Question

What is current density? Discuss.

Hint:

Remember the key difference: Current (\(I\)) is a scalar describing the total flow through a surface. Current Density (\(\vec{J}\)) is a vector describing the flow at a specific point. Think of it like mass vs. density.

Answer 1

Step 1: Understanding the Concept:
While electric current (\(I\)) is a scalar quantity that tells us the total amount of charge flowing through a cross-section of a conductor per unit time, it doesn't describe how the flow is distributed across that cross-section. Current density (\(\vec{J}\)) provides this microscopic description. It is a vector pointing in the direction of the flow of positive charge.
Step 2: Key Formula and Discussion:
Magnitude:
If a current \(I\) flows uniformly through a conductor with a cross-sectional area \(A\) perpendicular to the flow, the magnitude of the current density is:
\[ J = \frac{I}{A} \] Vector Nature:
Current density is a vector, \(\vec{J}\). The total current \(I\) through a surface \(S\) is the flux of the current density vector through that surface:
\[ I = \int_S \vec{J} \cdot d\vec{A} \] This integral form is more general and accounts for non-uniform current distribution and surfaces that are not perpendicular to the current flow.
Relation to Drift Velocity:
Current density is related to the microscopic properties of the charge carriers. If a conductor has \(n\) charge carriers per unit volume, each with charge \(q\) and moving with an average drift velocity \(\vec{v}_d\), the current density is given by:
\[ \vec{J} = nq\vec{v}_d \] For electrons, \(q = -e\), so \(\vec{J} = -ne\vec{v}_d\). Since the conventional current direction is opposite to the electron drift velocity, \(\vec{J}\) is in the direction of conventional current.
Relation to Ohm's Law (Microscopic Form):
Current density is directly related to the electric field (\(\vec{E}\)) within the conductor. This relationship is the microscopic or point form of Ohm's Law:
\[ \vec{J} = \sigma \vec{E} \] where \(\sigma\) is the electrical conductivity of the material (\(\sigma = 1/\rho\), where \(\rho\) is the resistivity). This equation states that the current density at a point is directly proportional to the electric field at that point.
Step 3: Final Answer:
Current density is the vector measure of electric current flow per unit area at a point. It is fundamental in relating macroscopic quantities like current to microscopic quantities like drift velocity and the local electric field.