# Speed of a charge in a magnetic field

Does speed of a charged particle change in a non-uniform magnetic field?

I know that a uniform magnetic field cannot change the $KE$ of the particle, i.e. $\frac{1}{2}mv^{2}$ is constant. And we are considering mass to be constant (i.e. not considering relativistic effects). Therefore $v$ or speed is constant.

I also know that the velocity of the particle has to change, as there will be a change in the direction. The charged particle is displaced unequally in equal intervals of time but covers equal distances in equal intervals of time.

But what will happen when the charge moves through a non-uniform magnetic field? I suppose the results should be similar. Or are they?

EDIT

@user3814483 -

$$\frac{mv^{2}}{r}=qvB$$ ; $$\therefore \frac{mv}{r}=qvB$$ $$or$$ $$r=\frac{mv}{qB}=\frac{p}{qB}$$

So, a stronger magnetic field strength will just make the radius smaller. So, the motion of the $p^{+}$ will be somewhat like this?

Short answer: even in non-uniform fields, the speed won't change, but the guiding center can drift with some velocity.

In a magnetic field (uniform or otherwise), the force on a charged particle is:

$$F = q\; \vec{v} \times \vec{B(x)}$$

The direction of this force will be tangential to the velocity vector because of the cross product (the result of the cross product is mutually orthogonal to the vectors $\vec{v}$ and $\vec{B}$). So the force will only act to change the direction of the particle's velocity. The individual velocity components will change, as you say, but the speed cannot change because magnetic fields do no work.

As an aside, in a non-uniform field the guiding center of the particle can drift. The guiding center is easily understood by considering gyration of a charged particle in a uniform field. It will gyrate about a center with some radius. That centroid is called the guiding center.

In a non-uniform field, the motion of the charged particle will look like a cycloid instead of a circle, because in regions of higher field the particle will have a tighter radius than in regions of lower field. This ultimately results in a whole drift of the particle's guiding center. This is called the Grad-B drift.

• In this answer, Lok said that speed will change when magnetic field's strength will change : Lok's answer
– user49111
Commented Sep 27, 2014 at 7:33
• Because of the EM radiation during the circular motion the particle loos energy and it is always a spiral. Commented Sep 27, 2014 at 9:19
• @TuPapi Right, I'm neglecting synchrotron radiation. That has nothing to do with uniform vs. nonuniform fields, though, in both cases the electron will lose some energy. The effect is only important for energetic electrons. Commented Sep 27, 2014 at 17:20
• @TuPapi yes, the motion will be as you've drawn it. You can keep drawing the trajectory following your same logic, and you should find that the trajectory for long periods of time will start looking like a cycloid. The wikipedia link in my post shows the trajectory (See the first figure, part D). Commented Sep 28, 2014 at 0:45
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