I Ampere's circuital law -- Monopole thought experiment

Summary
I have been struggling to get this thought experiment, precisely the part where charges are getting cancelled out
source: Optics by Eugene Hecht
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vanhees71

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Don't struggle with this thought experiment. I've no clue what Hecht is after (not only with this example for overly confusing students with some strange didactical ideas).

As all fundamental laws the Maxwell equations grew out from many observatikns and mathematical analysis. For magnetostatics, the electric and manetic field components completely decouple, and thus you can concentrate on the magnetic field only. It obeys the two equations (in Heaviside-Lorentz units)
$$\vec{\nabla} \cdot \vec{B}=0, \quad \vec{\nabla} \times \vec{B}=\vec{j}/c.$$
The first equation tells you that there are no manetic monopoles. The second that the vortices of ##\vec{B}## are currents of electric charges.

One can derive the solution of this set of eqs. using Helmholtz's fundamental theorem of vector calculus, finally resulting in Biot-Savart's law,
$$\vec{B}(\vec{r})=\frac{1}{4 \pi c} \int_{\mathbb{R}^3} \mathrm{d}^3 r' \vec{j}(\vec{r}') \times \frac{\vec{x}-\vec{x}'}{|\vec{x}-\vec{x}'|^3}.$$
Solve the integral for an infinetely thin wire with a current ##i##,
$$\vec{j}(\vec{r}')=I \vec{e}_3 \delta(x_1) \delta(x_2).$$
 
That's really interesting, I thought Biot-Savart's law was formulated because of experimental results and I initially planned to read till diffraction guess I'll go for another book after this Electromagnetic theory.
 

vanhees71

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Sure, historically Biot-Savart and Ampere have deduced their equations from experiments. Maxwell has found that the corresponding equation, which reads in modern form
$$\vec{\nabla} \times \vec{B}=\frac{1}{c} \vec{j}$$
cannot be correct, supposed electric charge is conserved, i.e., the continuity equation
$$\partial_t \rho + \vec{\nabla} \cdot \vec{j}=0$$
strictly holds.

To get matters right, he realized that this problem is solved by including what he called (for reasons which are obsolete today) the "displacement current",
$$\vec{\nabla} \times \vec{B}=\frac{1}{c} (\vec{j}+\partial_t \vec{E}),$$
because then taking the divergence of this equation leads, together with Gauss's Law for the electric field,
$$\vec{\nabla} \cdot \vec{E}=\rho.$$
Some good books on classical optics is

A. Sommerfeld, Lectures on Theoretical Physics, vol. 4, Optics, Academic Press (1954)
G. R. Fowles, Introduction to modern optics, Dover (1989)
M. Born, E. Wolf, Principles of optics, Cambridge Univsersity Press (1999)
 

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