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Upon reading Griffiths' Introduction to Electrodynamics the author equates current density $ \mathbf J $ with the product of the volumetric charge density and velocity: $$ \mathbf J = \rho \mathbf v. $$ (He presents a similar relationship between $\mathbf K$ and $\sigma$). Question: How exactly is $\mathbf v$ defined? At first I thought that $\mathbf v$ was a new field, independent of $\rho$ But this cannot be the case as result of the continuity equation: $$ \nabla \cdot (\rho \mathbf v) = - \frac{\partial \rho}{\partial t}. $$ I believe that the continuity equation alone is not enough to determine $\mathbf v$. I also suspect that one must also impose $|\mathbf v| < c$ , where $c$ is the speed of the EM waves. Is this enough to determine $\mathbf v$? What additional conditions on $\mathbf v$ are necessary to define it? This seems like a problem in fluid or continuum mechanics, an area in which I have no experience.

Thanks in advance.

armoredchihuahua
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2 Answers2

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Electric current can be defined on different levels. Firstly, let me restrict the discussion to classical transport, since things may get rather different in quantum domain, and bridging between classical and quantum descriptions of transport phenomena can be somewhat complex. (E.g., see a discussion in this answer.)

Boltzmann equation
On the classical level, the most complete description of electron transport is obtained using non-equilibrium statistical physics, that is Boltzmann equation for the destribution function of momenta/velocities and positions: $f(\mathbf{v}, \mathbf{r},t)$: $$ \mathbf{j}=\int d^3\mathbf{v} d^3\mathbf{r}\mathbf{v}f(\mathbf{v}, \mathbf{r},t). $$ Note that the distribution function is usually normalized to the number of particles, so the current is that of the whole system: $$ \int d^3\mathbf{v} d^3\mathbf{r}f(\mathbf{v}, \mathbf{r},t)=N $$

Hydrodynamic description
Hydrodynamic description is a simplification, which is obtained from the Boltzmann equation under the assumption of local thermodynamic equilibrium: that is, we assume that temperature, concentration, potentials and other mean and external variables vary slowly in space, so at every point the system can be described by a Boltzmann distribution. (See, e.g., the references in this answer for more details.)

We then arrive to usual hydrodynamic equations, among which we have Navie-Stokes equation and the continuity equation $$ \nabla\cdot \mathbf{\rho\mathbf{v}}+\frac{\partial \rho}{\partial t}=0 $$ Thus, the current can be defined naturally as $\mathbf{j}=\rho\mathbf{v}$, where it is understood that the velocity and the concentration fields are related via the hydrodynamic equations.

Drude approach
Finally, the ad-hoc approach adopted in elementary texts is simply writing the current as $\mathbf{j} = \rho\mathbf{v}$, where the velocity is assemed some kind of mean "drift" velocity, determined by Newton-like equations of motion (see, e.g., this derivation). This intuitive approach was historically rather successful - see Drude model. It can be seen as a linearized version of the hydrodynamic model, which assumes constant electron density.

Roger V.
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V, here is the velocity of charges or the drift velocity $V_d$ for electrons and other charged particles. The average velocity of charged particles in a material due to an electric field is known as drift velocity.

It can be represented as

$$V_d={\frac I {neA}}$$

Where I$$ is the current through the cross section,

$n$ is the free electron density per cubic area,

$e$ is needless to say, charge of the charged particle, and

$A$ is the area of the cross-section.

Please upvote if it helps.

Ritzthephysibeast
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