Electric power and energy Knowledge-Based Study

Study notes

1. Work and Electrical Potential Energy

   When an electric current passes through a conductor, electric charges move under the influence of an electric field established by a potential difference. The work $W$ done by the electric source in moving a quantity of charge $q$ across a potential difference $V$ is given by:

   $$W = qV$$

   By definition, electric current is the rate of flow of charge, $I = \frac{q}{t}$, which means $q = It$. Substituting this into the work equation yields the total electrical energy transferred or dissipated in a circuit component over time $t$:

   $$E = VIt$$

   In purely resistive components, this electrical energy is converted entirely into internal thermal energy, a phenomenon known as Joule heating.

   Rate of Energy Transfer

   Electric power ($P$) is defined as the rate at which electrical energy is transferred, transformed, or dissipated in an electrical circuit per unit of time:

   $$P = \frac{E}{t}$$

   Substituting the energy expression $E = VIt$ into the power equation provides the fundamental relation for electric power:

   $$P = VI$$

   The standard SI unit of power is the watt (W), where $1\text{ W} = 1\text{ J/s} = 1\text{ V}\cdot\text{A}$.

   Equivalent Formulations Using Ohm's Law

   For an ohmic resistor with resistance $R$, Ohm's law states that $V = IR$. Substituting this relation into $P = VI$ produces two very useful alternate equations:

   • In terms of current and resistance: $$P = I^2R$$
   • In terms of voltage and resistance: $$P = \frac{V^2}{R}$$

   Each form is particularly advantageous depending on which circuit parameter remains constant.

   Calculating Energy Consumption

   Because power is the rate of energy consumption, the total energy $E$ consumed by a device operating at a steady power $P$ for a duration $t$ is:

   $$E = Pt = VIt = I^2Rt = \frac{V^2}{R}t$$

   While the SI unit of energy is the joule (J), domestic and commercial electricity meters measure energy in kilowatt-hours (kWh) due to the large magnitudes involved.

   The Kilowatt-Hour

   One kilowatt-hour represents the amount of energy delivered by a power of one kilowatt operating continuously for one hour:

   $$1\text{ kWh} = (1000\text{ W}) \times (3600\text{ s}) = 3.6 \times 10^6\text{ J}$$

   Components in Series

   In a series circuit, the current $I$ flowing through each component is identical. Therefore, the formulation $P = I^2R$ reveals that the power dissipated is directly proportional to the resistance:

   $$P \propto R \quad (\text{for constant } I)$$

   A larger resistor dissipates more power and, in the case of light bulbs, glows brighter when connected in series with smaller resistors.

   Components in Parallel

   In a parallel circuit, each branch experiences the same potential difference $V$. Using the formulation $P = \frac{V^2}{R}$, power dissipation is inversely proportional to resistance:

   $$P \propto \frac{1}{R} \quad (\text{for constant } V)$$

   Consequently, when identical voltage is applied, the branch with lower resistance draws greater current and dissipates significantly more power.