# Using Ampère's on balloon shape

In an assignment I'm looking at for exam preparation I am asked to use Maxwell's corrected version of Ampère's law (the integral form) $$\oint_{loop}\vec{B}\cdot d\vec{l} = \mu_0 \left(I_{encl} + \epsilon_0 \frac{d}{dt} \int_{surface} \vec{E} \cdot d\vec{a}\right)$$
to find the magnetic field in the point $p_1$ in the distance $r<R$ above the wire. There is a current, $I(t)$, running through the wire, the capacitor is circular and has a radius $R$, and the loop is a circle around the wire.

I have to calculate it using this complicated balloon shape given (see the figure below), which I have some troubles with.

What I'm thinking is that the wire is piercing the balloon three different places - to the far right and at the capacitor twice. The piercing on the far right would contribute with $\mu_0I$ from the first part of the equation, but I'm not sure what to the with the other two.

Then comes the surface integral of the $\vec{E}$ field, which I'm not sure how to incorporate either.

Can you come with some tips or explanation that could help me better understand how to use the law on the surface?

The $I_{\rm encl}$ term you have to be careful with because although a current $I$ enters on the left hand side of amperian surface there are two places where it is leaving.
So in effect the current term is found by relating it to the ratio if the areas $\frac{\pi r^2}{\pi R^2}$.
For the second term on the right hand side you are only looking to integrate between $r$ and $R$ as that is the only part of your amperian surface where the electric file is perpendicular to the amperian surface.
Assuming an ideal capacitor you can relate the electric field between the plates to the surface charge density $(=\frac{\rm charge}{area})$ on the plates.