How Can the Lorentz Force Be Used to Solve the Magnetic Mirror Problem?

[Rafimah]

Homework Statement

Hello!

I am currently working on the following problem:

A charged particle with a perpendicular velocity $ v_{perp,0} at z = 0 moves along the axis of a magnetic mirror with a magnetic field strength given by $ B = B_0 (1+\alpha(t)^2 z^2) where $\alpha$ is a function of time.

a) What are the two adiabatic invariants for the electron in the mirror?

b) What is the equation of motion in the z direction?

c) If $\alpha$ is constant in time, what are the solutions for z(t) and $v_z(t)$ if $v_z = v_{z,0} $ at z = 0 ?

d) Assuming $\alpha$ changes slowly in time, then what is an expression for the amplitude of the oscillation in z and of the velocity of the particle when it passes through z = 0 as a function of $\alpha$, $B_0$ and of $v_{z,0} ?

Relevant Equations

Lorentz equation, mu and J adiabatic invariants (given in a)

a) I know the invariants are $\mu = \frac{0.5*m*v_{perp}^2{B} $ and $J = v_{parallel} x

b) I used the invariance of $\mu$ to get the following equation:

$$ v_{perp}^2 = v_{perp,0}^2(1+\alpha(t)^2 z^2) $$

I am thinking of using the Lorentz force to get $v_z$, but I'm not so clear on how to go about that.

Once I have $v_z$ properly, I believe I know how to do the rest. Just change it to \frac{dz}{dt} and integrate to get z(t), and take the derivative to get v(t).

Thanks!

 

[mark726]

Hello,

Thank you for sharing your thoughts on the invariants and your approach to finding the solution. Using the invariance of $\mu$ is a good starting point and it seems like you have a good understanding of how to use it to find the equation for $v_{perp}$.

As for using the Lorentz force to find $v_z$, you are correct in thinking that it can be used to find the velocity in the z-direction. The Lorentz force is given by $F = q(v \times B)$, where $q$ is the charge of the particle, $v$ is the velocity and $B$ is the magnetic field. Since we know the invariants, we can use them to find the velocity components and then use the Lorentz force to find $v_z$.

Once you have $v_z$, you can follow the rest of your approach to find the solution. It seems like you have a good understanding of how to solve this problem. Let me know if you have any further questions or need any clarification.