MadSci Network: Physics
Query:

Re: Imagine a thin wire of good tensile strength is kept in a magnetic field

Date: Fri Mar 16 12:50:00 2007
Posted By: Aaron J. Redd, Post-doc/Fellow, Plasma Physics and Controlled Nuclear Fusion, University of Washington
Area of science: Physics
ID: 1168522570.Ph
Message:

Siddhartha,

You have asked a very good set of questions, but left out some important
information, so I will try to help you find the answers you seek.

Vibrating wires and induced currents are two very different types of
physics problems, but they do have one thing in common: the boundary
conditions almost completely determine the answer to the question.  In the
case of a vibrating wire, the tension in the wire determines the
fundamental vibration frequency, but you also need to determine if the ends
are fixed, if the vibration is the fundamental mode, and so on.  In the
case of induced currents, the actual amount of current will be determined
by the rest of the circuit, the circuit structure and the impedances of the
circuit elements.  If you haven't already read an introductory physics
text, then that would be a good starting point.  If you want to read more
about vibrations in wires and other one-dimensional systems, a college
sophmore-level or junior-level mechanics textbook should give you some good
information (such as "Mechanics, 3rd Ed." by K. Symon, Chapter 8). 
Similarly, the induced current in moving/vibrating structure in a magnetic
field is a favorite example in college electricity&magnetism textbooks,
both in physics and in electrical engineering (such as "Electromagnetic
Fields, 2nd Ed." by R. Wangsness, Chapter 17).

Now, an example problem based on your questions might be as follows:
Suppose that there is a wire of length L and cross-sectional area A
immersed in a magnetic field of strength B.  This wire has a mass density
of "delta", an electrical resistivity of "rho", and is kept at some tension
T by a perfectly rigid and perfectly conducting frame (the wire ends are
fixed).

Then, the fundamental vibration frequency "omega" of the wire is:
omega = sqrt( T / delta*A*A )
where "sqrt" indicates a square root of the stuff in parentheses.

To simplify things a bit further, let's assume that the wire is vibrating
in its fundamental mode, and that we can choose a coordinate system that
has the wire extending along the y-direction (ends at (0,0,0) and (0,L,0))
and vibrating only in the x-direction.  Then, it's pretty simple to get the
x-position of any part of the wire (a function of y) versus time:
x(y,t) = X * sin( pi*y/L ) * sin( omega*t )
where X is the maximum amplitude of the vibration at the middle of the
wire, and "pi" is the usual 3.14159....

You asked about the induced current.  Simplifying again, assume that the
magnetic field is uniform and simply pointing in the z-direction.  Then,
there is a circuit loop made by the wire and conducting frame, and the
amount of magnetic flux enclosed by this loop will vary with time as the
wire vibrates.  This variation will produce an electromotive force (that
is, and induced voltage) in the circuit loop.  The amplitude of this EMF is
given by:
EMF = 2*B*X*L*omega/pi
In this simple example, the induced current is found by taking this EMF and
dividing by the resistance of the wire.  After some algebra, you will get
the somewhat simple answer of:
I = (2*B*X/pi*rho) * sqrt(T/delta)
So, the wire length and cross-sectional area drop out, but we need to know
the density and resistivity of the wire, and the tension T.  We also don't
know the amplitude X, which is a measure of "how hard" the wire was plucked
to start its vibrations (or, if you prefer, a measure of the energy stored
in the mechanical vibrations).

So, what are the complications?  In the real world, no frame is perfectly
conducting, or perfectly rigid.  So there will be some electrical
resistance and inductance in the frame, which will result in a smaller
sinusoidally varying current "I", and the current lagging behind the wire
vibration by some finite phase angle.  If other wires are present, the
frame will provide a mechanical coupling, and they too will begin to
vibrate, and get induced currents.

Perhaps the most serious complication is the unstated assumption:  that the
mechanical energy of the vibrating wire is much larger than the electrical
energy of the induced current in the circuit.  If the resistivity of the
wire and other circuit elements are very small, then the electrical energy
is relatively large, and you need to reconsider the derivation of the
expressions for the vibrating wire: In the usual derivation, the wire mass
tries to keep the wire moving away from the center, but the wire tension
provides a "restoring force", pulling the wire back in again.  In this
case, with large electromagnetic energies, the "restoring forces" are the
wire tension and the magnetic "pressure" on the wire, and this will have an
impact on the solutions you're allowed to have.

I hope that this discussion has been of help to you.  Good luck with this
problem!

Aaron J. Redd


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