# What makes current density a vector?

The current density vector is defined as a vector whose magnitude is the electric current per cross-sectional area at a given point in space, its direction being that of the motion of the positive charges at this point.

this is a quote from wikipedia and if I'm not wrong, it says that: $$||J|| = \frac{I}{A} = \sqrt{J_x^2+J_y^2+J_z^2}$$

Now, how does it make sense that current density is a vector when both current and area have no relation to vectors?

edit: I am looking for a mathematical definition. I get that the direction of current flow has a direction so it can be a vector, but how can you get a vector from two scalars? and btw I am very unexperienced with the topic so sorry if I get some stuff wrong.

• you should at least quote the whole sentence, which continues with ", its direction being that of the motion of the positive charges at this point. "
– fqq
Jun 28 '20 at 14:42
• Oriented areas do have a lot to do with vectors. Jun 28 '20 at 14:45
• Jun 28 '20 at 14:50

Current density $$\bf J$$ is related to current $$I$$ by $$dI={\bf J\cdot dA}$$. To get the current through a finite area, integrate $$I$$ over the area.
For a wire, you can write $$\textbf{I}$$ (which is a vector in this situation) as $$I\,d\textbf{l}$$ where $$d\textbf{l}$$ is a vector that is tangent to the wire, in the direction of the current. So $$\textbf{J}$$ is a vector because the current itself is a vector. Normally, we don't need current in vector form if it is flowing through a wire, since there are only two directions it can flow in. But current density is used to analyze more complicated two or three dimensional situations, in which case current can flow in all directions, it has to be a vector.