electromagnetism - Symmetry in electricity and magnetism due to magnetic monopoles

I was wondering about the differences between electricity and magnetism in the context of Maxwell's equations. When I thought over it, I came to the conclusion that the only difference between the two is that magnetic monopoles do not exist. Is this right?

Next one. Now I searched for the equations with magnetic monopoles and found them at Wikipedia. They seem quite symmetrical (except the constants of course), except two major differences:

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  It is $\nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}} {\partial t} - \mu_0\mathbf{j}_{\mathrm m}$, but $\nabla \times \mathbf{B} = \mu_0 \epsilon_0 \frac{\partial \mathbf{E}} {\partial t} + \mu_0 \mathbf{j}_{\mathrm e}$. This means that the induced "magnetic emf" (if I may call it that) is produced by changing electric fields and currents in the exact opposite sense (I mean direction) to the counterpart phenomenon of Electromagnetic Induction. Why so?? Is there a lenz law for "magnetic emf" induction also??

  
  


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  Also, the lorentz force on magnetic charges $\mathbf {F}={}q_\mathrm m (\mathbf {B} - {\mathbf v} \times \frac {\mathbf {E}} {c^2})$. Why this minus sign in the force on magnetic charges that does not appear in the lorentz force on electric charges.

  
  

Answer

Both of these follow from desirable properties of this hypothetical magnetic charge, namely:

1. Magnetic charge is conserved.



2. Magnetic field lines radiate outwards from positive magnetic charges.


3. The net force between two magnetic charges moving at constant speed along parallel tracks is less than that between two stationary charges.

All three of these properties hold for electric charges. The last one may not be as familiar, but it basically works as follows: if we have a positive electric charge moving at constant velocity, it generates a magnetic field in addition to its electric field. A second positive electric charge moving parallel to the first one will therefore experience a magnetic force, and if you work out the directions, this force works out to be attractive. Thus, the net force between the two charges (electric and magnetic together) is less than the magnitude of the force they would exert on each other if they were at rest. [ASIDE: This can also be thought of in terms of the transformation properties of forces between different reference frames in special relativity, if you prefer to think of it that way.]

Now, the conservation of electric charge can be written in terms of the continuity equation: $$ \vec{\nabla} \cdot \vec{j}_e + \frac{ \partial \rho_e}{\partial t} = 0 $$ Note that this can be derived from Ampère's Law and Gauss's Law ($\epsilon_0 \vec{\nabla} \cdot \vec{E} = \rho_e$), using the fact that the divergence of a curl is always zero: $$ 0 = \vec{\nabla} \cdot (\vec{\nabla} \times \vec{B}) = \mu_0 \vec{\nabla} \cdot \vec{j}_e + \mu_0 \frac{\partial ( \epsilon_0 \vec{\nabla} \cdot \vec{E})}{\partial t} = \mu_0 \left( \vec{\nabla} \cdot \vec{j}_e + \frac{\partial \rho_e}{\partial t} \right) $$ If we want to extend Maxwell's equations to magnetic charges, we need to have a magnetic version of Gauss's Law and add in a magnetic current term to Faraday's Law: $$ \vec{\nabla} \cdot \vec{B} = \alpha \rho_m \qquad \vec{\nabla} \times \vec{E} = \beta \vec{j}_m - \frac{\partial \vec{B}}{\partial t} , $$ where $\alpha$ and $\beta$ are arbitrary proportionality factors. But if we try to derive a continuity equation for magnetic charge from these two facts (as we did above for electric charge), we get $$ \beta \vec{\nabla} \cdot \vec{j}_m - \alpha \frac{\partial \rho_m}{\partial t} = 0, $$ and this is equivalent to the continuity equation if and only if $\alpha = - \beta$. Beyond this, the choice of $\alpha$ is to some degree arbitrary; different values correspond to different choices of which type of magnetic charge we call "positive", and what units we use to measure it. If we want to have magnetic field lines radiating away from "positive" magnetic charges, then we will want $\alpha > 0$; the usual choice in MKS units is to pick $\alpha = \mu_0$ (and $\beta = -\mu_0$), as you have in your equations above.

This negative sign in the magnetic current term Faraday's Law then implies that the electric field lines created by a moving magnetic charge will obey a "left-hand rule" instead of a "right-hand rule". In other words, the direction of $\vec{E}$ created by a moving magnetic charge would be opposite the direction of $\vec{B}$ created by a moving electric charge. If we still want two magnetic charges moving along parallel tracks to exhibit a lesser force than what they feel when at rest, then we must also flip the sign of the $\vec{v} \times \vec{E}$ term in the Lorentz force law to compensate for this flip.