Refer to the diagram. An ideal battery is connected to three resistors and one c
ID: 1621503 • Letter: R
Question
Refer to the diagram. An ideal battery is connected to three resistors and one capacitor, with the switch initially open, as shown. Here are the known values: epsilon, R_1, R_2, R_3, C Answer the following, giving expressions that use only known values. a. The switch is closed. Find the maximum charge that will be stored on the capacitor. b. Suppose the switch has been closed for a long time; now it is re-opened. Find the total power being produced at the moment when the energy stored on the capacitor is 2/3 of its maximum value.Explanation / Answer
The moment the charge on the capacitor is maximum the current through the branch containing it is 0 and hence voltage across capacitor is equal to that dropped across R2.
So V = R2*E/(R1+R2)
And hence charge, q0 = CV = R2*E*C/(R1+R2)
As energy stored in capacitor as a function of time is given by
U = (q02/2C)e-2t/RC = U0e-2t/RC
Where R = R1 + R2
Hence power
P = dU/dt = U0(-2/RC)(e-2t/RC) = -(q02/RC2)(e-2t/RC)
And as per given condition
2U0/3 = U0e-2t/RC
e-2t/RC = 2/3
And hence
P = (q02/RC2)(2/3) =(2/3)* {R2*E*C/(R1+R2)}2*{1/(R1+R2)C2} = (2/3)(E2){R22/(R1+R2)3}
P = (2/3)(E2){R22/(R1+R2)3}