A capacitor of capacitance C is given a charge go. At time t = 0 it is connected to an uncharged capacitor of equal capacitance through a resistance R. Find the charge on the first capacitor and the second capacitor as a function of time t. Also plot the corresponding q-t graphs.
Answers
GIVEN :
A capacitor of capacitance C , charge qo.
At time t = 0 it is connected to an uncharged capacitor of equal capacitance through a resistance R.
TO FIND :
◆Find the charge on the first capacitor and the second capacitor as a function of time t.
◆Plot the corresponding q-t graphs.
SOLUTION :
◆Charge on the first capacitor when connected to second capacitor
( q0-q)
◆The current i,
i= dq/dt
◆Kirchoff's voltage law in the circuit gives,
q/C + iR - (q0-q) / C = 0
q/ C + iR = (q0 - q)/ C
R dq/dt = (q0 - 2q )/C
Dq / (q0 - 2q) = dt / RC
◆Solving further,
-1/2 [ln (q0 - 2q) ] = t/RC
◆Here, At t=0 charge on second capacitor is 0.
◆Thus,equation becomes
Ln (q0 - 2q )/ q0 = -2t/RC
◆Taking exponential ,
1- 2q/q0 = e ^(-2t/RC)
2q/q0 = 1 - e ^(-2t/RC)
q = q0 /2 [ 1- e ^(-2t/RC)]
ANSWER :
Charge on 2nd capacitor ,
q = q0 /2 [ 1- e ^(-2t/RC)]
FOR Graph, refer attachment below,
