Line current density into a surface integral

Question

I'm struggling with understanding how one can generally use the a known line current density $vec{K}$ of a single loop of current in order to calculate the magnetic field of an object with a surface, like a cylinder. In other words, if I know $vec{K} = alpha hat{varphi}$ on some cylinder, meaning it is like a solenoid with loops of current around it, how can one use $vec{K}$ to get a measure for the current around the entirer cylinder? If I'm not wrong, in this case $dI=K Rdvarphi$, and not $dI = Kdz$ because the current goes in a circle and not up. Am I right? How does one generally choose the direction in which if one multiplies the line current density, one gets the current?

Puk · Answer

When we speak of current, we always speak (sometimes implicitly) of the current flowing through some oriented surface (i.e. a surface with a choice of normal vector direction). The answer you are looking for will depend on the choice of this surface in general.
For example, you might choose a flat surface intersecting the entire cylinder at $varphi = 0^circ$, with the normal vector $hat{n}$ pointing along $hat{varphi}$. This surface intersects the cylinder along a straight line $ell$ at $r = R$ and $varphi = 0^circ$ that is as long as the cylinder (say $L$). The current is
$$intlimits_{ell}{dz vec{K}·hat{n}} = intlimits_{ell}{dz alpha} = alpha L.$$
Update: When calculating the magnetic field distribution, the "total" current is not necessarily useful, at least not always. You would typically use the surface integral version of the Biot-Savart law:
$$vec{B}(vec{r})=frac{mu_0}{4pi}iintlimits_S{dA'frac{vec{K}·(vec{r} -vec{r}')}{left|vec{r}-vec{r}'right|^3}}.$$

Line current density into a surface integral

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