学习麦克斯韦方程组 

从麦克斯韦方程组导出磁场的波动方程.

以自然洛伦兹-亥维赛单位输入麦克斯韦方程组.

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X Inactivate[{gaussE = Div[\[ScriptCapitalE][x, y, z, t], {x, y, z}] == \[Rho][x, y, z, t], gaussB = Div[\[ScriptCapitalB][x, y, z, t], {x, y, z}] == 0, faraday = Curl[\[ScriptCapitalE][x, y, z, t], {x, y, z}] == -D[\[ScriptCapitalB][x, y, z, t], t], ampere = Curl[\[ScriptCapitalB][x, y, z, t], {x, y, z}] == j[x, y, z, t] + D[\[ScriptCapitalE][x, y, z, t], t]}, Div | Curl | D];

创建清晰格式的方程组表格.

显示完整的 Wolfram 语言输入隐藏输入

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X Inactivate[{gaussE = Div[\[ScriptCapitalE][x, y, z, t], {x, y, z}] == \[Rho][x, y, z, t], gaussB = Div[\[ScriptCapitalB][x, y, z, t], {x, y, z}] == 0, faraday = Curl[\[ScriptCapitalE][x, y, z, t], {x, y, z}] == -D[\[ScriptCapitalB][x, y, z, t], t], ampere = Curl[\[ScriptCapitalB][x, y, z, t], {x, y, z}] == j[x, y, z, t] + D[\[ScriptCapitalE][x, y, z, t], t]}, Div | Curl | D];; Grid[Transpose[{{"Gauss's Law for Electricity", "Gauss's Law for Magnetism", "Faraday's Law", "Ampere's Law"}, %}], Alignment -> Left, Dividers -> All, Spacings -> {4, 2}] // TraditionalForm

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在真空（ 和 ）状态提取安培定律的旋度.

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X Inactive[Curl][#, {x, y, z}] & /@ ampere /. j -> (0 &)

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替换微分顺序.

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X Inactive[Curl][#, {x, y, z}] & /@ ampere /. j -> (0 &); % /. Inactivate[Curl[D[v_, t_], x_] :> D[Curl[v, x], t]]

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代入法拉第定律.

In[5]:=

X Inactivate[{gaussE = Div[\[ScriptCapitalE][x, y, z, t], {x, y, z}] == \[Rho][x, y, z, t], gaussB = Div[\[ScriptCapitalB][x, y, z, t], {x, y, z}] == 0, faraday = Curl[\[ScriptCapitalE][x, y, z, t], {x, y, z}] == -D[\[ScriptCapitalB][x, y, z, t], t], ampere = Curl[\[ScriptCapitalB][x, y, z, t], {x, y, z}] == j[x, y, z, t] + D[\[ScriptCapitalE][x, y, z, t], t]}, Div | Curl | D];; % /. Inactivate[Curl[D[v_, t_], x_] :> D[Curl[v, x], t]]; % /. Solve[faraday, Inactive[Curl][\[ScriptCapitalE][x, y, z, t], {x, y, z}]][[1]]

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激活磁场波动方程中的方程结果.

In[6]:=

X Inactivate[{gaussE = Div[\[ScriptCapitalE][x, y, z, t], {x, y, z}] == \[Rho][x, y, z, t], gaussB = Div[\[ScriptCapitalB][x, y, z, t], {x, y, z}] == 0, faraday = Curl[\[ScriptCapitalE][x, y, z, t], {x, y, z}] == -D[\[ScriptCapitalB][x, y, z, t], t], ampere = Curl[\[ScriptCapitalB][x, y, z, t], {x, y, z}] == j[x, y, z, t] + D[\[ScriptCapitalE][x, y, z, t], t]}, Div | Curl | D];; % /. Solve[faraday, Inactive[Curl][\[ScriptCapitalE][x, y, z, t], {x, y, z}]][[1]]; wave = Activate[%]

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用传统符号表示的方程.

方程的平面波解可以用简单的替代验证.

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X wave /. \[ScriptCapitalB] -> (A Exp[{kx, ky, kz}.{#1, #2, #3} - I Sqrt[kx^2 + ky^2 + kz^2] #4] &) // Simplify

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