Does $\int_C B \cdot dr$ have any physical meaning when $C$ is *not* a closed loop?

Question

Question. Does the line integral $\int_C \textbf{B} \cdot d\textbf{r}$ have any physical meaning when $C$ is not a closed loop? That is, I know that Ampere's law asserts that in the case of a static electric field, the line integral of the magnetic field around a closed loop is proportional to the electric current flowing through the loop. But I'm wondering if the line integral has any meaning when the path is just an arbitrary path through the magnetic field, and specifically not a closed loop?

Context. I am a mathematics grad student, TA'ing a vector calculus course. My professor has assigned a recitation activity where we are to calculate several such line integrals with several paths through the magnetic field generated by a wire. So I'm just wondering if there is any physical meaning to such things, as that will help illustrate the purpose of these integrals!

Thanks!

Answer 1

Not really.

I mean, you can force some artificial meaning onto it: for example, if you need to calculate an integral over a stick perpendicular to the current carrying wire, $I_1$, you can write something like that: $I_1 + I_2 = \mu_0 J$, where $J$ is current. $I_2$ is very easy to calculate, it is $\frac{2\pi-\theta}{2\pi}\mu_0 I$, therefore you find $I_1 = \frac{\theta}{2\pi} \mu_0 $.

It can be generalized, by the way.

You can use such integrals as auxiliary ones. But still, it's quite artificial. It's like if you, for some reason, need to calculate one complex integral of magnetic field, you can close the loop and calculate the easier one. But question rises why would you calculate the integral in the first place.