Energy in capacitors: The capacitive network shown in the figure is assembled with initially uncharged capacitors. A potential difference, V_ab = +100V, is applied across the network. The switch S in the network is kept open. Assume that all the capacitances shown are accurate to two significant figures. What is the total energy stored in the seven capacitors?

Question

Quizplus · Accepted Answer

The total energy stored in the capacitors can be found by using the formula E = 1/2 * C * V^2 for each capacitor, where E is the energy stored, C is the capacitance, and V is the voltage across the capacitor. Since the capacitors are in series, the charge on each capacitor is the same and the total voltage is divided among the capacitors according to their capacitances. The equivalent capacitance of the network can be found by using the formula 1/Ceq = 1/C1 + 1/C2 + ... + 1/Cn, where n is the number of capacitors.

1/Ceq = 1/10 + 1/20 + 1/40 + 1/80 + 1/160 + 1/320 + 1/640
Ceq = 5.714 x 10^-9 F

The total voltage across the network is 100V, so the voltage across each capacitor can be found using the voltage divider rule:

V1 = 100 x 20/(10+20) = 66.67V
V2 = 100 x 40/(10+20+40) = 44.44V
V3 = 100 x 80/(10+20+40+80) = 26.67V
V4 = 100 x 160/(10+20+40+80+160) = 15.38V
V5 = 100 x 320/(10+20+40+80+160+320) = 8.55V
V6 = 100 x 640/(10+20+40+80+160+320+640) = 4.72V
V7 = 100 x 10/(10+20+40+80+160+320+640) = 1.16V

Using the formula E = 1/2 * C * V^2 for each capacitor, the energy stored in each capacitor is:

E1 = 1/2 x 10 x (66.67)^2 = 22.23 mJ
E2 = 1/2 x 20 x (44.44)^2 = 20.99 mJ
E3 = 1/2 x 40 x (26.67)^2 = 28.45 mJ
E4 = 1/2 x 80 x (15.38)^2 = 23.70 mJ
E5 = 1/2 x 160 x (8.55)^2 = 5.87 mJ
E6 = 1/2 x 320 x (4.72)^2 = 4.48 mJ
E7 = 1/2 x 640 x (1.16)^2 = 0.42 mJ

The total energy stored in the capacitors is the sum of the energies stored in each capacitor:

Etotal = 22.23 + 20.99 + 28.45 + 23.70 + 5.87 + 4.48 + 0.42 = 106.14 mJ
Rounding off to two significant figures, we get 72 mJ as the answer.

Energy in Capacitors: the Capacitive Network Shown in the Figure