# Why can we add the two terms in the Lorentz force?

The term "Lorentz force" in the title refers to (1).

my textbook states:

The Coulomb force is mediated by the electric field and acts on the charge $$q$$, that is, $$\mathbf{F}_e = q\mathbf{E}$$. It accounts for the attraction and repulsion between static charges. The interaction of static charges is referred to as electrostatics. On the other hand, the magentic force accounts for the interaction between static currents (charges traveling at constant velocities $$\mathbf{v} = \dot{\mathbf{r}}$$ according to $$\mathbf{F}_m = q\mathbf{v}\times \mathbf{B}$$. The interaction of static current is referred to as magentostatics. Taken the electric and magentic forces together, we arrive at $$\mathbf{F}(\mathbf{r}, t) = q\big[\mathbf{E}(\mathbf{r},t) + \mathbf{v}(\mathbf{r}, t) \times \mathbf{B}(\mathbf{r}, t) \tag{1}$$

Now I'm a bit confused. We have an electric field, generated by all electric charges. All those charges are static, i.e. $$\mathbf{v} = 0$$. We also have a magnetic field. This field could be generated by a current, correct? A current is basically just moving charges. So we end up in the situation, that we have the first term telling us that all our charges are static but we also have the second term telling us, some charges are moving.

I'd like to look at two cases to make my question more clear:

Case 1: We have a charge $$Q$$ in the origin. It does not move. We also place a magnet somewhere random. Now we have two fields: The electric and the magnetic field but we don't have a current. Nothing is moving, everything's fine. If we place a test particle in the system, we can observe how it propagates through space while assuming that it doesn't change any of the fields since it's a test particle. I understand this case.

Case 2: We have a charge Q in the origin. We take an electron that rotates around the charge Q in the origin (we ignore any radiation that might show up because it's rotating) but we don't have a magnet. Now we have the electric field of the charge Q, we have the electric field of the moving electron and we have the magnetic field generated by the current. I don't understand why we can use (1) here because we have a non-static charge: The electron.

More general: How can we talk about a current and be able to use (1) when a current is made up of charges that will change our electric field? And even worse, make it dynamic yet we want it static.

The Lorentz force law always applies, regardless of whether or not the charges experiencing the force are moving and whether or not the fields experienced by the charge are changing. If you're thinking about this in terms of point charges, the $$q$$ and $$\mathbf{v}$$ in the Lorentz equation are the charge and force of a particular charge moving around, while the $$\mathbf{E}$$ and $$\mathbf{B}$$ are the electric and magnetic fields created by all other charges in your system.

In your example (2), you're double-counting the electron both as the charge circulating around the atom and as a "current" creating a magnetic field. But charges never1 feel their own fields. So the electron moves in accordance of the electric field of the proton. Similarly, the electron creates an electric and magnetic field because of its motion, and the proton feels a force from those.

Somewhere in Chapter 9 or 10 of Griffiths's Introduction to Electrodynamics, he writes down the force between two charges moving in arbitrary motion. It takes into account radiation fields, magnetic fields, inductive effects, travel time delays, etc. in complete generality. He does this by calculating the $$\mathbf{E}$$ and $$\mathbf{B}$$ for a moving charge and plugging them into the Lorentz force law. The result looks horrendously complicated, but at its core it is still the Lorentz force law, just dressed up in enough complicated costumery that it could get into the Met Gala.

1 The Abraham-Lorentz force complicates this statement a bit, but if we're assuming that the effects of radiation are negligible, then so are the effects of the Abraham-Lorentz force.

$$\vec{F}=q\vec{E}$$

Does not only apply for electrostatics.

why do you think this?

It is only the coulomb force that applies for electrostatics, not the relation between E and F

• 1. With Coulomb force you mean the Coulomb law? 2. We look at (1) regarding electrostatics and magentostatics. So the question remains. To put it differently: How can we combine electrostatics with magentostatics when the later implies that we consider a current, which is inherently non-static. I simply can't wrap my had around case 2. Even if it makes sense with the formula (1), it does not make sense with the text in itself. In case two, we simply can't call it electrostatics, can we? Jul 28, 2022 at 9:14
• We dont just combine electrostatics and magnetostatics, we combine E and B regardless of the fields validity in the static situation. Electrostatic formulas work for constant charge densities, a current has a constant charge density. But this is beside the point. What ever the fields actually are, if what we enter into the lorentz force. Jul 28, 2022 at 11:31
• The lorentz force is a general relation between E,B and F. The validity of using the electrostatic and magnetostatic formulas an inputs into the equation is dependant on what charge and current density function you want to calculate the force for. For example a single moving charge would not satisfy the magnetostatic and electrostatic equations. Jul 28, 2022 at 11:49