Set 53 Problem number 4

Problem

An initially uncharged capacitor has capacitance 7 `microC/Volt.  It is placed in series with a resistance of 50 Ohms and a 37 volt source, and the switch is closed. When the capacitor charge reaches 38 `microC, how much current will be flowing through the circuit?

Solution

A capacitor which holds 7 `microC/Volt, when holding 38 `microC, will be at a potential difference of ( 38 `microC)/( 7 `microC/Volt) = 5.428 Volts.

• Since this voltage results from charge built by the current on the capacitor, it will oppose that of the source, so that the total voltage across the resistor will be the 37 volts of the source, less the 5.428 Volts across the capacitor, or 31.57 Volts.
• This voltage across a 50 ohm resistor will result in a current of ( 31.57 V)/( 50 ohm) = .6314 Amps.

Generalized Solution

A capacitor is characterized by its capacitance C, which is its charge / voltage ratio. 

• If this capacitor holds charge Q, then its voltage Vcap is such that C = Q / Vcap, so that Vcap = Q / C.

When the charge is built by the current from the source, its voltage will resist that of the source, since the positive terminal of the source tends to move positive charges to the plate of the capacitor to which it is most directly connected and the negative terminal to the plate connected to it.

• Thus if Vs is the voltage of the source, the voltage across the resistor will be Vr = Vs - Vcap.
• It follows that the current through the resistor is I = (Vs - Vcap) / R.

Note that as charge builds on the capacitor, the voltage across the resistor decreases and so therefore does the current.  This causes charge to build on the capacitor more and more slowly.