A household lamp is connected to a 60 Hz alternating-current (AC) source that…

MATHEMATICS · Multi-part · A Level · Mathematics · Integration · Integration

Problem

A household lamp is connected to a 60 Hz alternating-current (AC) source that delivers an instantaneous current I(t)=I0sin⁡(2πft),I(t)=I_{0}\sin(2\pi ft), where I0=0.70 AI_{0}=0.70\,\text{A} and f=60 Hzf=60\,\text{Hz}. Consider the time interval from t=0t=0 to t=T/2t=T/2, where TT is the period of the AC signal. (a) Determine the total electric charge Δq\Delta q that passes through the filament during this half-cycle. (b) Show algebraically that the net charge transported over a full cycle is zero. (c) Using the free-electron density n=8.5×1028 m−3n=8.5\times10^{28}\,\text{m}^{-3} and a filament cross-sectional area of 7.0×10−9 m27.0\times10^{-9}\,\text{m}^{2}, calculate the maximum instantaneous drift speed of the conduction electrons. (d) Explain why the filament still heats up even though the average current over a full cycle is zero.

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