A load carries voltage and current

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Electricity, Magnetism & Electromagnetism · Foundations of mathematical language

Mathematical Methods for Physics, Ch Complex numbers, the complex exponential and phasors

Problem

A load carries voltage

V(t) = 8.0\cos(\omega t)\,\mathrm{V}

and current

I(t) = 3.0\cos(\omega t - \pi/4)\,\mathrm{A}

. With the phasor convention

A\cos(\omega t+\phi)=\mathrm{Re}[\tilde A e^{-i\omega t}]

and

\tilde A = Ae^{-i\phi}

, the complex amplitudes are

\tilde V = 8.0\,\mathrm{V}

and

\tilde I = 3.0\,e^{i\pi/4}\,\mathrm{A}

. Someone argues: "the instantaneous power is

p(t)=V(t)I(t)

, and since

V(t)=\mathrm{Re}[\tilde V e^{-i\omega t}]

and

I(t)=\mathrm{Re}[\tilde I e^{-i\omega t}]

, it follows that

p(t)=\mathrm{Re}[\tilde V\tilde I\,e^{-2i\omega t}]

; averaged over a cycle this is zero because

e^{-2i\omega t}

has zero mean, so the mean power delivered to this load is

0\,\mathrm{W}

." ((a)) Explain the error in this reasoning. ((b)) Compute the correct mean power delivered to the load over one cycle.

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