A source supplying a voltage \(V\) charges a capacitor \(C\) through a resistance \(R\). Calculate the energy supplied by the source, that dissipated by the resistor, and that stored in the capacitor, after an infinite time, You should find that one half of the energy is dissipated in \(R\), and that the other half is stored in \(C\).

Understanding how resistors impact energy in an electric circuit is crucial for grasping the basic principles of electronics. Resistors are components that impede the flow of electric current, and in the process, they convert some of the electrical energy passing through them into heat. This phenomenon is known as energy dissipation.

When a voltage source charges a capacitor through a resistor, the resistor's opposition to current flow causes energy to be lost in the form of heat. Mathematically, the amount of energy (\(E_{\text{resistor}}\)) dissipated by the resistor is quantified as the integral of the power over time. The power, which is the rate of energy conversion, is given by the product of current (\(I\)) and voltage (\(V\)) across the resistor. Over an infinite period, as the capacitor charges, the current decreases and eventually stops when the capacitor is fully charged.

In this scenario, since the resistor and capacitor are in series, the energy dissipated in the resistor is exactly half of the total energy supplied by the voltage source. This can be a counterintuitive idea because it might seem that all the energy from the source should be stored in the capacitor. However, due to the natural resistance of the resistor, only half of this energy makes it to the capacitor; the other half is irreversibly converted to heat in the resistor itself.

Capacitors play a different role from resistors in a circuit; they are devices specifically designed to store energy in an electric field. The energy (\(E_{\text{capacitor}}\) ) stored in a capacitor is a function of the capacitance (\(C\) )—a measure of the capacitor’s ability to store charge—and the voltage (\(V\) ) across its plates.

For a fully charged capacitor the energy stored can be expressed with the formula: \( E_{\text{capacitor}} = \frac{1}{2} C V^2 \). This equation tells us that the stored energy is proportional to the square of the voltage across the capacitor and directly proportional to its capacitance. When a capacitor charges through a resistor, it gradually accumulates energy until it reaches a point where it can no longer accept charge. At that point, the current ceases and the capacitor retains the energy in the form of an electric field between its plates.

Intuitively, one might expect that after an infinite amount of time, the capacitor would have absorbed all of the energy from the source. However, as we've learned, some energy is always lost due to the resistance in the circuit. Thus, the energy conserved within the capacitor is only half of what was supplied by the voltage source, with the other half dissipated as heat by the resistor.

The principle of energy conservation remains an essential tenet in electrical circuits. According to this principle, energy cannot be created or destroyed in an isolated system; it can only be transferred or transformed from one form to another.

When analyzing a circuit that includes a voltage source, resistor, and capacitor, we apply the concept of energy conservation by noting that the energy supplied by the source (\(E_{\text{source}}\) ) ultimately distributes between the resistor and the capacitor. As previously discussed, the resistor dissipates half of this energy as heat (\(E_{\text{resistor}}\) ), while the other half is stored in the capacitor (\(E_{\text{capacitor}}\) ) as an electric field.

This distribution might initially seem like an anomaly - where did half the energy go? The 'lost' energy is not really lost; it is simply transferred into the environment as thermal energy. This heat generated by the resistor is a byproduct of converting electrical energy into another form. The rigorous adherence to the energy conservation principle in electric circuits allows us to predict and verify outcomes, such as the equal distribution of energy between heat dissipation and electric potential storage, when a capacitor is charged indefinitely through a resistor.