Faraday's Induction Law Notation

Question

I am confused as to the notation used in a course I'm taking on physical optics.

I have presented 2 variants of Faraday's Law, combined with the full set of Maxwell's equations.

The first formulation for Faraday's Law is the following: $$\oint_C E \cdot dC = - \int\int\ \frac{\partial B}{\partial t} \cdot dS$$ where I believe the subscript C refers to the curve C and the dC term in the integral refers to infinitesimal tangent vectors to C, while the dS term refers to normal vectors to the surface.

My misunderstanding here is why do we have two integrals if there is a single variable we are integrating over in the right-hand side.

Also, there is a formulation of Gauss' Electric Law that I'm confused about: $$ \oint\int_S \mu B \cdot dS = 0 $$

with S this time a closed surface and $\mu$ the magnetic permeability of the medium.

Was S before not a closed surface? Why did we use a contour integral for the first integration now and why didn't we previously?

Finally, there is another formulation of Faraday's Law that seems to be equivalent, but I have no idea why: $$ \nabla \times E = - \mu \frac{\partial H}{\partial t} $$

I do not understand intuitively what the "curl" is so that I can brain check that the formulas mean the same thing.

Answer 1

Faraday's law is the statement that "the electromotive force around a closed path is equal to the negative of the time rate of change of the magnetic flux enclosed by the path".

In its integral form, you are equating walking along a circle, adding up the E field component along that circle C with the magnetic field through the surface traces by C. Here is called S. So for the RHS, you go to all the places on the surface, examine the B field piercing that surface, and it will equal the E-field sum of the LHS.

The differential form is indeed equivalent, but instead of tracing out a large surface, you are zooming in the differential element, an infinitely small surface surrounded by an infinitely small path.

Curl, along with is dual, divergence, is a convenient way to decompose a field. Taking a differential element, the divergence measures part of the field that pierces the element, and the curl measures the part of the field that flows circularly around the element.

Lastly, I've never seen Gauss's electric law in that form. But it is the statement that if you add up the perpendicular electric field component along and surface, you will gain information about the charge content within the surface. If you want me to comment on that specific form you presented, please give a reference, so that we'll have more context.

You can ask for more clarification in the comments if you need further explanation.