# Ampere's circuital law

I'm having trouble understanding the surface used for Ampere's Circuital Law. In classes, we've been using simple circular or rectangular loops. Is the surface supposed to be area of the circle or the rectangle? Or is it a simple case of the law?

A example would be the surface used for displacement current. This one is just lost on me. The surface looks like a pot enclosing the wire and the capacitor. Does the law specify a closed loop of a surface? Is the line integral taken on the mouth of the pot-like surface or the bottom ($S_1$ below)? And is the enclosed current in the smaller mouth or the larger bottom surface ($S_2$)?

• You are actually free choose any surface you want, as long as you choose a surface whose edge corresponds to the closed loop of current. No matter what surface you choose, the integral will evaluate to the same result. This freedom is what allows us to arbitrary choose the simplest surface possible to evaluate the integral whenever we can. Feb 22, 2014 at 0:17
• Okay, that edit made everything clear. The diagram in the book made it look like there were two boundary loops for S2. Feb 22, 2014 at 16:00
• Very closely related physics.stackexchange.com/questions/159550/… Jul 12, 2015 at 13:07

For reference, Ampère's law with Maxwell's correction in integral form: $$\oint_{\partial S} \mathbf{B} \cdot d\mathbf{l} = \mu_0 I_{enc} + \mu_0 \epsilon_0 \iint_S \frac{\partial\mathbf{E}}{\partial t} \cdot d\mathbf{a}$$
The value of the right-hand side is independent of the surface chosen. To see this, suppose two surfaces, $S_1$ and $S_2$, both have the same boundary $\partial S$. Then take the difference in the right-hand side evaluated on the two surfaces. You will end up with a term of the form $\mu_0 I + \mu_0 \epsilon_0 \iint \partial\mathbf{E}/\partial t \cdot d\mathbf{a}$ evaluated on a closed surface. Using Gauss's law, the second term can be converted into $\mu_0 \partial Q/\partial t$ where $Q$ is the charge enclosed. But $\mu_0(I + \partial Q/\partial t) = 0$ by charge conservation.