# Uncharged particle in a magnetic field suddenly gets charged

Sorcerer
What happens to the velocity?

Say we have a piece of space debris in free fall. It has a particular velocity at a given time. Then say at some later time it's bombarded with ionic gas or something that causes it to become charged. Does the velocity change?

My guess would be yes, because the Lorentz Force law (in the simplest terms) is FL = q(E + v x B), and obviously when there is no charge, q is zero, right? So FL = 0, right? So whatever velocity it has prior to being charged, I'm assuming when it gains a charge, suddenly FL ≠ 0, which means the force just changed, and so the velocity must instantaneously change as well, correct?

If that's right, in what ways would the velocity change, and how dramatic a change in velocity is realistic in Earth's magnetosphere? Could we get a particle to suddenly be shot out into interplanetary space? If I'm not mistaken, it's possible for such a particle to start following a helical path when it becomes charged, or is that wrong?

Thanks for the replies.

Homework Helper
which means the force just changed, and so the velocity must instantaneously change as well, correct?
If the force changes, the acceleration changes immediately, not the velocity.

[One assumes that the charge is added gently -- an adjacent charged particle travelling at the same instantaneous velocity sticks to our previously uncharged particle and the two begin travelling together in a circular or helical path]

Sorcerer
If the force changes, the acceleration changes immediately, not the velocity.

[One assumes that the charge is added gently -- an adjacent charged particle travelling at the same instantaneous velocity sticks to our previously uncharged particle and the two begin travelling together in a circular or helical path]
Well yes the acceleration changes, but an acceleration is a change in velocity over time, is it not? Anyway, that aside, let's assume the uncharged particle is traveling at a constant velocity (no acceleration) and at a particular angle with respect to the B field lines. And let's say the particles (or whatever) that cause the particle to become charged travel at any velocity. What would the equations of motion be? (I'm trying to find some function or relation that describes exactly how the motion of the particle would change that is dependent upon: the initial velocity of the particle; the velocity of whatever caused the particle to become charged; the direction of the magnetic field lines; the strength of the magnetic field; and time. Does such a thing exist?)

How would I mathematically describe the situation of an uncharged particle moving through a magnetic field gaining a charge at some point in time?

Anything you can give will help, even if it's just some vocabulary to google.

Homework Helper
Well yes the acceleration changes, but an acceleration is a change in velocity over time, is it not? Anyway, that aside, let's assume the uncharged particle is traveling at a constant velocity (no acceleration) and at a particular angle with respect to the B field lines. And let's say the particles (or whatever) that cause the particle to become charged travel at any velocity.
You spoke of an instantaneous change in velocity as a result of an applied force. That was incorrect. Forces cause accelerations. Accelerations are not instantaneous changes in velocity.

Now you want to contemplate an inelastic collision between a charged and an uncharged particle. That results in an impulsive change in velocity. The resulting combined particle has a mass, a charge and a velocity. There is no problem in computing its ensuing trajectory.

Staff Emeritus
What happens to the velocity?

Say we have a piece of space debris in free fall. It has a particular velocity at a given time. Then say at some later time it's bombarded with ionic gas or something that causes it to become charged. Does the velocity change?

My guess would be yes, because the Lorentz Force law (in the simplest terms) is FL = q(E + v x B), and obviously when there is no charge, q is zero, right? So FL = 0, right? So whatever velocity it has prior to being charged, I'm assuming when it gains a charge, suddenly FL ≠ 0, which means the force just changed, and so the velocity must instantaneously change as well, correct?

If that's right, in what ways would the velocity change, and how dramatic a change in velocity is realistic in Earth's magnetosphere? Could we get a particle to suddenly be shot out into interplanetary space? If I'm not mistaken, it's possible for such a particle to start following a helical path when it becomes charged, or is that wrong?

Thanks for the replies.

How is this different than the standard General Physics question of a charge particle moving with a certain velocity entering a region with some uniform magnetic field? Isn't this technically the identical problem?

