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Magnetic Mirror Problem

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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} ?
Homework 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!
 

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