MadSci Network: Physics |
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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