In Gauss's law we have continuous charge densities for lines, surfaces and volumes. However, regarding current density, Wikipedia, only have a surface current density. Don't we also have line and volume current densities for Ampère's circuital law?
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The charge density that appears in Gauss's Law is a volume charge density.
The singular charge densities used in relation to Gauss's Law are mathematical models of volume charge densities for situations where the charge distribution has a small extent in one or more dimension. Everything in actual fact is a volume charge density. Similarly, every current density is a volume current density defined over a surface.
One could define current densities that have small extents in one or more dimension. However, the current density in Ampere's Law is a volume current density defined at a surface.
garyp
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Current density and charge density are two different quantities. Charge density is defind for line, surface and volume but current density is defined as curret divided by area
Explorer
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However, regarding current density, Wikipedia, only have a surface current density. Don't we also have line and volume current densities for Ampère's circuital law?
The current density $\mathrm{J}$ is not a surface current density in that sense (line, surface, and volume charge densities are scalars while current density is a vector). That is, current density isn't the current density on a surface but rather through a surface normal to $\mathrm{J}$. It has units of $\mathrm{\frac{A}{m^2}}$ so that the flux of the current density through some surface gives the current through that surface.
In fact, there is a surface current density $\mathrm{K}$ (see, for example, Surface current and current density) which is used for currents confined to a surface and has units of $\mathrm{\frac{A}{m}}$ since, in this case, the 'surface' the current is through is a line.
Alfred Centauri
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