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* Page found: Maxwell- Gleichungen im Vakuum (eq math.2110.7)

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TeX (original user input): 

\begin{align}
& 0=\frac{\partial }{\partial t}\left( {{\varepsilon }_{0}}{{\nabla }_{r}}\cdot \bar{E}-\rho  \right)={{\varepsilon }_{0}}{{\nabla }_{r}}\cdot \dot{\bar{E}}-\dot{\rho }=\frac{{{\varepsilon }_{0}}}{{{a}_{2}}}\nabla \cdot \left( \nabla \times \bar{B}-{{\mu }_{0}}\bar{j} \right)-\dot{\rho } \\
& \frac{{{\varepsilon }_{0}}}{{{a}_{2}}}\nabla \cdot \nabla \times \bar{B}=0 \\
& \Rightarrow \frac{{{\varepsilon }_{0}}}{{{a}_{2}}}\nabla \cdot \left( {{\mu }_{0}}\bar{j} \right)-\dot{\rho }=0 \\
& \Rightarrow {{a}_{2}}={{\varepsilon }_{0}}{{\mu }_{0}} \\
\end{align}

TeX (checked): 

{\begin{aligned}&0={\frac {\partial }{\partial t}}\left({{\varepsilon }_{0}}{{\nabla }_{r}}\cdot {\bar {E}}-\rho \right)={{\varepsilon }_{0}}{{\nabla }_{r}}\cdot {\dot {\bar {E}}}-{\dot {\rho }}={\frac {{\varepsilon }_{0}}{{a}_{2}}}\nabla \cdot \left(\nabla \times {\bar {B}}-{{\mu }_{0}}{\bar {j}}\right)-{\dot {\rho }}\\&{\frac {{\varepsilon }_{0}}{{a}_{2}}}\nabla \cdot \nabla \times {\bar {B}}=0\\&\Rightarrow {\frac {{\varepsilon }_{0}}{{a}_{2}}}\nabla \cdot \left({{\mu }_{0}}{\bar {j}}\right)-{\dot {\rho }}=0\\&\Rightarrow {{a}_{2}}={{\varepsilon }_{0}}{{\mu }_{0}}\\\end{aligned}}

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MathML (38.222 KB / 4.229 KB) :

0 = ∂ ∂ ⁡ t ⁢ ( ε 0 ⁢ ∇ r ⋅ E ¯ - ρ ) = ε 0 ⁢ ∇ r ⋅ E ¯ ˙ - ρ ˙ = ε 0 a 2 ⁢ ∇ ⋅ ( ∇ × B ¯ - μ 0 ⁢ j ¯ ) - ρ ˙ ε 0 a 2 ⁢ ∇ ⋅ ∇ × B ¯ = 0 ⇒ ε 0 a 2 ⁢ ∇ ⋅ ( μ 0 ⁢ j ¯ ) - ρ ˙ = 0 ⇒ a 2 = ε 0 ⁢ μ 0 missing-subexpression 0 𝑡 ⋅ subscript 𝜀 0 subscript ∇ 𝑟 ¯ 𝐸 𝜌 ⋅ subscript 𝜀 0 subscript ∇ 𝑟 ˙ ¯ 𝐸 ˙ 𝜌 ⋅ subscript 𝜀 0 subscript 𝑎 2 ∇ ∇ ¯ 𝐵 subscript 𝜇 0 ¯ 𝑗 ˙ 𝜌 missing-subexpression ⋅ subscript 𝜀 0 subscript 𝑎 2 ∇ ∇ ¯ 𝐵 0 missing-subexpression ⇒ absent ⋅ subscript 𝜀 0 subscript 𝑎 2 ∇ subscript 𝜇 0 ¯ 𝑗 ˙ 𝜌 0 missing-subexpression ⇒ absent subscript 𝑎 2 subscript 𝜀 0 subscript 𝜇 0 {\displaystyle{\displaystyle\begin{aligned} &\displaystyle 0=\frac{\partial}{% \partial t}\left({{\varepsilon}_{0}}{{\nabla}_{r}}\cdot\bar{E}-\rho\right)={{% \varepsilon}_{0}}{{\nabla}_{r}}\cdot\dot{\bar{E}}-\dot{\rho}=\frac{{{% \varepsilon}_{0}}}{{{a}_{2}}}\nabla\cdot\left(\nabla\times\bar{B}-{{\mu}_{0}}% \bar{j}\right)-\dot{\rho}\\ &\displaystyle\frac{{{\varepsilon}_{0}}}{{{a}_{2}}}\nabla\cdot\nabla\times\bar% {B}=0\\ &\displaystyle\Rightarrow\frac{{{\varepsilon}_{0}}}{{{a}_{2}}}\nabla\cdot\left% ({{\mu}_{0}}\bar{j}\right)-\dot{\rho}=0\\ &\displaystyle\Rightarrow{{a}_{2}}={{\varepsilon}_{0}}{{\mu}_{0}}\\ \end{aligned}}}

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MathML (4.186 KB / 530 B) :

0=∂∂t(ε0∇r⋅E¯−ρ)=ε0∇r⋅E¯˙−ρ˙=ε0a2∇⋅(∇×B¯−μ0j¯)−ρ˙ε0a2∇⋅∇×B¯=0⇒ε0a2∇⋅(μ0j¯)−ρ˙=0⇒a2=ε0μ0

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Calculated based on the variables occurring on the entire Maxwell- Gleichungen im Vakuum page

Identifiers

  • t
  • ε0
  • r
  • E¯
  • ρ
  • ε0
  • r
  • E¯˙
  • ρ˙
  • ε0
  • a2
  • B¯
  • μ0
  • j¯
  • ρ˙
  • ε0
  • a2
  • B¯
  • ε0
  • a2
  • μ0
  • j¯
  • ρ˙
  • a2
  • ε0
  • μ0

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