Review. One insulated conductor from a household extension...

Review. One insulated conductor from a household extension cord has a mass per length of 19.0 g/m. A section of this conductor is held under tension between two clamps. A subsection is located in a magnetic field of magnitude 15.3 mT directed perpendicular to the length of the cord. When the cord carries an AC current of 9.00 A at a frequency of 60.0 Hz, it vibrates in resonance in its simplest standing-wave vibration mode. (a) Determine the relationship that must be satisfied between the separation d of the clamps and the tension T in the cord. (b) Determine one possible combination of values for these variables.

Key Concepts

Standing Waves on a Tensioned String

When a string or wire is fixed at both ends and set into vibration, standing wave patterns are established. These patterns, characterized by nodes and antinodes, have resonant frequencies that depend on the length of the string and the boundary conditions. In the simplest (fundamental) mode, the wavelength is twice the length of the string, and the frequency is given by f = (1/2L)?(T/?), where T is the tension and ? is the mass per unit length.

Wave Speed on a Stretched String

The speed of a wave traveling along a stretched string is determined by the tension in the string and its linear mass density. This relationship is expressed as v = ?(T/?). The wave speed, in turn, directly influences the frequencies of the standing waves that can be established on the string, as the wavelengths are fixed by the clamp separation and the boundary conditions.

Forcing and Resonance in Oscillatory Systems

Resonance occurs when an external periodic force drives a system at a frequency that matches one of its natural frequencies, resulting in large amplitude oscillations. In the context of a current carrying conductor in a magnetic field, a time-varying current produces a periodic driving force (via the magnetic Lorentz force) that can excite standing waves in the conductor if the driving frequency coincides with a resonant frequency defined by the system's tension, length, and mass density.

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