Re: [Phys-l] drop a metal cylinder through a solenoid

[John Denker <jsd@av8n.com>]

On 03/24/2012 12:19 PM, Carl Mungan wrote:
> Suppose the metal bob is a cube of side d. Magnetic field is B. Bob
> is initially raised up to 90 degrees, a distance of L above the
> lowest point of its swing (ie. the arm length).
>
> I would start by saying motional emf is v*B*d where v is speed of the
> bob at any point. If the damping is small, v isn't much different
> than regular pendulum, sqrt(2*g*L*cos(theta)) where theta is angle of
> pendulum relative to vertical.
>
> Now it gets trickier. That emf must correspond to some charge
> separation between opposite edges of the bob (edges of area d2 that
> are a distance d apart).

Agreed.

> Perhaps I can model it as a parallel-plate capacitor.

Gack!

It might be easier to analyze things in the frame instantaneously
comoving with the bob.  In this frame, there is an electric field
plus the (now irrelevant) magnetic field.  We now have a dielectric
object in an applied electric field.  Infinite dielectric constant.

This problem is doable analytically in selected geometries.  It is
easy to do numerically in general.  Relaxation methods work fine.

>  I'll also need the drift velocity to get the time during which the charges move.

That seems unnecessarily microscopic.  I reckon the macroscopic 
resistivity is all we need to know.

For simplicity, assume the RC time constant is tiny compared to the 
timescale over which the bob speed changes.  (This assumption can be 
lifted if necessary.)

> that will only get me power

To find the energy, integrate dt.  Integrate over one cycle.
Divide by the stored energy to find 2π/Q.

1/Q is the fractional energy loss per radian.  2π/Q is the fractional
energy loss per cycle.