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Current, $I$, is generalised as: $$I=\iint_{A}^{} \vec{J}\cdot d\vec{A}$$

I know that current density always points in the direction of flow of positive charge. I wonder if the infinitesimal element, $d\vec{A}$, always points in the same side as the current density. Also, I wonder if current can have a negative quantity and what does it mean?

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  • $\begingroup$ The area element points out of a surface. A negative current density would indicate (conventional) current flow into a surface. $\endgroup$ – Rob Jeffries May 25 '15 at 15:25
  • $\begingroup$ @Rob How about if it is a current flowing in a wire? Thanks $\endgroup$ – James Harroid May 25 '15 at 15:29
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    $\begingroup$ @RobJeffries The area element points out of a closed surface. For an open surface (cross-section of a wire, for example), the direction of positive orientation is chosen arbitrary, but of course once chosen it cannot be changed. So, yes, current can be negative, depending on whether the net current flows along or against the chosen direction of orientation. $\endgroup$ – garyp May 25 '15 at 16:25
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You just choose a direction for $\vec A$. It can be at random. It is not important. You just have to remember the choice you make. Then if the current turns out to be negative, you know that the current flows the other way.

That is all. Negative current just means that it flows opposite to whatever area direction choice you made.

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There are several ways to define the direction of a surface $A$. If the surface is closed the vector $d \vec A$ by convention points out of the surface. If the surface is open then the boundary of the surface is some curve, let us call it $C$ the direction of $d \vec A$ is then given to point in the direction of the right hand rule. If you have a negative current, it simply means that the current points in the opposite direction to $d \vec A$. For a closed surface this means that charge will be entering it (if positive) or leaving it (if negative). Overall the charge in the surface will increase. For an open surface a negative current means that it is flowing through the surface in the opposite direction. In the case of a wire, a negative current indicates that the current is in the opposite direction to the direction you have assigned $d \vec A$

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  • $\begingroup$ The Right Hand Rule has no bearing on this question. The choice of orientation is arbitrary for open surfaces. $\endgroup$ – garyp May 25 '15 at 16:26
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I think, the important point is that for a given current density the surface is not fixed. You need to choose the surface yourself in order to define the current, that is, for a given current density you can get different currents for different choices of the surface . If dealing with a circuit, for example, the surface might be any surface being pierced by the wire (not necessarily orthogonal to the wire, the scalar product in the integration takes care of that).

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  • $\begingroup$ Does this answer either of the questions? I think they were "what defines the direction of dA?" and "what does a negative current mean?" $\endgroup$ – NoethersOneRing Apr 29 '17 at 21:56
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    $\begingroup$ The question was "I wonder if the infinitesimal element, dA, always points in the same side as the current density." My answer is: No, it doesn't because "for a given current density the surface is not fixed. You need to choose the surface yourself in order to define the current". $\endgroup$ – Photon Apr 30 '17 at 5:46

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