Maxwell's Equations in Potential Form

Maxwell's Laws are:
\[\mathbf{\nabla} \cdot \mathbf{E}=\frac{\rho}{\epsilon}\]
\[\mathbf{\nabla} \times \mathbf{E}=- \mu \frac{\partial H}{\partial t}\]
\[\mathbf{\nabla} \cdot \mathbf{H}=0\]
\[\mathbf{\nabla} \times \mathbf{H}=- \epsilon \frac{\partial E}{\partial t}+ \mathbf{J}\]
  is solenoidal since  
\[\mathbf{\nabla} \cdot \mathbf{H}=0\]
  hence we can write  
\[\mathbf{H} = \mathbf{\nabla} \times \mathbf{A}\]
  for some vector  
  called the magnetic vector potential.
Subsitutute the last equation into (2) to obtain
\[\mathbf{\nabla} \times \mathbf{E}=- mu \frac{\partial }{\partial t}(\mathbf{\nabla } \times \mathbf{A})\]

Rearranging this gives
\[\mathbf{\nabla} \times (\mathbf{E}+ \mu \frac{\partial \mathbf{A}}{\partial t})=\mathbf{0}\]

Hence we can write
\[\mathbf{E}+ \mu \frac{\partial \mathbf{A}}{\partial t}=-\mathbf{\nabla} \phi \rightarrow \mathbf{E}=- \mu \frac{\partial \mathbf{A}}{\partial t}-\mathbf{\nabla} \phi\]

for some scalar function  
\[\mathbf{A}, \phi\]
\[\mathbf{A'}= \mathbf{A} + \mathbf{\nabla} \psi , \phi'=\phi- \frac{\partial \psi}{\partial t}\]
\[\mathbf{A}, \phi\]
also satisy  
\[\mathbf{H} = \mathbf{\nabla} \times \mathbf{A;}\]
\[\mathbf{E}=- \frac{\partial \mathbf{A'}}{\partial t}-\mathbf{\nabla} \phi'\]

Substitute the last equation into (1) above to give
\[\mathbf{\nabla} \cdot (- \frac{\partial \mathbf{A}}{\partial t}-\mathbf{\nabla} \phi)= -\nabla^2 \phi- \frac{\partial }{\partial t}(\mathbf{\nabla} \cdot \mathbf{A})=\frac{\rho}{\epsilon}\]

Now substitute  
\[\mathbf{H} = \mathbf{\nabla} \times \mathbf{A}\]
\[\mathbf{E}=- \frac{\partial \mathbf{A'}}{\partial t}-\mathbf{\nabla} \phi'\]
  into (4) to obtain  
\[\mathbf{\nabla} \times (\mathbf{\nabla} \times \mathbf{A})= - \epsilon \frac{\partial}{\partial t}(- \frac{\partial \mathbf{A'}}{\partial t}-\mathbf{\nabla} \phi)= \mathbf{J}\]

Use the vector identity  
\[\mathbf{\nabla} \times (\mathbf{\nabla} \times \mathbf{A})=\mathbf{\nabla} ( \mathbf{\nabla} \cdot \mathbf{A}) -\nabla^2 \mathbf{A} \]
  to obtain
\[\mathbf{\nabla} ( \mathbf{\nabla} \cdot \mathbf{A}) -\nabla^2 \mathbf{A} + \epsilon \frac{\partial}{\partial t}(- \frac{\partial \mathbf{A'}}{\partial t}-\mathbf{\nabla} \phi)= \mathbf{J}\]
\[\mathbf{\nabla} \cdot \mathbf{A}\]
  is arbitrary, so we can define  
\[\mathbf{\nabla} \cdot \mathbf{A} =-\epsilon \frac{\partial \phi}{\partial t}\]
. Subtitute this into (5) to obtain
\[\mathbf{\nabla} (- \frac{\partial \phi}{\partial t}) -\nabla^2 \mathbf{A} + \epsilon \frac{\partial}{\partial t}(- \frac{\partial \mathbf{A'}}{\partial t}-\mathbf{\nabla} \phi)= \mathbf{J}\]
\[\mathbf{\nabla} (-\epsilon \frac{\partial \phi}{\partial t})=-(-\epsilon \frac{\partial \phi}{\partial t} ( \mathbf{\nabla} \phi\]
, we can write, after some simplification,  
\[\nabla^2 \mathbf{A}- \epsilon \frac{partial^2 \mathbf{A}}{\partial t^2 } = - \mathbf{J} \]

Now substitute  
\[\mathbf{\nabla} \cdot \mathbf{A} =-\epsilon \frac{\partial \phi}{\partial t}\]
\[ -\nabla^2 \phi- \frac{\partial }{\partial t}(\mathbf{\nabla} \cdot \mathbf{A})=\frac{\rho}{\epsilon}\]

\[-\nabla^2 \phi- \frac{\partial}{\partial t} (- \epsilon \frac{\partial \phi}{\partial t}) = \frac{ \rho}{\epsilon} \rightarrow \nabla^2 \phi- \epsilon \frac{\partial^2 \phi}{\partial t^2}) =- \frac{ \rho}{\epsilon} {\epsilon}\]

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