- #1

olgerm

Gold Member

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## Main Question or Discussion Point

These are Maxwell´s equations in potential formulation:

∇

∇

In coulomb gauge in every point and at any time DIV(A)=[PLAIN]https://upload.wikimedia.org/math/4/4/1/44131cc26bd9db464d0edb7459ccca84.png. [Broken] Am I right?

Where could I find Maxwell´s equations in terms of potentials without vector operator?

How must ROT (same as curl) be generalized to make the equations describe EM-field in D-dimensional space equally with these equation

##\begin{cases}

& \sum_{i=1}^D(\frac{\partial E_i}{\partial x_i})=\rho \frac{1}{{\epsilon_0}} \\

& \frac{\partial E_a}{\partial t}=\sum_{i=1}^D(\frac{\partial B_{[i;a]}}{\partial x_i})-J_a \\

& \frac{\partial B_{[a;b]}}{\partial t}=\frac{\partial E_b}{\partial x_a}-\frac{\partial E_a}{\partial x_b}\\

& \frac{\partial B_{[a;b]}}{\partial x_c}+\frac{\partial B_{[b;c]}}{\partial x_a}+\frac{\partial B_{[c;a]}}{\partial x_b}=0

\end{cases}##

,which are in terms of E and B?

φ is electripotentialfield.

E is electricvectorfield.

A is magneticpotentialvectorfield.

B is magneticvectorfield.

ρ is electriccharge density.

∇

^{2}φ = DIV(grad(φ)) . Am I right?∇

^{2}A = ROT(ROT(A))=ROT(B)=grad(DIV(A))-Laplace(A) . Am I right?In coulomb gauge in every point and at any time DIV(A)=[PLAIN]https://upload.wikimedia.org/math/4/4/1/44131cc26bd9db464d0edb7459ccca84.png. [Broken] Am I right?

Where could I find Maxwell´s equations in terms of potentials without vector operator?

How must ROT (same as curl) be generalized to make the equations describe EM-field in D-dimensional space equally with these equation

##\begin{cases}

& \sum_{i=1}^D(\frac{\partial E_i}{\partial x_i})=\rho \frac{1}{{\epsilon_0}} \\

& \frac{\partial E_a}{\partial t}=\sum_{i=1}^D(\frac{\partial B_{[i;a]}}{\partial x_i})-J_a \\

& \frac{\partial B_{[a;b]}}{\partial t}=\frac{\partial E_b}{\partial x_a}-\frac{\partial E_a}{\partial x_b}\\

& \frac{\partial B_{[a;b]}}{\partial x_c}+\frac{\partial B_{[b;c]}}{\partial x_a}+\frac{\partial B_{[c;a]}}{\partial x_b}=0

\end{cases}##

,which are in terms of E and B?

φ is electripotentialfield.

E is electricvectorfield.

A is magneticpotentialvectorfield.

B is magneticvectorfield.

ρ is electriccharge density.

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