- #1

unified

- 40

- 0

Angular momentum density stored in an electromagnetic field: $$\vec{l}_{em} = \epsilon_0[\vec{r} \times (\vec{E} \times \vec{B})]$$

Electric field of an electric charge: $$\frac{q_e}{4\pi\epsilon_0}\frac{r - r'}{|r - r'|^3}$$

Magnetic field of a magnetic charge: $$\frac{\mu_0q_m}{4\pi}\frac{r - r'}{|r - r'|^3}$$

Question Statement:

Suppose we have an electric change and a hypothetical magnetic charge with the fields as given in the relevant equations. Now, let's calculate the total angular momentum stored in the electromagnetic field generated by an electric and magnetic charge separated by a distance d. Suppose we orient our coordinate system so that the electric charge is located at the origin (0, 0, 0) and the magnetic charge is located at (0, 0, d). Now, we are finding the angular momentum relative to the location of the electric charge since we have put it at the origin. If you do the calculation, you will end up with the following easily computable integral. $$\vec{L}_{em} = \frac{q_eq_m\mu_0d}{16\pi^2}\int_V\frac{rsin^3{\theta}drd{\theta}d\phi}{[r^2 -2rcos{\theta}d + d^2]^{3/2}}\hat{z} = \frac{q_eq_m\mu_0}{4\pi}\hat{z}$$

Now, suppose that we placed the electric charge at (0, 0, -d/2) and the magnetic charge at (0, 0, d/2), and we still compute the angular momentum relative to the location of the electric charge, so in the angular momentum density formula $$\vec{r} = x\hat{x} + y\hat{y} + (z + \frac{d}{2})\hat{z}$$

I won't post it here, but if you do the same calculation you will get an integral that is divergent. In both cases, we computed the angular momentum relative the location of the electric charge. The only difference is that in the first example, we placed the electric charge at the origin, and in the second example, we placed the electric charge at (0, 0, -d/2). We have changed nothing physical about the situation, but we get the widely agreed upon finite answer in the former example, and a divergent integral in the latter example.

Does angular momentum depend on the coordinate system, and if so does that mean it's not an inherent physical property of the system?