Homework Help: Particle motion in a magnetic ﬁeld

1. Apr 7, 2013

Phyrrus

1. The problem statement, all variables and given/known data
Assume the earth's magnetic field is almost homogeneous with direction along the z-axis, with a small inhomogeneous modification which make the field lines converge towards the z-axis. Also ignore relativistic and gravitational effects.
.
First assume the magnetic ﬁeld, B = B$_{0}$ = B$_{0}$k, to be time independent and homogeneous, with
k as a unit vector in the z-direction. A particle with charge q and mass m is moving in this ﬁeld.
Initially, at time t = 0 the particle has velocity v$_{0}$, with u$_{0}$ as the component in the z-direction and
w0 as the component in the x; y-plane.
a) Write the vector form of the equation of motion of the particle and show that it has solutions of
the form
r(t) = ρ$_{0}$(cos $\omega$$_{0}$ti + sin $\omega$$_{0}$0tj) + v$_{z}$t k
Determine the constants ρ$_{0}$, ω$_{0}$ and v$_{0}$ in terms of the initial velocity and magnetic ﬁeld strength B0.

2. Relevant equations
F=q(E+v$\times$B)

3. The attempt at a solution
E=0
a=(q/m)(v$_{0}$$\times$B$_{0}$)

v$_{0}$=(w$_{0}$,u$_{0}$)
where w$_{0}$=|w$_{0}$|cos$\vartheta$i+w$_{0}$sin$\vartheta$j)

therefore v$_{0}$$\times$B$_{0}$ = B$_{0}$|w$_{0}$|sin$\vartheta$i-B$_{0}$w$_{0}$cos$\vartheta$j

2. Apr 7, 2013

Staff: Mentor

What is ϑ?
For (a), all you have to do is to calculate the acceleration vector based on the given formula for r(t), and show that it satisfies the equation for the Lorentz force.

3. Apr 8, 2013

Phyrrus

The $\vartheta$ was the angle used to separate w$_{0}$ into x and y components.

But I'm sorry, can you please elaborate more? Do you mean all I need to do is differentiate r(t) twice and equate coefficients?

4. Apr 8, 2013

Staff: Mentor

"show that it has solutions of the form" -> show that [equation] is a solution -> show that [equation] has the Lorentz force as acceleration.

With the calculated v0 x B0, right.

5. Apr 8, 2013

Phyrrus

Ok, but how do I calculate v$_{0}$$\times$B$_{0}$ when v$_{0}$=(w$_{0}$,u$_{0}$) ? Don't I need to separate w$_{0}$ into x and y components?

6. Apr 8, 2013

Staff: Mentor

I would convert everything to polar coordinates, but splitting them in components in Cartesian coordinates is possible, too.

7. Apr 11, 2013

Phyrrus

I'm sorry, I still don't quite get it

8. Apr 11, 2013

Staff: Mentor

Did you do that? What did you get?

9. Apr 11, 2013

Phyrrus

I'm not going to lie, I couldn't do it.

10. Apr 11, 2013

Staff: Mentor

What did you get? Where did you run into problems?
Those are obvious follow-up questions, you could have answered them without an extra post from me...

11. Apr 11, 2013

Phyrrus

I'm still not entirely sure what I am supposed to do. Is this right?

d^2r/dt^2 = -$\omega_{0}$$^{2}$$\rho_{0}$(cos$\omega_{0}$t i + sin$\omega_{0}$ j)

Now we can say that, the above acceleration expression is equal to the Lorentz acceleration of (q/m)*(v$_{0}$$\times$B$_{0}$?

But now how do I split v$_{0}$$into i,j,k components? I tried splitting it into components before with the angle theta, but that was obviously wrong. 12. Apr 11, 2013 mfb Staff: Mentor \vec{v} \times \vec{B} = \begin{pmatrix} v_j B_0 \\ -v_i B_0 \\ 0 \end{pmatrix} where I used that the B-field has a component in k-direction only. v_i and v_j are just the i- and j-component of $\frac{d\vec{r}}{dt}$ 13. Apr 11, 2013 Phyrrus Ok, so then v[itex]_{j}$=cos($\omega$$_{0}$t) and -v$_{i}$=sin($\omega$$_{0}$t)

And then the constants -$\omega$$_{0}$$^{2}$$\rho$$_{0}$=(q/m)*B$_{0}$

14. Apr 11, 2013

Staff: Mentor

I think there are prefactors ρ_0 ω_0 missing, but the concept is right.

15. Apr 11, 2013

Phyrrus

So once I find the prefactors, I am given a 3 set of relations from which I can solve for the 3 constants?

16. Apr 11, 2013

Staff: Mentor

Right.

17. Apr 12, 2013

Phyrrus

Ok, thanks for all your help. How do I find them?

18. Apr 12, 2013

Staff: Mentor

Find what?
It is all in the thread now, you just have to combine it.