The electromagnetic field equations as known until 1860 (i.e. before Maxwell)
and especially Ampere's law
$$\vec{\nabla}\times\vec{B}=\mu_0\vec{J} \tag{1}$$
were found from experiments with static or slowly varying electric and
magnetic fields.
However, Maxwell noticed that there must be something wrong.
By applying the divergence operator ($\vec{\nabla}\cdot$) to Ampere's law (1)
the conclusion is
$$0=\vec{\nabla}\cdot\vec{J} \tag{2}$$
This is obviously in contradiction with
the conservation law for electric charge (written in differential form as a continuity equation):
$$0=\vec{\nabla}\cdot\vec{J}+\frac{\partial\rho}{\partial t} \tag{3}$$
Therefore Maxwell had two choices:
- Ampere's law (1) is wrong, and charge conservation (3) is correct.
- Ampere's law (1) is correct, and charge conservation (3) is wrong.
Maxwell decided for the first option, i.e. Ampere's law is wrong,
because he had no doubt in the correctness of charge conservation.
He modified Ampere's law by adding a term ($\mu_0\epsilon_0\frac{\partial\vec{E}}{\partial t}$)
and got what was later called the Ampere-Maxwell law:
$$\vec{\nabla}\times\vec{B}=\mu_0\vec{J}+\mu_0\epsilon_0\frac{\partial\vec{E}}{\partial t} \tag{4}$$
He chose this particular additional term, because when you now apply the
divergence operator ($\vec{\nabla}\cdot$) to (4) and also use Gauss's law
$$\epsilon_0\vec{\nabla}\cdot\vec{E}=\rho \tag{5}$$
then you get the charge conservation law (3) as it should be.
So Maxwell's statement was: Ampere's law (1) is actually only an approximation
of the true Maxwell-Ampere law (4).
Usually this approximation is valid because $\epsilon_0=8.854\cdot 10^{-12}$ F/m
is a very small constant
and therefore the additional term $\mu_0\epsilon_0\frac{\partial\vec{E}}{\partial t}$
is neglectable.
But this approximation becomes invalid when the electric field $\vec{E}$
varies very fast with time $t$, and hence the term
$\mu_0\epsilon_0\frac{\partial\vec{E}}{\partial t}$
is no longer neglectable. This happens in high frequency electromagnetic waves,
but did not happen in all the electromagnetic experiments done before 1860
with static or slowly varying electromagnetic fields.