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Physics LibreTexts

13.2: Alternating Voltage across a Capacitor

FIGURE XIII.3

+

-

 

 
  File:C:/Users/DELMAR~1/AppData/Local/Temp/msohtmlclip1/01/clip_image020.gif

 

 

 

                                                           

                                               

 

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.

 

 

 

FIGURE XIII.4

I

V

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