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 $dI=K Rd\varphi$, 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?

Answer 1

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

$$\int\limits_{\ell}{dz\ \vec{K}·\hat{n}} = \int\limits_{\ell}{dz\ \alpha} = \alpha L.$$

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:

$$\vec{B}(\vec{r})=\frac{\mu_0}{4\pi}\iint\limits_S{dA'\frac{\vec{K}·(\vec{r} -\vec{r}')}{\left|\vec{r}-\vec{r}'\right|^3}}.$$