Brian Bi

## What is the difference between magnetic charge and electric charge?

In a deep sense, there is really no difference except the names we use.

• Electric charges are sources of electric fields.
• Magnetic charges are sources of magnetic fields.
• An electric field exerts a force on an electric charge, which is proportional to the magnitude of both the field and the charge, and parallel to the field.
• A magnetic field exerts a force on a magnetic charge, which is  proportional to the magnitude of both the field and the charge, and  parallel to the field.
• Therefore, opposite electric charges attract and like electric charges repel, and likewise opposite magnetic charges attract and like magnetic charges repel.
• Moving electric charges generate magnetic fields.
• Moving magnetic charges generate electric fields.
• Magnetic fields exert forces on electric charges that are perpendicular to both the field and the charge's velocity, and proportional to the magnitude of both the field and the charge.
• Electric fields exert forces on magnetic charges that are perpendicular to both the field and the charge's velocity, and proportional to the magnitude of both the field and the charge.
• Therefore, parallel electric currents attract and antiparallel electric currents repel, and likewise parallel magnetic currents attract and antiparallel magnetic currents repel.
• The energy stored in the electric field is proportional to the square of the field's magnitude.
• The energy stored in the magnetic field is proportional to the square of the field's magnitude.
• A changing electric field is accompanied by a magnetic field.
• A changing magnetic field is accompanied by an electric field.

All the above facts can be traced to the fact that the hypothetical Maxwell's equations in the presence of magnetic charges:

\begin{align} \nabla \cdot \mathbf{E} &= k \rho_e \\\nabla \cdot \mathbf{B} &= k \rho_m \\\nabla \times \mathbf{E} &= -\frac{1}{c} \frac{\partial\mathbf{B}}{\partial t} - \frac{k}{c} \mathbf{J}_m \\\nabla \times \mathbf{B} &= \frac{k}{c} \mathbf{J}_e + \frac{1}{c} \frac{\partial\mathbf{E}}{\partial t } \end{align}

combined with the Lorentz force law in the presence of magnetic charges:$\mathbf{F} = q_e\left(\mathbf{E} + \frac{\mathbf{v}}{c} \times \mathbf{B}\right) + q_m\left(\mathbf{B} - \frac{\mathbf{v}}{c} \times \mathbf{E}\right)$are invariant under a duality transformation of the following form:

\begin{align} q_e' &\leftarrow q_e \cos \theta + q_m \sin \theta \\q_m' &\leftarrow q_m \cos \theta - q_e \sin \theta \\\mathbf{E}' &\leftarrow \mathbf{E} \cos \theta + \mathbf{B} \sin \theta \\\mathbf{B}' &\leftarrow \mathbf{B} \cos \theta - \mathbf{E} \sin \theta \end{align}

(Note: Since SI and cgs-Gaussian units both suck, I am using a hybrid unit system, which is described here. The magnetic field is scaled up by a factor of $c$, and $k$ is $4\pi$ times Coulomb's constant.)

In other words, the electromagnetic charge of a particle can be thought of as a point on an E-M plane. Purely electrically charged particles lie along the x axis, and purely magnetically charged particles lie along the y axis. (A neutrino, which has no electric or magnetic charge, would lie at the origin.) If we rotated all particles uniformly by some fixed angle, everything would keep on going like it did before. For example, we may choose $\theta$ to be 90 degrees, which would change a particle of E/M charge $(q_e, q_m)$ to one with E/M charge $(q_m, -q_e)$. In other words it would turn an electric monopole (such as an electron) into a magnetic monopole and vice versa.

So, if somehow all the charged particles in the universe were somehow replaced with magnetic monopoles, we wouldn't be able to notice. So in that deep sense, there is no difference between electric and magnetic charge.

What is interesting is that all particles that are known to humanity seem to lie along a single line through the origin on the E-M plane. We can choose the E axis however we want, and the duality transformations guarantee that the laws of electromagnetism take the same form. For convenience, we choose our E axis along that line. Measured relative to that choice of axes, then, all electromagnetically charged particles we know of are purely electrically charged, and have no magnetic charge. You could just as well choose the M axis to be along that line, though. Or you could choose both the E and M axes to not be along that line, in which case all the charged particles we know of have a combination of electric and magnetic charge (but the ratio is the same for all of them; they all lie along a line in the E-M plane, a fact which is not changed by our choice of axes).

In summary, the difference is purely one of convention. To distinguish between electric and magnetic charge, we define the electron to be electrically charged, not magnetically charged. That's the real answer.