Simple Nature - Light and Matter

Figure aa shows a useful way to visualize what’s going on. Whena circuit is oscillating at a frequency ω, we use points in the plane torepresent sinusoidal functions with various phases and amplitudes.self-check IWhich of the following functions can be represented in this way? cos(6t−4), cos 2 t, tan t ⊲ Answer, p. 926aa / Representing functionswith points in polar coordinates.ab / Adding two sinusoidalfunctions.The simplest examples of how to visualize this in polar coordinatesare ones like cos ωt + cos ωt = 2 cos ωt, where everything hasthe same phase, so all the points lie along a single line in the polarplot, and addition is just like adding numbers on the number line.The less trivial example cos ωt + sin ωt = √ 2 sin(ωt + π/4), can bevisualized as in figure ab.Figure ab suggests that all of this can be tied together nicelyif we identify our plane with the plane of complex numbers. Forexample, the complex numbers 1 and i represent the functions sin ωtand cos ωt. In figure z, for example, the voltage across the capacitoris a sine wave multiplied by a number that gives its amplitude, sowe associate that function with a number Ṽ lying on the real axis.Its magnitude, |Ṽ |, gives the amplitude in units of volts, while itsargument arg Ṽ , gives its phase angle, which is zero. The currentis a multiple of a sine wave, so we identify it with a number Ĩlying on the imaginary axis. We have arg Ĩ = 90◦ , and |Ĩ| is theamplitude of the current, in units of amperes. But comparing withour result above, we have |Ĩ| = ωC|Ṽ |. Bringing together the phaseand magnitude information, we have Ĩ = iωCṼ . This looks verymuch like Ohm’s law, so we writewhere the quantityĨ =ṼZ C,Z C = − i , [impedance of a capacitor]ωChaving units of ohms, is called the impedance of the capacitor atthis frequency.It makes sense that the impedance becomes infinite at zero frequency.Zero frequency means that it would take an infinite timebefore the voltage would change by any amount. In other words,this is like a situation where the capacitor has been connected acrossthe terminals of a battery and been allowed to settle down to a statewhere there is constant charge on both terminals. Since the electricfields between the plates are constant, there is no energy beingadded to or taken out of the field. A capacitor that can’t exchangeenergy with any other circuit component is nothing more than abroken (open) circuit.Note that we have two types of complex numbers: those thatrepresent sinusoidal functions of time, and those that represent608 Chapter 10 Fields