===
RL 電路

<math>
\begin{align}&
\frac\epsilon R-i=\frac\epsilon Re^\frac{-Rt}L
\\&
i=\frac\epsilon R\left(1-e^\frac{-Rt}L\right)
\end{align}
</math>