Re: [Phys-l] Magnetic force and work

["Bob Sciamanda" <trebor@winbeam.com>]

I think I need to point out the important difference between the "cyclotron" 
(circular) motion of a charged particle injected into a B field, and the 
motion which I described of the carriers of an emf-driven current in a wire, 
in the presence of a B field:

The circular cyclotron motion results because the only force on the particle 
is the qVxB force, which cannot change the particle's speed; it can only 
turn it into a circular path, with its previous speed preserved.

In the case of the current carrying wire, an emf maintains the (average) 
drift velocity component along the wire direction "no matter what".  When 
the qVxB force acts in a perpendicular direction the overall kinematic 
effect is only to add a new perpendicular component to the carrier's 
velocity, because the emf continually "restores" the drift velocity 
component along the wire, re-plenishing the kinetic energy of this drift 
motion which is going into the new perpendicular motion.  The particle 
(average) motion is not a circle, but a straight line making an angle PHI 
with the wire direction, where tan(PHI) = u/v   ( the ratio of the new 
perpendicular velocity component to the emf-maintained drift velocity 
component).

As already implied, this model speaks only of a carrier's average motion, 
abstracting from its erratic "thermal" excursions.

Bob Sciamanda
Physics, Edinboro Univ of PA (Em)
http://www.winbeam.com/~trebor/
trebor@velocity.net
----- Original Message ----- 
From: "Bob Sciamanda" <trebor@winbeam.com>
To: "Forum for Physics Educators" <phys-l@carnot.physics.buffalo.edu>
Sent: Sunday, March 26, 2006 5:19 PM
Subject: Re: [Phys-l] Magnetic force and work


> Here is my appreciation of the interaction of a B field with an emf driven
> current in a wire:
>
> General notation: [x] specifies that x is a vector.
> Let [i], [j] and [k] be the unit x,y,z vectors : to the right, up the 
> page,
> and out of the page, respectively.
>
> The wire lies along the x axis with the current consisting of positive
> carriers moving to the right with a
> drift velocity [v] = v[i].  The magnetic field is everywhere [B]
> = -B[k]:into the page.  The qVxB force gives the carriers an additional
> velocity component u[j], up the page. Thus, the (positive) carriers in the
> wire have a resultant velocity [w] = v[i] + u[j], with components to the
> right, and up the page.
>
> This resultant velocity is in the first quadrant of the xy plane making an
> angle TH with the y axis, where Tan(TH)=v/u.  Note that TH is also the 
> angle
> between the total magnetic force [w]x[B] and the y axis [j].
>
> The rate at which the magnetic force does work on a charge carrier is:
> P = q*( [w]x[B] ) DOT [w] = q*( [w]x[B] ) DOT ( v[i] + u[j] )
>
> Performing the DOT product:
>
> P = q* | [w]x[B] | * (-v*Cos(TH) + u*Sin(TH) )
>
> This is ostensibly zero, because v/u = tan(TH), by construction. The
> magnetic force does no net work, but acts as a "go-between" to enable the
> external emf agent to do the work.
>
> The first term is power taken from the external agent; the second term is
> power given to the current carriers.  One might say that the magnetic 
> field
> delivers energy to the [j] motion of the carriers, but it gets that energy
> from the [i] motion of the carriers - which energy ultimately comes from
> whatever emf is driving the [i] current.
>
> Bob Sciamanda
> Physics, Edinboro Univ of PA (Em)
> http://www.winbeam.com/~trebor/
> trebor@velocity.net
>
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