Changing magnetic field (magnetic mirror)

[Kara386]

Homework Statement

I think this is a 'magnetic mirror' question - the field lines converge on the z axis.

For a particle moving into an area of increasing field strength, where the field lines converge. Assume the ##B## field is rotationally symmetric about the ##z## axis, with ##B = B_z \hat{z}## being the main field, ##B_r## pointing inwards and ##B_φ = 0## (cylindrical coordinates). Use the fact that this field has ##div B = 0## to find ##B_r##, and show that ##\frac{v_p^2}{B}## is a conserved quantity by considering the Lorentz force.

Homework Equations

The Attempt at a Solution

I'm a little rusty on cylindrical co-ords, but I think the field can be written as
##B = B_r \hat{r} + B_z \hat{z}##
In cylindrical co-ordinates the divergence of the field is (using the fact it's divergence free):
##\frac{1}{r}\frac{\partial{(r B_r)}}{\partial r} + \frac{\partial B_z}{\partial z} = 0##

With a converging field clearly ##B_z = B(z)##, and using the product rule on the first term:

##\frac{\partial B_r}{\partial r} + B_r + \frac{\partial B_z}{\partial z} = 0##

Which gives

##B_r = -\frac{\partial B_r}{\partial r} - \frac{\partial B_z}{\partial z}##

The equation of motion for this system is

##m\frac{dv}{dt} = q(v \times B) ##
Which can be calculated from the determinant

##m\frac{dv}{dt} = q
\left| \begin{array}{ccc}
\hat{r} & \hat{\theta} & \hat{z} \\
v_r & v_{\theta} & v_z \\
B_r & 0 & B_z\end{array} \right|##

##= qv_{\theta} B_z \hat{r} - \hat{\theta} q(vB_z - v_zB_r) - \hat{z} q v_{\theta} B_r##

Can't see how to show from there that the quantity ##\frac{v_p^2}{B}## is conserved. Not even sure that determinant is right! Thanks for any help! :)

 

[Fred Wright]

First, your expression for ## B_r ## is incorrect. You dropped a factor of r. Since the change in the field with respect to r is much greater than the change in the field with respect to z you can show that ## B_r## is approximately proportional to r.
Secondly, look up the expressions for velocity in cylindrical coordinates and plug them into your determinant.
Third, for ##\frac{{v_p}^2}{B}## to be a conserved quantity it must be a constant of motion. That is, $$ \frac{d({\frac{{v_p}^2}{B}})}{dt}=0$$

 

[Kara386]

First, your expression for ## B_r ## is incorrect. You dropped a factor of r. Since the change in the field with respect to r is much greater than the change in the field with respect to z you can show that ## B_r## is approximately proportional to r.
Secondly, look up the expressions for velocity in cylindrical coordinates and plug them into your determinant.
Third, for ##\frac{{v_p}^2}{B}## to be a conserved quantity it must be a constant of motion. That is, $$ \frac{d({\frac{{v_p}^2}{B}})}{dt}=0$$

Yes, I made lots of mistakes, I got there in the end though. Thanks anyway for your response! :)