# What is the origin of Lorenz force?

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• patric44
In summary: The formula is$$\frac{{\partial}{\partial t}}{\partial x}\mathbf{F}(x,t)=\frac{{\partial}{\partial x}}{\partial y}\mathbf{G}(x,t)+\frac{{\partial}{\partial y}}{\partial z}\mathbf{B}(x,t),$$where ##\mathbf{F},\mathbf{G}## are 3-vector fields, and ##\mathbf{B}## is a 4-vector field. The Lorenz-Lorenz formula is a special case of the more general Maxwell's equations.The Lorentz-Lorenz formula can be written

#### patric44

hi guys
one of my friends asked me about the origin of the electromagnetic induction, I know somehow that its related to the Lorenz force as the electrons in the conduction band of say the coil interact with the magnetic field ,which results in separation of electrons and positive ions which creates a net potential difference and so on (correct me if I am wrong), but then why do charged particles interact with magnetic fields in such a way? in another words what is the fundamental reason of Lorenz force?

another question:
I am also assuming that this interaction with the magnetic field has nothing to do with the interaction with the magnetic dipole moments resultant from the intrinsic spin of these particles, isn't that correct?

$$\mathbf{F}=q(\mathbf{E}+\mathbf{v}\times \mathbf{B})$$

anuttarasammyak said:
$$\mathbf{F}=q(\mathbf{E}+\mathbf{v}\times \mathbf{B})$$
yes

If you find no problem on the first term, a tricky solution is to transfer to the frame of reference where v=0. The second term vanishes there. Though E also changes you can get force from it and then go back to the original frame of reference transferring the force. Your issue is rooted in relativity of electromagnetism.

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BvU and patric44
In covariant form Lorentz force is
$$\frac{dp^\alpha}{d\tau}=q u_\beta F^{\alpha\beta}$$
where u is 4-velocity of particle and F is electromagnetic antisymmetric tensor whose independent 6 components are 3 of electric field and 3 of magnetic field. Here the first term and the second term of Lorentz force are treated in a same manner.

patric44
One should first emphasize that it's Lorentz force and not Lorenz force. It goes back to the Dutch physicist Hendrik Antoon Lorentz, who discovered "classical electron theory", i.e., he started the modern understanding of electromagnetic phenomena as the interaction of charged particles ("electrons") with the electromagnetic field.

One should also emphasize that there is one Lorentz force, which is (in SI units)
$$\frac{\mathrm{d} \vec{p}}{\mathrm{d} t}=\vec{F}=q (\vec{E}+\vec{v} \times \vec{B}). \qquad (*)$$
In manifest covariant form the equation of motion for a charged particle in an electromagnetic field (neglecting the notorious problem of "radiation reaction") is given in #5. Note that these equations can be split in temporal and spatial components as
$$\frac{\mathrm{d} p^0}{\mathrm{d} \tau}=q \vec{E} \cdot \vec{u}, \quad \frac{\mathrm{d} \vec{p}}{\mathrm{d} \vec{\tau}}=q (u^0 \vec{E}+\vec{u} \times \vec{B})=\frac{q}{\sqrt{1-\beta^2}} (\vec{E}+\vec{v} \times \vec{B}),$$
where ##\beta=|\vec{v}|/c## and ##\gamma=1/\sqrt{1-\beta^2}##. Multiplying the latter equation by ##1/\gamma## leads to (*). One should note that
$$p^{\mu}=m u^{\mu}=m \frac{\mathrm{d} x^{\mu}}{\mathrm{d} \tau}$$
is a four-vector with the proper time of the particle defined by ##\mathrm{d} \tau=\mathrm{d} t \sqrt{1-\beta^2}##.

Ludwik Lorenz was a Danish physicist. The similarity of their names is indeed confusing and sometimes to the disadvantage of Lorenz, who nowadays is rightfully credited as the discoverer of the advantage of the Lorenz gauge. In many older textbooks they called it "Lorentz gauge", but Lorentz used it some years later than Lorenz, and so it's more to the historical facts to call it Lorenz gauge.

The name of both phycists occurs in the theory of dielectrics in the socalled "Lorentz-Lorenz formula".

BvU and patric44