Definition of a line charge with Dirac delta function Is the following statement correct for a line charge distribution $λ(x)$?
$$ρ(\mathbf r)=λ(x)δ(y)δ(z)$$
If yes - what does it say?
 A: 
$$\rho(\mathbf{r})=\lambda(x)\delta(y)\delta(z)$$
describes a charge density in the form of a (possibly infinite, depends on what your allowed x values are in the system) line in 3D space, where $\lambda(x)$ is the linear charge density as a function of x. The delta functions indicates the charge density is concentrated at one point in the yz plane, but extended in the x axis
For a 1D description of the above, you will simply use
$$\rho(x)=\lambda(x)$$
A: The factor δ(y) indicates that the charge distribtution is non-zero only for y=0, i.e. on the zx plane; likewise, δ(z) that it is non-zero only on the xy plane. Thefore, the product δ(y)δ(x) Indiactes that the charge distribution in non-zero on the intersection of zx and xy planes, i.e. the x-axis. Then, the function λ(x) defines the actual form of the charge density on the x-axis. Not that, since, δ functions are distributions in the mathematical sence, this charge distrubtion makes sence only under 2-D integrals over y and z for a given x. Such a proccedure will reduce ρ(r) to λ(x).
