What contradiction is found by using Ampere's circuital law to obtain the magnetic field during the charging of a capacitor?

(N/A) When applying Ampere's circuital law to a loop enclosing a charging capacitor, we encounter a contradiction.
Consider a loop of radius $r$ outside the capacitor plates as shown in figure $(a)$. Applying Ampere's circuital law gives $\oint \vec{B} \cdot d\vec{l} = \mu_0 i(t)$, which yields a non-zero magnetic field $B = \frac{\mu_0 i(t)}{2\pi r}$.
Now, consider a surface bounded by the same loop but passing through the region between the capacitor plates, as shown in figures $(b)$ and $(c)$. Since there is no conduction current flowing between the plates, the current enclosed by this surface is zero $(\sum I = 0)$.
Applying Ampere's circuital law to this surface gives $\oint \vec{B} \cdot d\vec{l} = \mu_0 (0) = 0$, which implies $B = 0$.
This is a contradiction because the magnetic field at the same point $P$ cannot be both non-zero and zero. This inconsistency led Maxwell to propose the existence of displacement current $i_d = \epsilon_0 \frac{d\Phi_E}{dt}$ to complete the law.