Microscopic Lorentz force density

Question

Consider a system of $N$ point charges $q_k$ which follow trajectories $\boldsymbol r_k(t)$. The microscopic charge and current densities of this system of $N$ charges are given by

$$\xi (\mathbf{r},t) = \sum_{k=1}^N q_k \delta(\mathbf{r}-\mathbf{r}_k(t)) \tag{1}$$

$$\boldsymbol j (\mathbf{r},t) = \sum_{k=1}^N q_k \mathbf{v}_k(t)\delta(\mathbf{r}-\mathbf{r}_k(t)) \tag{2}$$

All books I have read say that the microscopic Lorentz force density is

$$\boldsymbol f_{micro} = \xi \boldsymbol e + \boldsymbol j \times \boldsymbol b \tag{3}$$

where $\boldsymbol e$ and $\boldsymbol b$ are the (total) microscopic electric and magnetic fields.

I have two questions:

1. I know that the net electromagnetic force on a given point charge, let's say $q_3$, is calculated from the e.m. field due to all others charges, $q_1,q_2,q_4,q_5...$, at the position of $q_3$. HOWEVER, in equation (3) the e.m. field $(\boldsymbol e,\boldsymbol b)$ is supposed to be the TOTAL e.m. field due to ALL charges and external sources. But just at the position of $q_3$ the field created by $q_3$ is singular. So it seems to me that the Lorentz force is not well defined by equation (3).

   [If instead of the MICROSCOPIC fields $(\boldsymbol e,\boldsymbol b)$, we use the LOCAL (or molecular) fields, i.e. the fields due to all charges minus the one on which the force is calculated, then equation (3) perhaps will work, I am not sure. Anyway, the concept of LOCAL field is never used in this context; it is mainly used in relation to molecular polarizability in dielectrics, $\boldsymbol p =\alpha \boldsymbol E_{local}$].

2. The net electromagnetic force on the distribution $(\boldsymbol \xi,\boldsymbol j)$ given by (1)-(2) is:

   $$\boldsymbol F = \int_V (\xi \boldsymbol e + \boldsymbol j \times \boldsymbol b) dV \tag{4}$$

   where $V$ is a region of space containing $(\boldsymbol \xi,\boldsymbol j)$. Equation (4) seems to be free of the ambiguities of equation (3): since a body cannot exert a net force on itself (internal forces cancel each other), we can use in equation (4) the EXTERNAL FIELDS $\boldsymbol E_{ext}$ and $\boldsymbol B_{ext}$. Then equation (4) would be well defined.

   BUT it seems to me that the above is NOT true, because the magnetic forces DO NOT obey Newton's Third Law (see, for example, Griffiths, Introduction to Electrodynamics (4th edition, 2013), section 8.2.1 "Newton's Third Law in Electrodynamics"). This means that the internal forces do not cancel, and therefore the mechanical momentum in not conserved. This is the reason why in order to restore momentum conservation in electrodynamics, we need to take into account the momentum carried by the fields themselves. So we must keep the TOTAL e.m. field $(\boldsymbol e,\boldsymbol b)$ in the integral equation (4), and thus we run into the same issues as with the differential version (3).

I would like to clarify these ideas. Specifically, I want to know the following:

If equation (3) is correct, what exactly are the fields $(\boldsymbol e,\boldsymbol b)$ appearing in it? If they are the true microscopic fields, then we must include all the e.m. sources in the universe, in particular, the microscopic fields due to the system of $N$ charges under consideration. But then the microscopic net field at the position of a given point charge is always singular, because the field created by the charge itself diverge precisely at the position of the charge.

Thanks.

@Woe: The problem persists as long as we are working with point charges. Substitution of (1)-(2) in equation (4) yields:

$$\boldsymbol F = \sum_{k=1}^N q_k \int_V \left[ \delta \left( \mathbf{r}-\mathbf{r}_k(t) \right) \boldsymbol e (\mathbf{r},t) + \mathbf{v}_k(t)\delta \left(\mathbf{r}-\mathbf{r}_k(t) \right) \times \boldsymbol b (\mathbf{r},t) \right] dV $$

$$\boldsymbol F = \sum_{k=1}^N q_k \left[ \boldsymbol e (\mathbf{r}_k(t),t) + \mathbf{v}_k(t) \times \boldsymbol b (\mathbf{r}_k(t),t) \right] \tag{5}$$

All fields in equation (5) are singular at $\mathbf{r}_k(t)$.

IF we POSTULATE that the electromagnetic force in macroscopic matter is given by

$$\boldsymbol F = \int_V (\rho \boldsymbol E + \boldsymbol J \times \boldsymbol B) dV \tag{6}$$

where $\boldsymbol E$, $\boldsymbol B$ are the MACROSCOPIC fields, and $\rho$, $\boldsymbol J$ are the MACROSCOPIC charge and current densities, then everything works fine. No singularities, no problems. BUT then I claim that we cannot properly derive the macroscopic Lorentz force (6) from the microscopic equations (3)-(4).

Answer 1

Microscopically speaking, the fields $\mathbf E$ and $\mathbf B$ that you refer to are a solution to Maxwell's equations $$ \nabla\times \mathbf B - \frac{1}{c^2}\frac{\partial \mathbf E}{\partial t} = \mu_0 \mathbf J \quad (1) $$ where $\mathbf J = \mathbf J_\text{ext} + \sum\limits_{i} q_i\mathbf v_i\delta(\mathbf r-\mathbf r_i)$ is the total current comprising both external contributions and one coming from the charged particles of the system $i=1,...,N$. Also, we add Faraday's equation $$ \nabla \times \mathbf E = -\frac{\partial \mathbf B}{\partial t} \quad (2) $$ to relate $\mathbf E$ and $\mathbf B$, so that we have a closed system of equations for the fields. The extra variables related to the charges that appear in $\mathbf J$ have their own independent equations of motion, which will also couple to $\mathbf E$ and $\mathbf B$. So, as far as this classical picture is valid, there is no ambiguity in this set of equations - they are all determined provided initial conditions. In reality, the electric field of Eq.(1) only denotes the transverse part, and there is an extra longitudinal electric field that is not an independent variable because it is fully determined by charge distribution - both internal and external -, given by the following solution to Poisson's equation $$ \mathbf E_\parallel(\mathbf r,t) = \frac{1}{4\pi\epsilon_0} \int d^3r' \frac{\rho(\mathbf r',t)}{|\mathbf r - \mathbf r'|}, \quad (3) $$ You can check that $\nabla \cdot \mathbf E_\parallel = \rho/\epsilon_0$. This introduces no new independent variables, hence the system is closed and the problem is, in theory, solved. Lorentz force is then calculated with the solutions $\mathbf E$ and $\mathbf B$ from the above equations.

Now you want to know about the precise value of the field at a point where a charge is located. To answer to that question, the above classical electrodynamics is not adequate and the mathematical framework of your question fails - for example, an expression like $\mathbf v_i\delta(\mathbf r-\mathbf r_i)$ makes little sense. You will need to promote the canonical variables of the charges to operators, quantize the electromagnetic field and construct a quantum field theory involving charges and photons. I see from you question that you are interested in classical electrodynamics, that fails to explain the divergence at the location of a charge. Quantum mechanically, the charge does not have a specific location, and is rather a matter wave, so that the divergence problem can be fixed. Classical theories, in reality, fail to explain the microscopic world.

To conclude, classical electrodynamics deals with smooth charge and current distributions $\rho$ and $\mathbf J$ that enter equations (1) and (3), which have closed and well determined solutions. True microscopic questions cannot be addressed.