I came across a problem that I cannot get my head around.
Consider two very small spherical metallic balls given charges $+Q$ and $-Q$. Assume that both can be approximated as point charges. Now, they are connected by a straight, finite, conducting wire. A current will flow in the wire until the charges on both balls become zero. Consider a point P on the perpendicular bisector of the wire, at a distance $r$ from the wire. My goal is to find the magnetic field at point P, when the current in the wire is $i$. The following figure illustrates the mentioned situation.
I will now use the Ampère-Maxwell equation to obtain an expression for the field.
I have constructed a circular loop of radius $r$ around the wire, to use the Ampère-Maxwell Law. Firstly, one must notice that the two charges produce an electric field everywhere in space. And since the balls are getting discharged, the electric field is actually changing. I have calculated the electric flux through the surface when the charges on the balls are $+q$ and $-q$ below.
Now, for the final substitution...
So I have obtained a neat result after all! But, I realized there was a problem.
Let me use the Biot-Savart Law to find the magnetic field created only due to the current in the wire. This is a relatively easier calculation since the formula for the field due a finite current carrying straight wire is already known.
The answer turns out to be the same.
First of all, is the answer correct? If not, where did I go wrong?
This is what I cannot understand. The Biot-Savart Law gives you the magnetic field created merely due to the current flowing in a conducting wire. On the other hand, the Ampère-Maxwell Law gives you the net field due to the current carrying wire and due to the induced magnetic field (caused by the changing electric field).
So how is it that I get the same answer in both cases? The Biot-Savart Law cannot account for induced fields, right?
Why does there seem to be an inconsistency in the two laws? Have I missed something, or used a formula where it is not applicable?