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I know that ${\vec E}=-{\vec \nabla} V$. If a potential is scalar, I can find its electric field strength. The question is how can I know if a potential is scalar? Do all point charges result in scalar potential? Multiple point charges in a space still gives scalar potential? Can you give examples when the potential is not scalar?

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    $\begingroup$ If the potential is not a scalar, what is $\nabla V$ supposed to mean? $\endgroup$ – ACuriousMind Jun 5 '15 at 14:49
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    $\begingroup$ The electric potential is additive. Addition of two scalars results in a scalar. Is that your question? Other than that, I think you should go back to the definition of potential energy and think about it for a while. $\endgroup$ – CuriousOne Jun 5 '15 at 14:50
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    $\begingroup$ Well gradient of a vector field is a rank 2 tensor field...its just not as useful here. $\endgroup$ – Triatticus Jun 5 '15 at 16:22
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There is always a scalar potential $V$ and a vector potential $\vec A$. The vector potential always determines the magnetic field $\vec B$ but if your magnetic fields are not changing in time then you don't need the vector potential to determine the electric field $\vec E.$

You can get $\vec E = -\vec \nabla V -\frac{\partial \vec A}{\partial t}$ and $\vec B=\vec \nabla \times \vec A.$

Then if your magnetic field isn't changing in time then you can pick a vector potential that isn't changing in time and then $\frac{\partial \vec A}{\partial t}$ is zero so you didn't need it.

$\vec B=\vec \nabla \times \vec A$ gives us $\vec \nabla \cdot\vec B=0$ and $\vec E = -\vec \nabla V -\frac{\partial \vec A}{\partial t}$ and $\vec B=\vec \nabla \times \vec A$ together gives us $\vec \nabla \times \vec E =-\frac{\partial \vec B}{\partial t}$

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If you have a vector function let's say $\bar E $, the you can prove that if:

1) $\nabla \times \bar E =0 $ then $\bar E =\nabla \cdot V $ where V is a scalar function.

Assume another vector field function B,then if:

2) $ \nabla \cdot \bar B =0 $ then $\bar B =\nabla \times \bar A $ where $\bar A $ is a vector potential function

The above all of course have mathematical proves. If non of the above two statements hold, then in general you can express a vector function as a function of a scalar and a vector function. From here, for the electric and magnetic field, one begins to find different potentials to describe the fields and from here comes the gauge invariance in EM.

Hope this helps.

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The term "scalar potential" is OK when applied to electrostatics, but it is somewhat messy when applied to electromagnetism in general, as "scalar potential" is actually not a scalar function, but a component of the 4-potential. So point charges at rest create electric field that can be described by a "scalar potential", but moving point charges create magnetic field as well, which cannot be described by the "scalar potential". Let me also note that the "vector potential" $\mathbf{A}$ can always be used to describe arbitrary electromagnetic field using the so called Weyl gauge.

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