A Difference between conduction & convection current density?

Could anyone please explain the difference between the conduction current density (J=σE) and the convection current density (J=ρvd)? I really appreciate any examples or applications to further elaborate these two theories.

Conduction current is something you would witness in a conducting material, such as a metal. It refers to the movement of current in the presence of an electric field and can be described by Ohm's law.

Convection current is current flow in an insulating medium. This, however, does not follow Ohm's law.

We can define current as the electric charge passing through an area per unit volume. [itex] I = \frac{dQ}{dt}[/itex] per unit time. Current density is the amount of current flowing through a surface per unit time [itex] J = \frac{\Delta I}{\Delta S}[/itex] with [itex]I = \int J \cdot dS [/itex].

In a convection current, we have a current flowing through an insulating medium [itex]\Delta I = \frac{\Delta Q}{\Delta t} = \rho\Delta S\frac{\Delta y}{\Delta t} = \rho\Delta S u_y [/itex] where S is the surface the current is passing through, y is the length along the velocity vector, and [itex] u_y[/itex] is the velocity vector. So we can express the convection current as [itex]J_y = \frac{\Delta I}{\Delta S}=\rho\cdot u_y [/itex]. Conduction current density will describe the ability for an electric field E to propagate through a medium, controlled by the proportionality constant sigma, or conductivity. So, both describe a "current", but it might be easier to replace the word "current" with "flow; convection describes the flow through an insulating medium, and conductivity describes flow through a conducting medium.

In the case of a convecton current, do we mean the case of an "insulating medium" by "dielectric of a capacitor?"
Does it experience an electric field?
What do we mean by "it does not follow Ohm's law?"

Ohm's law is [itex] V=IR[/itex]. Voltage is an electric potential difference between two points which is therefore related to electric field. So conduction current, which is a function of electric field ([itex]J=E\sigma[/itex]), follows this law. Convection current expresses a flow due to convection, for example, a current flow due to a temperature or density differential between points. So it is not related to electric field, and therefore does not follow Ohm's law. An insulating medium would be a medium which does not conduct current (at least for a convection definition). An example of a convection current would be air in a house. If the air is heated at the bottom of the house, and the air is cooler at the top, the warm air rises due to a temperature (or density) differential. So you could define a current of air as it rose, which would be a convection current.

I think there is some confusion here. ##\mathbf{J}=\sigma\mathbf{E}## describes charge flow in an Ohmic conductor. ##\mathbf{J}=\rho \mathbf{v}_{d}## is more general and can describe charge flow in any macroscopic situation with a drift velocity, including within a conductor.

I really appreciate any examples or applications to further elaborate these two theories.

Here is a simple example. Lets say you wanted to find the drift velocity ##v_{d}## of the charges in a conductor with conductivity ##\sigma## in a uniform electric field of magnitude ##E##. Then relating the two equations gives
$$\sigma E=\rho v_{d}$$
Since ##\rho## is the number of charge carriers in a given volume, it can be expressed as
$$\rho=\frac{\rho_{m}ne}{m}$$
where ##\rho_{m}## is the density of the material, ##m## is the molecular mass of the material, ##n## is the number of free charge carriers per atom, and ##e## is elementary charge. The drift velocity is then
$$v_{d}=\frac{\sigma mE}{\rho_{m}ne}$$

Could anyone please explain the difference between the conduction current density (J=σE) and the convection current density (J=ρvd)? I really appreciate any examples or applications to further elaborate these two theories.

Note: vd is the particles' average drift velocity.

Thanks!

Regarding electric conduction and convection currents, I see it in the following way when considering moving media. The electric conduction current is defined byI =σE where σ is the electrical conductivity of the medium andE is the electrical field measured in a system which is moving with the medium. As the medium itself moves with a certain velocity vwith respect to a stationary reference system the total electrical current with respect to the stationary reference system can be written as

Regarding electric conduction and convection currents, I see it in the following way when considering moving media. The electric conduction current is defined byI =σE where σ is the electrical conductivity of the medium andE is the electrical field measured in a system which is moving with the medium. As the medium itself moves with a certain velocity vwith respect to a stationary reference system the total electrical current with respect to the stationary reference system can be written as

J = I + ρv

where ρ is the charge density in the moving medium.

This is not a correct interpretation. ##v_{d}## is the drift velocity of the charge carriers which is measured with respect to a stationary conductor.

This is not a correct interpretation. ##v_{d}## is the drift velocity of the charge carriers which is measured with respect to a stationary conductor.

I think the OP made a mistake confusing something (or he/she should indicate what the terms in J = ρvd mean and where he found this equation). Convection currents are proportional to the charge density ρ. Even if ρ = 0, you can have conduction currents.

EDIT: As far as I remember, the term convection current is used when addressing the current density of plasmas.