Two capacitors ( C 1 = 25 µF and C 2 = 85 µF) and a battery (E = 20 V) are conne

Question

Two capacitors (C1 = 25 µF andC2 = 85 µF) and a battery (E = 20 V) are connected in a circuit witha switch S as shown. C1 is initiallyuncharged.

(a) The switch is set to connect C2 to thebattery. What is the charge Q2 onC2?

Q2 = C   

(b) The sw itch is now thrown to the left disconnectingC2 from the battery and connecting the twocapacitors. Calculate the resulting charges Q1'and Q2' on the positive plates ofC1 and C2, respectively,when equilibrium is achieved.

Q1' = C   

Q2' = C    

(c) Calculate the energy lost to the circuit after theswitch is thrown to the left and the charges have come to rest.

| D U | = J

Explanation / Answer

Given that
C1= 25 F = 25 x10-6 F
        C2= 85F = 85 x 10-6 F The battery connected in the circuit is E = 20 V
(a) The capacitors C1 and C2 areconnected in parallel,therefore,the potential difference across thetwo capacitors is same.        When the switch is set toconnect C2 to the battery,then the charge Q2on C2 is              Q2 = C2 * E                   = 85 * 10-6 * 20
    = 1700 * 10-6 C
                  = 1700 C

(b)When capacitor C2 is disconnected from thebattery and the two capacitors are connected,then the resultingcapacitance of the capacitors is         C = C1 +C2    = 25 + 85
           = 110F
           = 110* 10-6 F

   The potential difference across the twocapacitors is             Q2= C * E1 ==> E1=(Q2/C)                  = (1700 * 10-6/110 * 10-6)
                 = 15.45 V
The resulting charges Q1' and Q2' onthe positive plates of capacitors C1 and C2are            Q1' = C1 * E1
                 = 25 * 10-6 * 15.45
=386.36 * 10-6 C and
            Q2' = C2 * E1
                   = 85 * 10-6 * 15.45
   = 1313.25 * 10-6 C

(c)The energy lost to the circuit after the switch isthrown to the left and the charges have come to rest is               |U| = U - U1
Where    U = (1/2) * C2 *E2
                  = (1/2) * 85 * 10-6 * (20)2
                  = 17000* 10-6 J    and
               U1= (1/2) * C * E12
                   = (1/2) * 110 * 10-6 * (15.45)2
   = 13128.63 * 10-6J Substituting the values in above equation ,we get            |U| = 17000 * 10-6 - 13128.63 *10-6
                   = 3871.36 * 10-6 J

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