Is it necessary for EM fields to be dependent & co-exist in static conditions? I was having a discussion today with one my colleagues in the lab about the independence and co-existence of EM fields.$$$$
My argument: 


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*In time-varying fields: EM fields are necessary dependent, and they have to co-exist. They get coupled through the time derivatives that appears in Maxwell's equations.

*In static fields: EM fields are independent and each can exist separately or together. 
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His argument:


*

*In time-varying fields: (SIMILAR TO MINE)

*In static fields: EM fields must co-exist. (He is mentioning the spin of the electron about its axis of rotation, and the rotation of the electron around the nucleus and how that induces a magnetic field.)$$$$


Examples to clarify my question:


*

*A point charge at rest (for simplicity), have a static E-field around it pointing in the radial direction. (is there a magnetic field?)

*A DC current flowing in a conductor produces an M-field around the conductor. (is there an electric field?)

 A: If no charge is moving, there is no magnetic field. A point charge at rest has only an electric field, from "its" point of view.
However, electric and magnetic fields are not seperate, since someone moving with respect to the resting charge would see a magnetic field due to the behaviour of the fields under Lorentz transformations. You may (for some situations, e.g. the point charge) find frames where the magnetic or the electric field vanishes, but that is of little consequence.
A: One can solve Maxwell's equations not just when $E$ and $B$ are dependent, but when they are constant multiples of each other---even in non-static cases.
For example, if $E=-B$, then $-\nabla\cdot{}E=\nabla\cdot{}B=0$ so there are no sources. Faraday's Law says $\frac{\partial{E}}{\partial{t}}=\nabla\times{}E$ and Maxwell-Ampere says $\frac{\partial{E}}{\partial{t}}+J=-\nabla\times{}E$. But $\frac{\partial{E}}{\partial{t}}=\nabla\times{}E$ is a first order elliptic equation, and given initial data, can be solved forward in time at least for a short distance (conceivably singularities might turn up in finite time). Maxwell-Ampere forces $J=-2\nabla\times{}E$, so some sort of interesting (and time-changing) current density is necessary to maintain $E=-B$. But physically reasonable or not, solutions certainly exist.
From a more formal perspective, solving $\frac{\partial{E}}{\partial{t}}=\nabla\times{}E$ together with $\nabla\cdot{}E=0$ is precisely the same as solving $d\eta=0$ where $\eta$ is a self-dual 2-form on $\mathbb{R}^4$ with its Euclidean metric. Given any three pluriharmonic functions on $\mathbb{C}^2$, one can construct a solution to $d\eta=0$, $\eta\in\bigwedge{}^+$; thus many non-trivial, global, smooth solutions exist, and also many solutions with singularities.
In the static case, assume $E=fB$ for some function $f$. Maxwell's equations reduce to $\nabla\times{}E=0$ and $\nabla\cdot{}E+\left<\nabla\log\,f,\,E\right>=0$. This is an elliptic first-order system for $E$, so given any $f$ one can solve for $E$, at least if the singularities of $\log\,f$ are not too bad. Again one has many, many solutions, although source and current density must be non-zero unless $f=const$.
