Current Density

Also called J

Current density is the electric current per unit cross-sectional area at a point in a conductor, written J. It is a vector, it points along the direction positive carriers move, and its SI unit is the ampere per square metre.

Current tells you how much charge passes a whole cross-section each second. Current density tells you how that flow is distributed inside the material, which is what you need when the conductor is not uniform.

AP Physics C: Electricity and Magnetism builds it in three steps. EK 11.1.A.2.i relates it to carrier motion, with the relevant equation J⃗=nqv⃗d\vec{J} = nq\vec{v}_d. EK 11.1.A.2.ii states plainly that current density is a vector quantity. EK 11.1.A.2.iii says a potential difference across a conductor creates a field within it proportional to the resistivity and the current density:

E⃗=ρJ⃗\vec{E} = \rho \vec{J}

That form is printed on the C: E&M sheet, and it is the microscopic statement of Ohm's law: the material property ρ\rho links a field at a point to a flow at that point, with no reference to the length or shape of the wire.

Getting the current back. EK 11.1.A.3 says that where the current density is given as a function, the total current follows by integrating it over the area. The sheet prints that too:

I=∫J⃗⋅dA⃗I = \int \vec{J} \cdot d\vec{A}

The dot product matters. Only the component of J⃗\vec{J} through the surface counts, exactly as with electric flux.

Note that no AP sheet prints the nqv⃗dnq\vec{v}_d form, so that one is reconstructed from the CED rather than read off the booklet.

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