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Mathematics for Physics

41 Electric Currents

A net flow of electric charge through a conductor or region of space that is used to produce magnetic fields and to transport charge between circuit elements, where a conductor is a material whose charges are free to move.

Current as flux of \(\mathbf{J}\). The current through a surface is the flux of the current density. Current density is the current per unit area. This principle is used to pass from a volume description of moving charge to the current in a wire.

The current as flux of current density is

\[ I = \displaystyle\int\mathbf{J}\cdot d\mathbf{a} \]

where

  • \(I\) is the electric current.
  • \(\mathbf{J}\) is the volume current density.
  • \(d\mathbf{a}\) is the vector area element.

The line-current element. A line current is the current along a thin wire. This principle is used to write the Biot–Savart and Ampère laws for circuits.

The line-current element is

\[ I\,d\mathbf{l} \]

where

  • \(I\) is the current in the wire.
  • \(d\mathbf{l}\) is a directed length element along the wire.

The local form of Ohm’s law. In a linear conductor the current density is proportional to the electric field. This principle is used to relate microscopic flow to Ohm’s law.

Ohm’s law in local form is

\[ \mathbf{J} = \sigma\mathbf{E} \]

where

  • \(\mathbf{J}\) is the current density.
  • \(\sigma\) is the conductivity.
  • \(\mathbf{E}\) is the electric field.

Note: Also called current.