- #1

stoopkid

- 6

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

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

[itex]∇×\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}[/itex]

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, [itex]\vec{E}(t)[/itex], of a point charge, q, located at point [itex]\vec{r_{0}}[/itex], and moving with velocity [itex]\vec{v}[/itex], is given by:

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

Taking the time derivative, I arrive at:

[itex]\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}[/itex]

So this all implies that:

[itex]∇×\vec{B} = \mu_{0}\epsilon_{0}\frac{\partial\vec{E}}{\partial t}[/itex]

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:

[itex]∇×\vec{B} = \mu_{0}\vec{J}[/itex], (Ampere)

where [itex]\vec{J}[/itex] 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.:

[itex]∇×\vec{B} = \mu_{0}\vec{J} + \mu_{0}\epsilon_{0}\frac{\partial\vec{E}}{\partial t}[/itex], (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 [itex]\frac{\partial\vec{E}}{\partial t} = 0[/itex], 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?