# Testing Lorentz force on a moving macroscopic charged conductor?

Did anyone ever made an experiment with a moving macroscopic charged conductor in a magnetic field testing its deviation due to Lorentz force? I mean: what moves here is the massive charged conductor itself, not only the charges inside it. Would it be feasible with our technology? If it is then why does this force only appears in late 1800's in the scientific literature?

A macroscopic object with charge $$q$$ will not experience a magnetic force of the form $${\bf F}=q(\bf v\times B)$$ if it were to pass through a magnetic field $$\bf B$$

And of course if there is an electric field, then we have the full form of the Lorentz (electromagnetic) force $${\bf F}=q(\bf E+ v\times B)$$ and charged particles experience a local electromagnetic force according to this equation. That is, this equation does not work for macroscopic objects.

If we wanted to extend the Lorentz force law to macroscopic objects, we would need to take into account additional quantities like the polarization (density) $$\bf P$$ and magnetization (density) $$\bf M$$ to be sources of the electromagnetic field and this can severely complicate the situation.

In such a case, the Lorentz force$$^1$$ (density) takes on a much different form $${\bf F} = (\rho_f −\nabla\cdot {\bf P}){\bf E} + ({\bf J_f}+ \frac{\partial {\bf P}}{\partial t})\times \mu_0{\bf H} − (\nabla\cdot {\bf M}){\bf H} − (\frac{\partial {\bf M}}{\partial t})\times \epsilon_0 {\bf E}$$

where $$\rho_f$$ and $$\bf J_f$$ are free charge and current densities respectively, where $${\bf B}= \mu_0 {\bf H} + \bf M$$

Note that the equation for the Lorentz force density requires many assumptions to be valid, and an explanation of this and a derivation can be found here. This equation has been tested, but for very simple systems and the experiment appears to be consistent with the equations.

$$^1$$ Note that many authors have many other forms to this equation depending on the setup, assumptions, boundary conditions etc., and the equation above for the electromagnetic force density is in the most general form.

• So the Coulomb force applies both to the charges and as a result to the whole charged conductor but the same doesn't apply to the Lorence force? Why this difference? Commented Oct 31, 2021 at 7:12
• Yes, the Coulomb force does. The Lorentz force applied to particles and is not suited to macroscopic bodies. This is determined through experiment. Cheers. Commented Oct 31, 2021 at 8:22
• But they are indeed the same force seen from different frame of referecnces, as shown in Special Relativity. Commented Oct 31, 2021 at 8:32
• But be careful. The magnitude can differ by $\gamma$ (Lorentz factor). The relativistically covariant form of the Lorentz force takes the form $\frac{dp^\alpha}{d\tau}=qF^{\alpha\beta}u_{\beta}$ where $p$ is 4-momentum and $F$ is the electromagnetic tensor with $u$ being a 4-velocity. To learn more about this see any intro text to relativistic electrodynamics or if you like search for "covariant Lorentz force". Cheers. Commented Oct 31, 2021 at 8:54
• It doesn't. You asked if the Lorentz force is the same in different fames of reference. That is what I answered in the last comment. Cheers. Commented Oct 31, 2021 at 9:15