# Current and current density

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?

• The area element points out of a surface. A negative current density would indicate (conventional) current flow into a surface. – Rob Jeffries May 25 '15 at 15:25
• @Rob How about if it is a current flowing in a wire? Thanks – James Harroid May 25 '15 at 15:29
• @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. – garyp May 25 '15 at 16:25

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.
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 $d \vec A$