Where does the $\partial \vec{E}/\partial t$ term from Maxwell's equation go in Ampere's Law? One of Maxwell's Equations (ME) is:
$$\nabla\times\vec B = \mu_0\vec J+\epsilon_0\mu_0 \frac{\partial \vec E}{\partial t}.$$
While Ampere's Law (AL) is:
$$\nabla\times\vec B = \mu_0\vec J.$$
Griffiths E&M book derives that form of AL using the Biot-Savart Law and applying Stokes' theorem. Intuitively, it makes sense to me: a steady current is  going to give rise to a magnetic field around it. But then I have trouble reconciling it with the ME I posted above -- while AL seems to say that for a given steady current $J$ you get a straightforward $B$, the above ME seems to say that for a given $J$ you could get many combinations of $B$ and $E$.
How is this reconciled?
 A: You are right. These two equations cannot be valid at the same time. In particular, AL is only valid for steady current situation. Indeed, current induces a circulating magnetic field, as indicated by AL. However, current is not the only thing which can produce circulating magnetic field. The time-varying electric field can also produce magnetic field. This is what the first equations is saying.
You are also true that only given J, there are many combinations of B and E. However, note that Maxwell's equations contain 4 equations. Only using this equation, the EM field is not uniquely determined. In order to obtain the field, this equation need collaborate with other 3 Maxwell's equations.
In conclusion, ME cannot be reconciled with AL; rather, it is the correction of AL. Current is not enough to determine the value of magnetic field. In order to obtain magnetic field, one needs to solve 4 Maxwell's equations all-together.
Hope it helps^_^
A: 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.
A: I believe that Ampere's Law is wrong in some situations. When Maxwell looked into it, he discovered that Ampere's Law is not always true, so he modified it to get the Ampere-Maxwell Law.
According to my understanding, this can be shown if you have a wire with a current flow that is charging a capacitor. If you calculate the magnetic field between the capacitor plates simply using Ampere's Law, you will get zero, however, using the Biot-Savart Law, there will be a magnetic field. The Ampere-Maxwell Law corrects this.
Note: this is my understanding, I've just been learning about electromagnetism, so I'm not an expert.
A: It is mentioned in the book Introduction to electrodynamics that Ampere could not find the second term because such a thing is hard to detect in laboratory. But now as we all know (because of Maxwell) that changing electric field produces magnetic field.
If you take laplace transform of the second term in M.E. you will find that term is directly proportional to frequency. This equation is used when you wish to analyse the interface between two medium, or surface inside a dielectric. 
Air is not a conductor, right. So there is not current and hence no J in free space. Still we have changing electric and magnetic field waves (electromagnetic waves) in space due to which communication is possible.
