## Maxwell's Equations in Potential Form

Maxwell's Laws are:
$\mathbf{\nabla} \cdot \mathbf{E}=\frac{\rho}{\epsilon}$
(1)
$\mathbf{\nabla} \times \mathbf{E}=- \mu \frac{\partial H}{\partial t}$
(2)
$\mathbf{\nabla} \cdot \mathbf{H}=0$
(3)
$\mathbf{\nabla} \times \mathbf{H}=- \epsilon \frac{\partial E}{\partial t}+ \mathbf{J}$
(4)
$\mathbf{H}$
is solenoidal since
$\mathbf{\nabla} \cdot \mathbf{H}=0$
hence we can write
$\mathbf{H} = \mathbf{\nabla} \times \mathbf{A}$
for some vector
$\mathbf{A}$
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
$\phi$
.
Transform
$\mathbf{A}, \phi$
.

$\mathbf{A'}= \mathbf{A} + \mathbf{\nabla} \psi , \phi'=\phi- \frac{\partial \psi}{\partial t}$

;
$\mathbf{A}, \phi$
.
also satisy
$\mathbf{H} = \mathbf{\nabla} \times \mathbf{A;}$
and
$\mathbf{E}=- \frac{\partial \mathbf{A'}}{\partial t}-\mathbf{\nabla} \phi'$

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
$\mathbf{H} = \mathbf{\nabla} \times \mathbf{A}$
and
$\mathbf{E}=- \frac{\partial \mathbf{A'}}{\partial t}-\mathbf{\nabla} \phi'$
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
$\mathbf{\nabla} ( \mathbf{\nabla} \cdot \mathbf{A}) -\nabla^2 \mathbf{A} + \epsilon \frac{\partial}{\partial t}(- \frac{\partial \mathbf{A'}}{\partial t}-\mathbf{\nabla} \phi)= \mathbf{J}$
(5)
$\mathbf{\nabla} \cdot \mathbf{A}$
is arbitrary, so we can define
$\mathbf{\nabla} \cdot \mathbf{A} =-\epsilon \frac{\partial \phi}{\partial t}$
. Subtitute this into (5) to obtain
$\mathbf{\nabla} (- \frac{\partial \phi}{\partial t}) -\nabla^2 \mathbf{A} + \epsilon \frac{\partial}{\partial t}(- \frac{\partial \mathbf{A'}}{\partial t}-\mathbf{\nabla} \phi)= \mathbf{J}$
(5)
Since
$\mathbf{\nabla} (-\epsilon \frac{\partial \phi}{\partial t})=-(-\epsilon \frac{\partial \phi}{\partial t} ( \mathbf{\nabla} \phi$
, we can write, after some simplification,
$\nabla^2 \mathbf{A}- \epsilon \frac{partial^2 \mathbf{A}}{\partial t^2 } = - \mathbf{J}$

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

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