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Loxias
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Homework Statement
A bead of mass m and charge q is placed on a frictionless, rigid rod that is spun
about at one end at a constant rate w on the xy plane. There is a constant magnetic
field in space [tex] B = B_0\hat{z} [/tex]
Homework Equations
Write the Lagrangian for the system, use the generalized coordinate r (the
distance of the bead from the origin).
The Attempt at a Solution
I chose
[tex]
x = rcos(wt) ,
y = rsin(wt)
[/tex]
and from
[tex] v = rw [/tex]
we get
[tex] v = (wrcos(wt), wrsin(wt)) [/tex]
assuming vector potential
[tex] \vec{A} = B_0(0,x,0) [/tex]
and [tex] L = \frac{1}{2}m V^2 + qV\vec{A} [/tex]
I get
[tex] L = qB_0wr^2cos(wt)sin(wt) + \frac{1}{2}m (\dot{r}^2 +r^2w^2) [/tex]
deriving equations of motion:
[tex] m\ddot{r} = mrw^2 + 2rB_0qwcos(wt)sin(wt) [/tex]
which is good unit-wise.
My question is, did I derive everything right or did I forget something or misused the potential of magnetic field?
Thanks :)
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