A phasor diagram shows three arrows of the same frequency

PHYSICS · Multi-part · Fermi University Physics · Optics & Photonics · Wave propagation and light in media · Wave propagation and light in media

Optics and Photonics, Ch Harmonic waves, phasors and group velocity

Problem

A phasor diagram shows three arrows of the same frequency, all rotating together: A1=3.00A_1=3.00 units at 0∘0^\circ, A2=2.00A_2=2.00 units at 90∘90^\circ, and A3=4.00A_3=4.00 units at 180∘180^\circ, all angles measured anticlockwise from the reference direction; under convention CV3 an arrow at angle ϕ\phi represents, at a fixed point, the wave Acos⁡(ωt−ϕ)A\cos(\omega t-\phi). ((a)) Write the corresponding sum-of-cosines equation E(t)=A1cos⁡(ωt)+A2cos⁡(ωt−90∘)+A3cos⁡(ωt−180∘)E(t)=A_1\cos(\omega t)+A_2\cos(\omega t-90^\circ)+A_3\cos(\omega t-180^\circ), and translate the diagram into components: the sum of the horizontal projections and the sum of the vertical projections of the three arrows. ((b)) From these components find the single resultant arrow (amplitude and angle) and rewrite the sum as one cosine wave E(t)=E0cos⁡(ωt−ϕ)E(t)=E_0\cos(\omega t-\phi).

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