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hisacro
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Ampere's circuital law -- Monopole thought experiment
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In summary, the Maxwell equations state that the electric and magnetic fields obey two separate equations. These equations can be solved using vector calculus, yielding Biot-Savart's law. Maxwell found that this law could not be correct, and that the electric charge was actually conserved. By including the displacement current, he was able to solve the continuity equation and confirm that electric charge is actually conserved.
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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##,
$$\vec{j}(\vec{r}')=I \vec{e}_3 \delta(x_1) \delta(x_2).$$
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##,
$$\vec{j}(\vec{r}')=I \vec{e}_3 \delta(x_1) \delta(x_2).$$
- #3
hisacro
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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.
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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)
$$\vec{\nabla} \times \vec{B}=\frac{1}{c} \vec{j}$$
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)
1. What is Ampere's circuital law?
Ampere's circuital law is a fundamental principle in electromagnetism that relates the magnetic field around a closed loop to the electric current passing through that loop. It states that the line integral of the magnetic field around a closed loop is equal to the product of the electric current passing through the loop and the permeability of free space.
2. What is the monopole thought experiment?
The monopole thought experiment is a hypothetical scenario that involves a single magnetic charge, or monopole, instead of the usual dipole configuration of north and south poles. It is used to explore the implications of Ampere's circuital law in the absence of magnetic charges.
3. Why is the monopole thought experiment important?
The monopole thought experiment helps to illustrate the symmetry of Ampere's circuital law and the fact that it applies equally to electric and magnetic fields. It also highlights the fact that while electric charges can exist as isolated monopoles, magnetic charges have never been observed in nature.
4. How does the monopole thought experiment relate to Ampere's circuital law?
In the monopole thought experiment, the magnetic field around a closed loop is still related to the electric current passing through that loop, as stated by Ampere's circuital law. However, in the absence of magnetic charges, the field is entirely due to the motion of electric charges, and there is no separate contribution from magnetic charges.
5. What are the limitations of the monopole thought experiment?
The monopole thought experiment is a useful tool for understanding the implications of Ampere's circuital law, but it is purely hypothetical and does not reflect the reality of our universe. Magnetic charges have never been observed, and the existence of monopoles would require a significant revision of our understanding of electromagnetism.
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