# Radial force on charged particle in beam of positive ions

## Homework Statement

Many experiments in physics call for a beam of charged particles. The stability and “optics” of charged-particle beams are influenced by the electric and magnetic forces that the individual charged particles in the beam exert on one another. Consider a beam of positively charged ions with kinetic energy E0. At t = 0 the beam has a radius R0, a uniform charge density ρ0, and is traveling in the x-direction.

(a) Derive an expression for the radial force on a charged particle in the beam at some initial radial position r0 (note that there will be electric and magnetic forces acting on the particle).

(b) Apply F = ma to determine the radial velocity of the charged particle when its radial position has increased from r0 to 2r0. You’ll need to make one assumption here – the charge density of the particle beam remains uniform (but not constant in time!) as the beam expands.

## Homework Equations

(1) Fmag = Q(v x B)

(2) Fnet = Q[E + (v x B)]

## The Attempt at a Solution

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I am not sure how to treat this situation. Do the charges in the center of the beam act like a current-carrying wire of circular cross section? In this case the volume current density would be J = I/πR02 at t=0 and the magnetic field generated by the beam at the position of the particle would be B = μ0I/2πr0. Would the electric field be that of a line charge, E = kρ0/r0? Can I plug these fields into (2) to get the radial force?

Are any of these trains of thought going in the right direction? Thank you.

## Answers and Replies

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mfb
Mentor
Do the charges in the center of the beam act like a current-carrying wire of circular cross section?
Yes.
Note that you need the magnetic field within the beam, not the magnetic field outside.
Would the electric field be that of a line charge
Sure. Same thing here, you need it inside.

You can express I via the particle velocity and given constants.
Can I plug these fields into (2) to get the radial force?
Sure, once you have the correct fields.

Ok, so would the magnetic field within the beam just be due to Ienc=ρvA, so B = μ0ρ0r0/2? And E = kρ0/r0? Or am I missing a subtlety?

I'm also a bit unsure how to deal with the direction of the force; because the field is circulating and the velocity is in x, the direction of the magnetic component of the force is rotating?

mfb
Mentor
Something is still wrong with the electric field.

The field is constant in time, there is nothing rotating. It is orthogonal to the beam direction so there is a force on the electron (which is also constant in time - assume the outwards motion is negligible).

I applied Gauss's law to get the electric field: E(2πr0L) = Qenc/ε00πr0L/ε0 → E = ρ0r0/2ε0 (only inside the beam)

When I plug that into (2) I get F = ρ0πr0L(ρ0r0/2ε0 + ½μ0ρor0v2) (where v x B is just vB since the fields are orthogonal)

but I don't think there should be a factor of length L since it wasn't given in the problem statement. And how does E0 fit in?

EDIT: I haven't been sleeping enough lately... Q is just the charge on the particle itself.

Last edited:
mfb
Mentor
Q is just the charge on the particle itself.
Right.
And how does E0 fit in?
What is the relation between kinetic energy and speed?

You can simplify the expression for the force.

I can say ½v2 = E0/m and simplify to F = qρ0r0(1/2ε0 + E0μ0/m].

If I set F=ma and get a, v = √(2aΔx). Would 2r0-r0=r0 be Δx in this case, since I'm looking for the radial velocity?

mfb
Mentor
v = √(2aΔx)
Be careful with the different directions (along the beam and perpendicular) and your variables.

The beam will expand. How does that influence the force?

The force when the particle is at 2r0 is F = 2qρ0r0(1/2ε0 + E0μ0/m], and v = √(2ar0)?

mfb
Mentor
What happens to the charge density if the beam spreads?

Does the density go as

ρ'V' = ρ0V
ρ'π(2r0)2L=ρ0πr02L
ρ' = ρ0/4 ?

So the force is overall reduced by a factor of ½ at r=2r0?

mfb
Mentor
Right.

astrocytosis
Thank you! This was very helpful