Guide 21-3. Energy Conservation in Circuits with Capacitors

Circuits with resistors and capacitors provide a good opportunity to examine energy conservation in circuits. As the capacitor charges or discharges, the energy stored in the capacitor and the energy dissipated in the resistor both change; ye, the total energy provided by the battery remains constant (assuming negligible losses due to the battery's own internal resistance). We've seen that this fact of energy conservation is expressed in the loop rule. The sum of the potential differences around the circuit is 0.

 Charging Capacitor

Consider the circuit of a battery, bulb (resistor), and capacitor in Figure 1. The switch is initially open, and the capacitor is uncharged. There is no potential difference across the bulb or the capacitor, but there is a potential difference V_ab across the battery.  (We'll use V₁₂ to represent V₁ - V₂.) When the switch is closed as in Figure 2, current flows in the circuit and charges the capacitor. The left-hand plate of the capacitor becomes positively charged, while the right-hand plate acquires an equal negative charge. As charge builds up on the capacitor, so does the potential difference across it. The potential difference across the resistor also changes with time as the amount of current in the circuit changes. The current rises to a maximum so quickly after the switch is closed that one typically shows the initial current as a maximum as in Figure 3. Thereafter, the current falls off exponentially to 0. The charge on the capacitor, on the other hand, is 0 initially and builds to a maximum value. This is shown in Figure 4.

Figure 1 Figure 2 Figure 3 Figure 4
The loop rule as applied to the circuit of Figure 2 is:  Vab + Vef + Vcd = 0. Let's see which of these potential differences are positive and which are negative. Vab is positive, because point a is on the higher potential side of the battery. Vef is negative, because point e is on the low potential side of the bulb. Vcd is negative, because point c is on the side of the capacitor with negative charge. Now we'll plot a graph of all three potential differences vs. time from t = 0 until the capacitor is almost completely charged. In Figure 5, Vab (red line) is constant, since that's the potential difference across the battery. The potential difference Vcd across the capacitor is 0 initially when it's uncharged but increases negatively toward an asymptote at the value of -Vab. The potential difference Vef across the resistor is a negative maximum initially when the current is maximum but approaches 0 asymptotically. Here's an animation showing the potentials changing as a function of time.	Figure 5

 Discharging Capacitor

Now we'll look at a circuit composed of a bulb and capacitor in Figure 6. The switch is initially open, and the capacitor is fully charged. There is no potential difference across the bulb since there is no current, but there is potential difference across the capacitor since the capacitor is charged. When the switch is closed as in Figure 7, current (conventional) moves clockwise around the circuit to discharge the capacitor. This current passes through the resistor from right to left, since point f is at lower potential than point e.  The potential difference across the resistor changes with time as the potential difference across the capacitor changes.  The current is maximum when the switch is closed and gradually diminishes to 0 when the capacitor is fully discharged. Graphs of current vs. time and charge on the capacitor vs. time are shown in Figures 8 and 9.

Figure 6 Figure 7 Figure 8 Figure 9

The loop rule as applied to the circuit of Figure 7 is:  Vfe + Vdc = 0. Let's see which one of these potential differences is positive and which one is negative. Vfe is negative, because point f is on the low potential side of the bulb. (Conventional current flows from higher to lower potential.) Vdc is positive, because point d is on the side of the capacitor with positive charge. A graph of the two potential differences vs. time from t = 0 until the capacitor is almost completely discharged is shown in Figure 10. The potential difference Vdc across the capacitor is initially a maximum when it's fully charged but decreases asymptotically toward 0. The potential difference Vfe across the resistor is a negative maximum initially when the current is maximum but approaches 0 asymptotically. Note that at any time, the sum of the potential differences is 0. Here's an animation showing the potentials changing as a function of time.

Figure 10

 The Time Constant

The rate at which a capacitor of capacitance C charges and discharges in a series circuit with a resistor of resistance R is characterized by a quantity termed the time constant of the circuit. The time constant is given by the product of R and C. For a discharging capacitor, RC is the amount of time for the potential difference across the capacitor to fall to 0.37 of its initial value. For a charging capacitor, RC is the amount of time for the potential difference across the capacitor to rise to 0.63 of its final value. These results can be seen from the equations for potential difference as a function of time.  These equations are given without proof.

discharging capacitor:  V = V₀e^-t/RC

charging capacitor:  V = V₀(1-e^-t/RC)

One can show that the values of 0.37 and 0.63 come from substituting t = RC into the equations above. With that substitution, the equations become

discharging capacitor:  V = V₀e^-1 = 0.37V₀

charging capacitor:  V = V₀(1-e^-1) = 0.63V₀, where e is the base of the natural logarithms.  

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