(I am just beginning to understand this material, so the question may seem stupid). The equation for Lorentz Force: $\mathbf F=q(\mathbf E + \mathbf v \times \mathbf B)$ describes the force that acts on an electric charge $q$. Both the electric field and the magnetic field act on the charge (if it is moving with respect to the magnetic field). But is it to be understood that the electric field $\mathbf E$ that appears in the above equation is not the one that may be the result of electromagnetic induction?

I am asking this because I guess that the way a magnetic field affects a charged particle is totally captured by the second addendum in the definition of the Lorentz force i.e. $(\mathbf v \times \mathbf B)$, while the electric field $\mathbf E$ in the equation makes reference only to electric field due to the distribution of charges. I am correct or I am missing something? And if so, can we say that the total force that acts on a charged particle is simply $q\mathbf E$ if we define E by taking into account also the electric field generated by (or the way the electric field is affected by) a changing magnetic field? And if I am wrong, does this mean that a magnetic field acts in two ways on a charged particle: one: by creating an electric field, and two, in the way described by the Lorentz force? (but this second option seems wrong to me).


1 Answer 1


The Lorentz force expression involves the total $\mathbf{E}$ and the total $\mathbf{B}$.

Sometimes, you can have a situation where you start with only nonzero $\mathbf{E}$ or $\mathbf{B}$, but by changing its value, necessarily make the other field nonzero by induction. In that case, both $\mathbf{E}$ and $\mathbf{B}$ will be nonzero and both contribute to the Lorentz force. In fact, it doesn't really make sense to say "this $\mathbf{E}$ is really from a changing $\mathbf{B}$, so it doesn't count". Fields don't keep track of where they came from, they just have the values they do.

You might ask: doesn't this make electromagnetism rather complicated? After all, a changing $\mathbf{B}$ can induce a changing $\mathbf{E}$, which in turn can induce a changing $\mathbf{B}$, and so on infinitely. Do we really need to account for all this complication to figure out the force? Yes, we do! In introductory textbooks, you usually consider situations where the higher terms are small and hence can be neglected, but that's not true in general.


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