Question about Maxwell's equations

[stoopkid]

My question is essentially about Ampere's law. I went the long way about and evaluated the curl of the magnetic field, $\vec{B}$, of a point charge, q, located at position $\vec{r_{0}}$, and moving with velocity $\vec{v}$:

$\vec{B} = \frac{\mu_{0}q}{4\pi}\frac{\vec{v}×\vec{Δr}}{\left\|\vec{Δr}\right\|^{3}}$

After a lot of calculation, I wound up with the following:

$∇×\vec{B} = \frac{\mu_{0}q(\vec{v}\bullet\vec{Δr})}{(4/3)\pi\left\|\vec{Δr}\right\|^{5}}\vec{Δr} - \frac{\mu_{0}q}{4\pi\left\|\vec{Δr}\right\|^3}\vec{v}$

So I looked at the second term and I thought that looked a lot like current density, so I figured the first term must be the displacement current, so I went on to see what the time derivative of the electric field of a moving point charge would look like. The electric field, $\vec{E}(t)$, of a point charge, q, located at point $\vec{r_{0}}$, and moving with velocity $\vec{v}$, is given by:

$\vec{E}(t) = \frac{q}{4\pi\epsilon_{0}}\frac{\vec{Δr}}{\left\|\vec{Δr}\right\|^3}$.

Taking the time derivative, I arrive at:

$\frac{\partial\vec{E}(t)}{\partial t} = \frac{q(\vec{v}\bullet\vec{Δr})}{(4/3)\pi\epsilon_{0}\left\|\vec{Δr}\right\|^{5}}\vec{Δr} - \frac{q}{4\pi\epsilon_{0}\left\|\vec{Δr}\right\|^3}\vec{v}$

So this all implies that:
$∇×\vec{B} = \mu_{0}\epsilon_{0}\frac{\partial\vec{E}}{\partial t}$

Which looks good, but it's only have the equation it's supposed to be, and what's more, it's not the half I would have expected. Ampere's original law gave the following empirical relationship:

$∇×\vec{B} = \mu_{0}\vec{J}$, (Ampere)

where $\vec{J}$ is the current density. It was not until about 40 years later that Maxwell did all the tricks to figure out that you needed to add a second term (the displacement current), i.e.:

$∇×\vec{B} = \mu_{0}\vec{J} + \mu_{0}\epsilon_{0}\frac{\partial\vec{E}}{\partial t}$, (Maxwell's correction).

So I would've assumed that I would end up with the 1st version of the equation with only the first term, or maybe the corrected equation with both terms, but I didn't expect to end up with only the 2nd term of the corrected equation, especially because it was apparently some big thing that took a long time to figure out after the original version of Ampere's law, and yet it can apparently be derived directly from the definitions of the electric and magnetic fields, by simply taking the time derivative and the curl, respectively, with no added assumptions. I have been trying to understand how Maxwell came up with the correction and I always read/hear things about how it was an argument about symmetry, or he derived it with a thought experiment where a surface goes in between capacitor plates and attaches to an Amperian loop around a wire carrying current, and equating this to the magnetic field using a surface "pierced" by the current, but I'm not seeing how any of this is necessary.

I have sort of a guess as to where I'm going wrong, but this is where I'm unsure. Ampere's law was originally derived experimentally for a wire carrying a current, i.e. the current keeps flowing. So as point charges move, point charges are assumed to come and fill their place. This keeps the electric field constant, so $\frac{\partial\vec{E}}{\partial t} = 0$, but even though the total electric field remains constant, the individual charges are still moving, so there is still a magnetic field. The magnetic field of all these charges should be derivable from the sum of their individual charges, and I think that has some relation to the term that I thought looked like current density originally, but explains why there isn't a term for changing electric field. The equations I got apply only to a moving point charge, and not to the situation where there is a "net flow of current" (i.e. charge continually replaced; current through a wire; etc). Am I on the right track here?

 

[vanhees71]

The trouble is that you need Maxwell's equations to get the electromagnetic field of a moving point charge. So it's a bit of a circular argument to get the Ampere-Maxwell Law from these fields. I think Maxwell's equations are the starting point to describe electromagnetic phenomena. They can be deduced to a certain extent using relativistic classical field theory asking for the equation of motion of a massless spin-1 field. To come to such an idea you need a pretty great deal of mathematics and perhaps even quantum field theory to motivate to look for unitary representations of the Lorentz group. So I think, the best way for starting electromagnetism is to take the Maxwell equations as postulates and then explain the physical meaning of the various terms appearing in them.

The easiest way to evaluate the electromagnetic field of a moving point charge is to use the retarded potentials (retarded Green's function of the D'Alembert operator) and the manifestly covariant description of the current density of a point charge
$$J^{\mu}(x)=\int_{\mathbb{R}} \int \mathrm{d} \tau q \frac{\mathrm{d} y^{\mu}}{\mathrm{d} \tau} \delta^{(4)}[x-y(\tau)].$$
You'll end up with the Lienard-Wiechert potentials:

http://en.wikipedia.org/wiki/Liénard–Wiechert_potential

 

[Jano L.]

stoopkid, your second equation is valid at all points except the point where the moving particle is. Your result, the Maxwell equation without J is valid there, as J = 0 at those points.

At the point $\mathbf r_0$, careful calculation of curl of B for the field of point particle gives


$$
\nabla \times \mathbf B (\mathbf x) = \mu_0 q\mathbf v \delta(\mathbf x - \mathbf r_0) + \mu_0\epsilon_0 \frac{\partial \mathbf E}{\partial t}(\mathbf x).
$$

so the role of J is played by $q\mathbf v \delta(\mathbf x- \mathbf r_0)$.

vanhees71, there is more to understanding electromagnetism than just evaluating mathematical consequences of the Maxwell equations. For example, one may want to see where the Maxwell equations are coming from, how they can be inferred from special formulae based on experiments (magnetic of moving charge can be inferred from the field of straight current) or just to see whether Maxwell equations are consistent with such special cases.

 

[Menaus]

The point of all this is to understand how Maxwell came to the conclusion of displacement current? If so, then I don't believe that studying the equations in vector form will work, as they were not originally derived in vector form, but in quaternion form.

Heaviside contributed greatly to the transition to vectors from quanternions, but the result MAY mean that the equations do not hold their original flavor, if you know what I mean.

I suggest if you want to understand Maxwell's thought process, then you ought to get his original treatise and read it, and then play with his 20 equations with 20 unknowns... Have fun! xD

Also: I suggest trying to get hold of the first edition, as the next few editions are simpler, and therefor, it may not show Maxwell's original train of thought.

 

[WannabeNewton]

Symmetry is a very nice way to look at it though, seeing as how Maxwell's equations come naturally out of Coloumb's law and SR; of course if you're going for historical motivation then it won't be satisfactory.

 

[stoopkid]

Thank you guys very much for your answers! My motivation in all this is actually just to see what happens when you take the curl of the magnetic field, the long way, and I'm glad I did! I did the same for the electric field, and that was A LOT more straight-forward than this, but was very helpful in understanding both physics and the math behind it. On the contrary, I would not have been able to understand what was going on with the curl of the magnetic field without this help.

Jano's answer is definitely the "missing piece to my puzzle". There is so much going on with the calculations already that it's easy to forget to check when your function is undefined! I haven't done the calculation yet myself but I'm 99% sure that this is what I'm missing.