[Physics] Ampere’s circuital law and the type of surface to be used


What kind of a surface can we use for Ampere's circuital law?
I was taught that any enclosed surface can be used for Gauss's Law(something like a cube,a sphere)-essentially 3-D enclosed surfaces.

For Ampere's law, I have used a circular ring for calculation of Magnetic field due to wire(and also rectangular shapes for solenoids)-essentially 2-D enclosed surfaces like a ring,rectangle etc.

Now when I came to know about Maxwell's Displacement Current, the book uses a bucket kind of a shape like in the figure.enter image description here

I am confused.One side is open and the other is closed.(I apologize if this sounds dumb.I only have a basic idea about the law.I don't really understand the 'surface integral' part and how it's supposed to be used.)

I've seen this link.I don't understand what is said there.

Best Answer

Here is an annotated picture of a butterfly net which shows that all you need is a loop and an open surface which adjoins the loop.
So the closed loop can be any shape you like as can the open surface linked to it.

enter image description here

You can choose the shape of your surface to suit your problem and so for the ideal capacitor which has no edge effects choose a surface part of which is at right angles to the electric field. It makes the integration a lot easier.
This is because the electric field will only be present between the plates and if you choose the surface to be at right angles to the electric field, and hence the rate of change of electric field, then the integral will become $\mu_o \epsilon_o \frac {dE}{dt}A$ where A is the area of a capacitor plate.

$E=\frac {q}{\epsilon_o A}$ so the integral is $\mu_o I$ because $\frac{dq}{dt}=I$ which makes it the "more familiar" right hand side of Ampere's law

enter image description here

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