Zz.

Mentor
How would I mathematically describe the situation of an uncharged particle moving through a magnetic field gaining a charge at some point in time?
Anything you can give will help, even if it's just some vocabulary to google.
If you're going to do this in any physically realistic way, you're going to need some calculus.

Write the charge as a function of time, which will give you the force and hence the acceleration as a function of time. Now you can integrate that once to get the velocity as a function of time (don't forget the arbitrary constant - it corresponds to the initial velocity) and again to get the position as a function of time.

Mentor
Isn't this technically the identical problem?
It is.

Sorcerer
If you're going to do this in any physically realistic way, you're going to need some calculus.

Write the charge as a function of time, which will give you the force and hence the acceleration as a function of time. Now you can integrate that once to get the velocity as a function of time (don't forget the arbitrary constant - it corresponds to the initial velocity) and again to get the position as a function of time.
It's not the math that's my problem so much as the physics. I only know electromagnetism in a very, very basic way. I know what Maxwell's equations are; I know they're Lorentz invariant; I know they can be combined into a wave equation; I know that c is the reciprocal of the square root of the permeability and permittivity constants.. Beyond that I know very little else. I have not taken E and M and have more or less forgotten everything in second semester intro physics on it (I took a break from school due to health reasons after getting an A.S.).

But I'll look it up now that you've given me some direction.

How is this different than the standard General Physics question of a charge particle moving with a certain velocity entering a region with some uniform magnetic field? Isn't this technically the identical problem?

Zz.
It may be, but I don't know the set up. But what I'm really interested in is if the particle is already in the field, and has a particular trajectory, and then becomes charged. It's obviously going to change its trajectory, but what are the limitations to how dramatic (with respect to time) can this change be?

Sorcerer
You spoke of an instantaneous change in velocity as a result of an applied force. That was incorrect. Forces cause accelerations. Accelerations are not instantaneous changes in velocity.

Now you want to contemplate an inelastic collision between a charged and an uncharged particle. That results in an impulsive change in velocity. The resulting combined particle has a mass, a charge and a velocity. There is no problem in computing its ensuing trajectory.
Accelerations are changes in velocity, but I was under the impression that there was no theoretical limit on acceleration like there was with velocity?

Anyway, I want to see how dramatic the change in velocity can be over a very short time window in an earth-like magnetic field.

Staff Emeritus
It may be, but I don't know the set up. But what I'm really interested in is if the particle is already in the field, and has a particular trajectory, and then becomes charged. It's obviously going to change its trajectory, but what are the limitations to how dramatic (with respect to time) can this change be?

To know how "dramatic" the change can be, then you need to have a quantitative description of the initial condition AND you will have to calculate the effect! This can't be done by hand-waving argument.

For example, if the particle is initially moving PARALLEL to the magnetic field lines, then no matter how big of a charge the particle gets, it will NOT alter its trajectory because the magnetic Lorentz force in this situation is ZERO.

So if you want to have a quantitative answer to how dramatic the change is, you will have to supply (i) the amount of charge (ii) the strength of the magnetic field (iii) the profile of the magnetic field (is it uniform? is there a spatial gradient?) (iv) the initial velocity of the particle right before it got charged, and (v) the direction of this velocity with respect to the external magnetic field.

All of the above requires numbers to be able to answer your question on how "dramatic" the change will be.

Zz.

Homework Helper
Accelerations are changes in velocity, but I was under the impression that there was no theoretical limit on acceleration like there was with velocity?
F=ma. If the force is determined by the magnitude of the charge, the local magnetic field strength and the particle's velocity relative to that field then the force is finite and calculable. Hence, so is the acceleration.

Sorcerer
F=ma. If the force is determined by the magnitude of the charge and its velocity relative to the magnetic field then the force is finite and calculable.
I was under the impression that that equation was not Lorentz invariant. But if you mean the 4-vector version, in this case is m0 constant?

