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Stored Energy in LCR Circuit

Four resistors ( R1 = 60 Ohms, R2 = 220 Ohms, R3 = 330 Ohms, and R4 = 480 Ohms), an ideal inductor (L = 8 mH), and a capacitor (C = 250 microF) are connected to a battery (V = 9 V) through a switch as shown in the figure below.

The switch has been open for a long time before it is closed at t = 0. What is Ustored, the total stored energy in the circuit elements (not including the battery) a long time after the switch is closed? Ustored =

J

Explanation / Answer

"A long time after the switch is closed" means that the voltage across the inductor is 0 and the current through the capacitor, i3, is 0.

The voltage across the capacitor is;

V = (9 V){R2R4/(R2 + R4)}/{R2R4/(R2 + R4) + R1} = (9 V)(R2R4)/(R2R4 + R1R2 + R1R4) = 6.44 V

The energy stored in the capacitor is:

E = (1/2)CV² =125 × 10^-6(6.44)² = 5.18 × 10^-3 J

The current through the inductor is:

I = 6.44 V/220 ?

The energy in the inductor is:

E = (1/2)LI² = 4 × 10^-3(6.44/220)² = 3.43 × 10^-6 J

The total energy is:

5.1834 × 10^-3 J

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