What does B.dl indicate in Ampere's Law

[sawer]

I know that$$\oint \vec E \cdot \vec{dS}$$ in Gauss Law indicates electric flux.
$$\oint \vec E \cdot \vec{dS} = \frac{Q_{enc}}{\varepsilon_0}$$

But what does B.dl indicate in Ampere's Law?
$\oint \vec{B} \cdot \vec{dl}$ = ??

 

[vanhees71]

Such a line integral around a closed loop is the circulation of the vector field, here the magnetic field. The fundamental laws are the Maxwell equations in local form, and the Ampere-Maxwell Law reads (written in terms of the macroscopic laws in Heaviside-Lorentz units)
$$\vec{\nabla} \times \vec{H}-\frac{1}{c} \partial_t \vec{D}=\frac{1}{c} \vec{j}.$$
The integral form follows from integrating over a surface and using Stokes's integral theorem to change the curl part into a line integral along the boundary of the surface,
$$\int_{\partial F} \mathrm{d} \vec{r} \cdot \vec{H} = \frac{1}{c} \int_F \mathrm{d}^2 \vec{f} \cdot (\vec{j}+\partial_t \vec{D}).$$
For the static case, where $\partial_t \vec{D}=0$, the right-hand side is the total electric current running through the surface under consideration.

For the non-static case, it's misleading to interpret the $\partial_t \vec{D}$ term as "source" of the magnetic field. Here you need the full (retarded) solutions of Maxwell's equations to express the electromagnetic field in terms of their sources, which are the electric charge and current densities. See, e.g.,
https://en.wikipedia.org/wiki/Jefimenko's_equations

 

[gleem]

Or in short
$$\oint \mathbf{B}\cdot \mathbf{dl}=\mu _{0}\int _{S}\boldsymbol{\mathbf{J\cdot}}\boldsymbol{dA}$$
The line integral of B around any loop is equal to the total current crossing any surface bounded by that loop at least for nonmagnetic materials and J >> ∂D/∂t

 

[sawer]

Such a line integral around a closed loop is the circulation of the vector field, here the magnetic field.

Is there a special name for that, like in electric case, gauss law is equal to "electric flux".

$\oint \vec{B} \cdot \vec{dl}$ is not equal to magnetic flux, right?

 

[sawer]

The line integral of B around any loop is

Is there a special name for that? (Like electric flux or magnetic flux etc...)

 

[gleem]

I do not know of a special name for it.

However the related expression
$$\oint \mathbf{H\cdot dl}$$
is called the magnetomotance.

 

[robphy]

"circulation"'
(might be a neat website: http://mathinsight.org/line_integral_circulation )