There is a contradiction in the equations we have so far.

What we have so far? Here is the set of equations:

\[ \vec{\nabla} \cdot \vec{E} = \frac{\rho}{\epsilon_0} \tag{1}\] \[ \vec{\nabla} \cdot \vec{B} = 0 \tag{2} \] \[ \vec{\nabla} \times \vec{E} = - \frac{\partial \vec{B}}{\partial t} \tag{3}\] \[ \vec{\nabla} \times \vec{B} = \mu_0 \vec{j} \tag{4}\]

But there is a flaw in these equations - if we follow them to the letter, there is a contradiction for non-steady currents.

Sign of trouble 1

According to (4) (Ampere's law), integral of magnetic field \( \vec{B} \) around closed contour \(L\) equals total current passing through surface \(S\), where \(L\) is the boundary of \(S\). \(S\) can be any surface as long as its boundary is \(L\). \[ \oint_L{ \vec{B} \cdot \vec{dl} } = I \tag{5}\] Lets apply this law to following setup:

Current flows through parallel-plates capacitor.

There is parallel-plate capacitor and external source provides current \( i\). We know that DC current does not flow through the capacitor but AC does. Two questions come to mind:

What is magnetic field here?
There is vacuum between plates, how charges are getting through?

Answering first question: Following same procedure as we used when deriving Ampere's law. We consider two cases:

Surface S cuts through the wire.
Surface goes between plates of the capacitor, wire does not intersect the surface

If we follow (5) in both cases then:

Surface \(S_1\): \[ \oint_L{ \vec{B} \cdot \vec{dl} } = I \].
Surface \(S_2\): There are no charges moving through S_2 so \[ \oint_L{ \vec{B} \cdot \vec{dl} } = 0 \]

So flux of magnetic field is zero and is not zero at the same time! That can not be right!

Another sign of trouble

We can detect that something is wrong with our equations in a different way too.
Now calculate divergence for both sides of equation (4): \[ \nabla \cdot (\vec{\nabla} \times \vec{B}) = \mu_0 (\nabla \cdot \vec{j}) \] It is known that divergence of curl of any vector is 0 ( Proof). \[ \nabla \cdot (\vec{\nabla} \times \vec{B}) = 0\] So it should hold: \[ \nabla \cdot \vec{j} =0 \tag{6} \] Does it hold in all possible situations?
We considered continuity equation: \[ \frac{ \partial \rho(x,y,x,t)}{ \partial t } + \nabla \cdot \vec{j} = 0 \] and we can see that (6) holds only if \( \frac{ \partial \rho(x,y,x,t)}{ \partial t } = 0\).
It means that charge density does not change with time, there are no charges "piling up" anywhere and current is steady.

Equations above are valid only for steady (DC) currents.