# Induced Electric Fields and Faraday's Law

I'm an undergraduate student and we just covered Faraday's law. However, I am still confused conceptually about a few things:

1. Faraday's law states that $$\oint_{\partial\Sigma} \vec{E} \cdot \vec{dl} = -\frac{d}{dt} \int_\Sigma \vec{B} \cdot \vec{dA}$$. What I interpret this to be is that we choose a loop $$\partial \Sigma$$ and a surface $$\Sigma$$ bound by this loop, determine the rate change of magnetic flux through this surface , and this equals the line integral on the left. However, what necessitates that the rate change of flux through every surface bound by this loop is the same? Consider a scenario where I have a magnetic field which changes as $$|B| \propto \frac{1}{t}$$ in a certain region, and falls off as $$\frac{1}{t^2}$$ every where else; then if I have a loop completely within the first region, I may either chose a surface entirely within this first region, or I might chose one that is large enough to encompass both. Intuitively, I cannot see how $$\frac{\partial B}{\partial t}$$ and hence rate change of flux is the same in both cases. Hence I expect the left hand side to be different, which cannot be as it is unchanged.
2. We learnt in class that the direction in which we integrate around the loop, that is, the direction of $$\vec{dl}$$ is such that it forms a right-hand system with $$\vec{dA}$$ (This is where the surface chosen is uniform so that the $$\vec{dA}$$ points uniformly in the same direction.) What if I chose instead my surface non uniformly so that my $$\vec{dA}$$ vectors point in all sorts of directions? Then (conceptually) how would I fix the direction of $$\vec{dl}$$?

We don't really cover the differential forms of the laws in my course, so I don't really know if there's an answer there. I'm just looking for physical insight more than anything.

Thanks for any help. =)

1. The reason the rate change of flux through every surface bound by this loop is the same is in the Gauss's law for magnetism. Since the flux of $${\bf B}$$ through any closed surface is zero, it is enough to take two surfaces bound to the same loop. The flux through the resulting closed surface is zero and then, taking into account the different position of the loop with respect to the two surfaces, the two fluxes through the two individual surfaces are equal.
2. The convention to fix the direction on the loop is that once an orientation $${\bf N}$$ for the surface has been chosen, $$d\vec l$$ is such that the vector which forms a right-handed system with $${\bf N}$$ and $$d\vec l$$, and which is in the tangent plane of the surface touching the loop at the border, points toward the surface and not out of it. In a more pictorial way, a person following the loop with his head pointing towards the same direction of $${\bf N}$$ should have the surface on the left.