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13.2: Alternating Voltage across a Capacitor












At any time, the charge Q on the capacitor is related to the potential difference V across it by File:C:/Users/DELMAR~1/AppData/Local/Temp/msohtmlclip1/01/clip_image022.gif   If there is a current in the circuit, then Q is changing, and File:C:/Users/DELMAR~1/AppData/Local/Temp/msohtmlclip1/01/clip_image024.gif


Now suppose that an alternating voltage given by


                                                            File:C:/Users/DELMAR~1/AppData/Local/Temp/msohtmlclip1/01/clip_image026.gif                                                             13.2.1


is applied across the capacitor.


In that case the current is                    File:C:/Users/DELMAR~1/AppData/Local/Temp/msohtmlclip1/01/clip_image028.gif                                                      13.2.2


which can be written                          File:C:/Users/DELMAR~1/AppData/Local/Temp/msohtmlclip1/01/clip_image030.gif                                                             13.2.3


where the peak current is                    File:C:/Users/DELMAR~1/AppData/Local/Temp/msohtmlclip1/01/clip_image032.gif                                                                 13.2.4


and, of course File:C:/Users/DELMAR~1/AppData/Local/Temp/msohtmlclip1/01/clip_image034.gif


The quantity 1/(Cw) is called the capacitive reactance XC.  It is expressed in ohms (check the dimensions), and, the higher the frequency, the smaller the reactance.  (The frequency n is w/(2p).)


[When we come to deal with complex numbers, in the next and future sections, we shall incorporate a sign into the reactance.  We shall call the reactance of a capacitor File:C:/Users/DELMAR~1/AppData/Local/Temp/msohtmlclip1/01/clip_image036.gifrather than merely File:C:/Users/DELMAR~1/AppData/Local/Temp/msohtmlclip1/01/clip_image038.gif, and the minus sign will indicate to us that V lags behind I.  The reactance of an inductor will remain Lw, since V leads on I. ]


Comparison of equations 13.2.1 and 13.2.3 shows that the current and voltage are out of phase, and that V lags behind I by 90o, as shown in figure XIII.4.







File:C:/Users/DELMAR~1/AppData/Local/Temp/msohtmlclip1/01/clip_image039.gif            File:C:/Users/DELMAR~1/AppData/Local/Temp/msohtmlclip1/01/clip_image041.gif