📂Electrodynamics

Current and Current Density

Definition¹

The electric current is defined as the amount of charge that passes through a given point in a conductor per unit of time, denoted by $I$. Thus, a negative charge moving to the left and a positive charge moving to the right constitute an electric current of the same sign.

The amount of Coulomb passing per unit of time is called ampere.

$$1 [A] = 1 [C/s]$$

Explanation

Ampere was French, and the actual pronunciation is closer to [am-pear]. Hence, though it is called the Ampère’s law, when referring to units, it must be referred to as ampere.

The notation $I$ originates from the first letter of the intensity of current.

Line Current Density

The above figure shows a charge with a line charge density of $\lambda$ moving along a conductor at speed $\mathbf{v}$. Since distance = speed x time, the unit length is $v\Delta t$. The amount of charge within a unit length is obtained by multiplying the unit length by the line charge density.

$$\Delta q=\lambda v \Delta t$$

The electric current is the amount of charge that passes through in a unit of time, so the amount of charge passing through point $P$ during $\Delta t$ is:

$$I=\dfrac{\Delta q}{\Delta t}=\dfrac{\lambda v \Delta t}{\Delta t}=\lambda v$$

The electric current is a vector, so if we include its direction, it is represented as follows:

$$\mathbf{I}=\lambda \mathbf{v}$$

When electric current flows along a conductor, its direction is clear (parallel to the conductor) and need not be specifically mentioned. However, when dealing with electric currents flowing on surfaces or through volumes, the direction must be clearly stated. The Lorentz force experienced by a conductor through which electric current flows due to an external magnetic field $\mathbf{B}$ is:

$$\mathbf{F}_{\text{mag}}=\int (\mathbf{v} \times \mathbf{B} ) dq=\int (\mathbf{v} \times \mathbf{B} ) \lambda dl=\int (\mathbf{I} \times \mathbf{B}) dl$$

Here, since the directions of $\mathbf{I}$ and $d\mathbf{l}$ are the same:

$$\mathbf{F}_{\text{mag}} = \int I (d\mathbf{l} \times \mathbf{B})$$

Usually, since the current flowing through a conductor is constant, it can be taken outside the integral:

$$\mathbf{F}_{\text{mag}}=I \int (d\mathbf{l} \times \mathbf{B})$$

Surface Current Density

The current flowing on a surface is explained as surface current density $\mathbf{K}$. It is the current that flows across a unit width of length and is expressed as follows mathematically:

$$\mathbf{K}=\dfrac{d \mathbf{l}} {dl_\perp}$$

To put this concept into easier terms, since $\mathbf{I}=\dfrac{d\mathbf{q} }{dt}$:

$$\dfrac{d \mathbf{I} }{dl_{\perp}}=\dfrac{d^2 \mathbf{q}}{dl_{\perp} dt}$$

Therefore, the surface current density is the amount of charge that passes per unit time, per unit width of length. When the surface charge density is $\sigma$ and the velocity of charge is $\mathbf{v}$, the surface current density is:

$$\mathbf{K}=\sigma \mathbf{v}$$

The magnetic force experienced by a surface current due to an external magnetic field is:

$$\mathbf{F}_{\text{mag}}=\int(\mathbf{v}\times \mathbf{B})\sigma da=\int (\mathbf{K} \times \mathbf{B})da$$

This is the formula for the current seen earlier with the current $\mathbf{I}$ replaced with surface current density $\mathbf{K}$.

Volume Current Density

Likewise, when current flows through a certain space, it is explained as volume current density $\mathbf{J}$. It is the current that flows per unit area and is mathematically expressed as:

$$\mathbf{J}=\dfrac {d\mathbf{I}} {da_{\perp}}$$

Therefore, conversely, the current $I$ passing through surface $\mathcal{S}$ can generally be represented as follows:

$$I = \int_{\mathcal{S}}J da_{\perp} = \int_{\mathcal{S}}\mathbf{J}\cdot d\mathbf{a}$$

Then, according to the Divergence Theorem, the total amount of charge that exits volume $\mathcal{V}$ is as follows:

$$\oint_{\mathcal{S}}\mathbf{J}\cdot d\mathbf{a} = \int_{\mathcal{V}} (\nabla \cdot \mathbf{J}) d \tau$$

Similarly, since $\dfrac {d\mathbf{I}} {da_{\perp}}=\dfrac{d^2 \mathbf{q} } {da_{\perp}{dt}}$, the volume current density is the amount of charge that passes per unit of time, per unit area. If the volume charge density is $\rho$ and the speed of the charge is $\mathbf{v}$, the volume current density is:

$$\mathbf{J}=\rho \mathbf{v}$$

The magnetic force experienced by a volume current is:

$$\mathbf{F}_{\text{mag}}=\int (\mathbf{v} \times \mathbf{B} )\rho d\tau = \int (\mathbf{J} \times \mathbf{B} ) d\tau$$