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I know some of you might be tempted to just answer “No, the Lorentz force acting on a particle is in every point perpendicular to the velocity vector” but please help me with my doubt.

Let’s say we have an electron that for some undefined reason is already moving in a circle and thus has a magnetic moment m. Then it also has an initial tangential velocity which magnitude is v, and again, it has a centripetal acceleration with magnitude a1 = v^2/R.

If we now allow a magnetic field B perpendicular to the surface area within the circumference drawn by the trajectory of the electron, we should notice a Lorentz force immediately acting on the electron and pointing in the same direction of the initial basic centripetal force that used to mantain the uniform circular motion when no magnetic field was applied.

As a result, the electron will now experience a total centripetal force given by the sum of the Lorentz force and the initial force.

One could write (a2 = ma1 - evB)/m. At the beginning of the experiment though, the total centripetal acceleration was just a = v^2/R.

The magnitude of the tangential velocity must have changed then if the radius R stayed the same (imagine in fact, that the electron was circulating into a single coil).

Now, this phenomena doesn’t happen in real life (right?), so how does physics justify such a paradox? Increased velocity should also increase the magnitude of the Lorentz force, which again would increase the velocity of the electron and so on in an infinite loop. Free infinite current basically. What is the right math behind everything?

Thanks in advance for your patience.

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  • $\begingroup$ If the radius doesn't change, that means there's an additional force opposing your new Lorentz force. For example, if you confine the electron to a ring, what you're really doing is always exerting an external force to make sure the centripidal force never changes no matter how strong B gets. There's no other way to confine an electron to a ring, other than with forces! $\endgroup$ Commented May 9, 2021 at 23:34
  • $\begingroup$ Your second equation is incorrect. First, the added magnetic force is in the same direction as the force that causes the first acceleration so the terms should add. Also, since magnetic field is changed in time, induced electric field will be present and accelerate the particle. Thus when the magnetic field stops changing, the particle will have higher orbital velocity, due to effect of induced electric field. $\endgroup$ Commented May 10, 2021 at 0:24

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If the radius is going to stay the same as you require (which is fine), the centripetal force needs to stay the same. The initial force that was the agent of that centripetal force will change if some force is added or subtracted in that direction.

If (for simplicity) a string was supplying $F_C$, then you add a magnetic force in that's acting in the radial direction, $F_C$ will stay the same while $F_{string}$ decreases by exactly $F_B$. Initially, $F_C = F_{string1}$ , then $F_C = F_{string2} + F_B $.

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  • $\begingroup$ That's the most "satisfying" answer I got and despite me not being an expert (so i can't really judge) it feels like the more natural answer, but then why is that? Is the *F_string decreasing after the appearance of the Lorentz force an hypothesis we supposed when studying such phenomenon, which was then confirmed by experiments? To better explain what I'm trying to say, did we say at some point in history: "The Lorentz Force is always perpendicular to the velocity vector, so given the fact that it doesn't change the velocity magnitude, it must happen that the F_string decreases" ? $\endgroup$ Commented May 10, 2021 at 9:17
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If the electron is already rotating, the effect of the Lorentz force from an uniform and perpendicular $B$ is to decrease the radius. The magnitude of the velocity doesn't change and $a_c = \frac{V^2}{r}$ increases with a smaller radius.

If there is some restriction (forcing the movement only along a coil for example), that restriction translates in a outward (centrifugal) force that balance the Lorentz force.

We can think of a ball bearing without cage in a horizontal position with just one (charged) sphere inside. It is an ideal situation without any friction, and the ball doesn't roll but slips freely. There is a small clearance of some cents of millimeter between the ball and the bearing races.

After an initial kick, the ball keeps in a circular motion, touching the outer bearing race, that provides the centripetal force.

After turn on a perpendicular and uniform magnetic field, the Lorentz force adds an additional centripetal force. But the inner race doesn't allow a decrease of the radius. So the ball keeps moving with the same velocity, but now touching the inner race.

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  • $\begingroup$ But what happens if the radius is fixed? $\endgroup$ Commented May 10, 2021 at 9:21
  • $\begingroup$ In this case the movement (the acceleration) doesn't change because the net force (sum of Lorentz force plus whatever is restricting the displacement of the charge) doesn't change. $\endgroup$ Commented May 10, 2021 at 10:59
  • $\begingroup$ But the net force does change. We start with the unknown force that moves the electron in a circle and end up with the same force plus the Lorentz force. The radius can't change in this example so...I really don't understand. $\endgroup$ Commented May 10, 2021 at 12:54
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    $\begingroup$ I edited my answer. $\endgroup$ Commented May 10, 2021 at 15:27
  • $\begingroup$ I see and that's the answer I came up with myself this morning and also what my physics professor gave me (finally, he didn't respond before), so I assume it's the correct one. Thank you so much!!! I have been dying these days to figure it out! I'm probably just sort of stupid. $\endgroup$ Commented May 11, 2021 at 10:16

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