Question

Show that energy is conserved in a simple charging RC circuit. That is show that the total work done by the battery equals the final energy stored in the capacitor plus the energy dissipated in the resistor.

Answer 1

We know that when a capacitor charges from a source E, it stores energy of

This is derived without taking into consideration any resistances present in the circuit.

We also know that the battery does work W=QV to pump the charge Q. It is explained that the remaining is dissipated as heat in the circuit resistance.

It is clearly shown in the following picture

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.

he loop rule as applied to the circuit of Figure 2 is: V_ab + V_ef + V_cd = 0. Let's see which of these potential differences are positive and which are negative.

V_ab is positive, because point a is on the higher potential side of the battery.
V_ef is negative, because point e is on the low potential side of the bulb.
V_cd 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, V_ab (red line) is constant, since that's the potential difference across the battery. The potential difference V_cd across the capacitor is 0 initially when it's uncharged but increases negatively toward an asymptote at the value of -V_ab. The potential difference V_ef 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 three potential differences is 0.

Fig1 Fig-2

Fig-3

Fig-4