Anyway I'm currently trying to find some function for q(t) that I can integrate. But why could it be continuous? I'm assuming an uncharged particle becomes charged as soon as it loses an electron?

EDIT-
Stupid question: Could I just use the Lorentz force equation, and make q a function of t?

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Sorcerer
To know how "dramatic" the change can be, then you need to have a quantitative description of the initial condition AND you will have to calculate the effect! This can't be done by hand-waving argument.

For example, if the particle is initially moving PARALLEL to the magnetic field lines, then no matter how big of a charge the particle gets, it will NOT alter its trajectory because the magnetic Lorentz force in this situation is ZERO.

So if you want to have a quantitative answer to how dramatic the change is, you will have to supply (i) the amount of charge (ii) the strength of the magnetic field (iii) the profile of the magnetic field (is it uniform? is there a spatial gradient?) (iv) the initial velocity of the particle right before it got charged, and (v) the direction of this velocity with respect to the external magnetic field.

All of the above requires numbers to be able to answer your question on how "dramatic" the change will be.

Zz.
Thanks. I'll give you a run down of what I'm after using your criteria.

(1) I'm assuming the particle is moving perpendicular to the field lines, or if not, varying from near parallel to perpendicular (but never parallel).

(2) I've not yet figured out the amount of charge for this. Actually I'm wanting it to be a variable I can manipulate to see what happens.

(3) The strength of the field is Earth's at low earth orbit.

(4) Presumably that field is not constant with respect to space (I'm guessing that since Earth is roughly a sphere, the field lines would separate the further they got from one pole, and then curve back again).

(5) I want the initial velocity to be something I can also vary to see what happens, but I'm thinking on the order of the average low orbit space junk.

(6) I supposed the most drastic effect would come from an initial velocity perpendicular to the field lines, right? But I also would like to vary this to see what happens.

As I said, I'm very raw when it comes to E&M. Could you point me in the direction to where I'd find the equations to manipulate and to input these values? I know you guys don't like doing this sort of thing, but I really am not up to speed on the basics of electromagnetism. Sorry if this is annoying.

EDIT- stupid question, but isn't the Lorentz force equation exactly what I need, except change q to q(t)? So integrating Fdt should give momentum, which has v, right?

Homework Helper
I was under the impression that that equation was not Lorentz invariant.
Why do you care? Finite coordinate 3-acceleration is still finite (and smaller) when you account for relativistic corrections.

Sorcerer
Why do you care? Finite coordinate acceleration is still finite (and smaller) when you account for relativistic corrections.
I don't really. This scenario shouldn't even be relativistic. Anyway, so would the Lorentz force law be what I need to look at and study for this scenario? It has charge, the magnetic field, and velocity.

My problem is, there is a point where q(t1)=0 and another where q(t2) > 0.

I suppose I can solve for v and then get a function v(f(q)) and plug stuff in.

Homework Helper
My problem is, there is a point where q(t1)=0 and another where q(t2) > 0.
That's a discontinuous change in acceleration. It poses no problem for an integration to obtain velocity. It does not result in a discontinuous velocity profile.

Sorcerer
That's a discontinuous change in acceleration. It poses no problem for an integration to obtain velocity. It does not result in a discontinuous velocity profile.

So, if I took this:

## d\vec v = \frac{q(t)(\vec E + \vec v \times \vec B)}{m} dt ##

then went and learned vector calculus good enough to integrate this, I should get a function for v that allows me to plug in values for v and q (and assuming E and B are available online for low earth orbit), right?

EDIT- but I suppose q(t) isn't specific enough. I'd need some function for q(t), since over time the object should be getting more charge as it's bombarded with photons, electrons, plasma and so on.

Homework Helper
I should get a function for v that allows me to plug in values for v and q (and assuming E and B are available online for low earth orbit), right?
You have a differential equation that gives you ##\frac{\vec{dv(t)}}{dt}## in terms of a formula involving ##\vec{v(t)}##. But you've not accounted for the momentum carried in by the continuing(?!) impacts of charged particles on the object in question.

