- #1
JustinLevy
- 895
- 1
I've seen in textbooks before, there if there were monopoles, Maxwell's equations would look more "symmetric" between the E and B fields. Such that (in cgs units):
[tex]\begin{align*}
\nabla \cdot \mathbf{E} &= 4 \pi \rho_e \\
\nabla \cdot \mathbf{B} &= 4 \pi \rho_m \\
-\nabla \times \mathbf{E} &= \frac{1}{c}\frac{\partial \mathbf{B}} {\partial t} + \frac{4 \pi}{c}\mathbf{j}_m \\
\nabla \times \mathbf{B} &= \frac{1}{c}\frac{\partial \mathbf{E}} {\partial t} + \frac{4 \pi}{c} \mathbf{j}_e
\end{align*}
[/tex]
And the Lorentz force law becomes:
[tex]\mathbf{F}=q_e\left(\mathbf{E}+\frac{\mathbf{v}}{c}\times\mathbf{B}\right) + q_m\left(\mathbf{B}-\frac{\mathbf{v}}{c}\times\mathbf{E}\right)[/tex]
My question is, How can we write electrodynamics in a Lagrangian now?
Using non-relativistic momentum, and without magnetic monopoles it is:
define the electric and magnetic fields in terms of a scalar and vector potential as such
[tex] \mathbf{E} = -\nabla \phi - \frac{\partial}{\partial t} \mathbf{A}[/tex]
[tex] \mathbf{B} = \nabla \times \mathbf{A}[/tex]
Using four vector notation to collect and define some things
[tex] A^\mu = (\phi,A_x,A_y,A_z)[/tex] (the potential four-vector)
[tex] j^\mu = (\rho c,j_x,j_y,j_z)[/tex] (the current four-vector)
[tex] F_{\mu\nu} = \partial_\mu A_\nu - \partial_\nu A_\mu[/tex] (the electromagnetic tensor)
The lagrangian is then:
[tex] \mathcal{L} = \frac{1}{2} m \dot{\mathbf{x}}^2 + j^\mu A_\mu + \frac{1}{4} \int \mathrm{d}^3r \ F^{\mu\nu}F_{\mu\nu} [/tex]
Where the degrees of freedom to vary are [itex]x[/itex] (the particle position), and the potential components [itex]A^\mu[/itex].
(I may have messed up signs and constants in this one.)
How in the world would I write this for the case with magnetic monopoles?
Since I should recover this when there aren't monopoles, I'm guessing I just need to add stuff?
I can easily write the case where there aren't electric charges, and only magnetic charges by analogy (creating another set of four-vectors). But then I need to "couple" these four-vectors and the charges to both of them, in order for the combination to work. I can't figure out how to do it right. In particular, I need to get a force q_m . B How the heck do you write than in terms of the usual vector potential?
Does anyone know how to do this?
[tex]\begin{align*}
\nabla \cdot \mathbf{E} &= 4 \pi \rho_e \\
\nabla \cdot \mathbf{B} &= 4 \pi \rho_m \\
-\nabla \times \mathbf{E} &= \frac{1}{c}\frac{\partial \mathbf{B}} {\partial t} + \frac{4 \pi}{c}\mathbf{j}_m \\
\nabla \times \mathbf{B} &= \frac{1}{c}\frac{\partial \mathbf{E}} {\partial t} + \frac{4 \pi}{c} \mathbf{j}_e
\end{align*}
[/tex]
And the Lorentz force law becomes:
[tex]\mathbf{F}=q_e\left(\mathbf{E}+\frac{\mathbf{v}}{c}\times\mathbf{B}\right) + q_m\left(\mathbf{B}-\frac{\mathbf{v}}{c}\times\mathbf{E}\right)[/tex]
My question is, How can we write electrodynamics in a Lagrangian now?
Using non-relativistic momentum, and without magnetic monopoles it is:
define the electric and magnetic fields in terms of a scalar and vector potential as such
[tex] \mathbf{E} = -\nabla \phi - \frac{\partial}{\partial t} \mathbf{A}[/tex]
[tex] \mathbf{B} = \nabla \times \mathbf{A}[/tex]
Using four vector notation to collect and define some things
[tex] A^\mu = (\phi,A_x,A_y,A_z)[/tex] (the potential four-vector)
[tex] j^\mu = (\rho c,j_x,j_y,j_z)[/tex] (the current four-vector)
[tex] F_{\mu\nu} = \partial_\mu A_\nu - \partial_\nu A_\mu[/tex] (the electromagnetic tensor)
The lagrangian is then:
[tex] \mathcal{L} = \frac{1}{2} m \dot{\mathbf{x}}^2 + j^\mu A_\mu + \frac{1}{4} \int \mathrm{d}^3r \ F^{\mu\nu}F_{\mu\nu} [/tex]
Where the degrees of freedom to vary are [itex]x[/itex] (the particle position), and the potential components [itex]A^\mu[/itex].
(I may have messed up signs and constants in this one.)
How in the world would I write this for the case with magnetic monopoles?
Since I should recover this when there aren't monopoles, I'm guessing I just need to add stuff?
I can easily write the case where there aren't electric charges, and only magnetic charges by analogy (creating another set of four-vectors). But then I need to "couple" these four-vectors and the charges to both of them, in order for the combination to work. I can't figure out how to do it right. In particular, I need to get a force q_m . B How the heck do you write than in terms of the usual vector potential?
Does anyone know how to do this?