Sorcerer
You have a differential equation that gives you ##\frac{\vec{dv(t)}}{dt}## in terms of a formula involving ##\vec{v(t)}##. But you've not accounted for the momentum carried in by the continuing(?!) impacts of charged particles on the object in question.
Man this is getting very complicated.

Is it reasonable to set up a situation where the momentum of the incoming particle that causes the charge is negligible compared to the momentum of the uncharged particle, and yet the change in charge is very large?

(by the way thanks for trying to point me in the right direction)

Homework Helper
Man this is getting very complicated.

Is it reasonable to set up a situation where the momentum of the incoming particle that causes the charge is negligible compared to the momentum of the uncharged particle, and yet the change in charge is very large?

(by the way thanks for trying to point me in the right direction)
In my opinion, no. It is unreasonable. Just start with a large charge instead of trying to build one from scratch.

Edit: I do not see any possibility for spectacular behavior from a particle with a -100 charge that is hypothetically assembled from 100 electrons one at a time that is any different from the behavior of a hypothetical particle with a -100 charge that is simply assumed to exist.

Sorcerer
In my opinion, no. It is unreasonable. Just start with a large charge instead of trying to build one from scratch.
I guess I could do this, and then infer from it that an uncharged particle that gains charge should have a change in motion from constant velocity to whatever dv/dt would be, I'm assuming, since ZapperZ pointed out that it's an "identical problem?"

The reason I'm looking at this is that I saw a study where they described how tiny space debris could have dramatic changes in motion by becoming charged in low earth orbit, so I'm toying with this to gain more understanding on a level that isn't publishable in peer reviewed journals.

Homework Helper
Suppose that you have two particles. Same mass. Same charge. Same velocity. Same position. Same fields around them.

One of them was always charged. The other gained charge slowly over time. What difference do you expect in their subsequent behavior? Make your life simple. Work the math for an object with a fixed charge.

Sorcerer
Suppose that you have two particles. Same mass. Same charge. Same velocity. Same position. Same fields around them.

One of them was always charged. The other gained charge slowly over time. What difference do you expect in their subsequent behavior? Make your life simple. Work the math for an object with a fixed charge.
At the point in time where their charge is the same (assuming they aren't interacting with each other), I expect their motion to be the same. For the one slowly gaining charge, I expect its motion should approach the motion of the other as ti→tf.

Anyway, I will certainly look at this in the coming week, but the only problem I have is that the real world example I want to understand involves an uncharged object gaining charge and then subsequently changing its motion.

However, I must walk before I crawl. If I don't understand the easy scenario I certainly won't understand the complex one.

Mentor
However, I must walk before I crawl. If I don't understand the easy scenario I certainly won't understand the complex one.
By far the easiest case to analyze is the one that @ZapperZ mentioned in post #5: a charged particle enters a region of non-zero E and B fields. This is equivalent to (but more physically reasonable than) the particle magically acquiring a charge or to the E and B fields being magically turned on.

Even then, the simplest case is the one in which there is only an E field, not a B field. That's the case I was describing in #6, and you can solve it without using any vector calculus or having to solve a differential equation. Get this one down before you try anything harder.

Then you can try the case in which there is a B field but not an E field. This one is harder because the ##\vec{v}\times\vec{B}## term says that the force depends on the velocity, but of course the velocity is changing under the influence of the force. The force is perpendicular to the direction of motion so it doesn't change the speed, just the direction (this is one of those times when the difference between speed and velocity really matters) and the particle follows a circular path. It's possible to get through this with neither differential equations nor vector calculus (once you recognize that the force is perpendicular to the direction of motion, the problem is analogous to a circular orbit).

Understand these two, and then you can take on the time-varying charge case and the case in which both B and E fields are present. These won't contribute any more physical insight/understanding, just an opportunity to practice with more hairy differential equations and vector calculus.