Power in an Electric Circuit

Calculating properties of a network of resistors involves applying principles of electrical circuit analysis, primarily Ohm's law and the rules governing series and parallel resistor configurations. Here's a general approach:
1. **Ohm's Law**: This fundamental law states that the current flowing through a conductor between two points is directly proportional to the voltage across the two points and inversely proportional to the resistance. Mathematically, Ohm's law is expressed as:
\[V = I \times R\]
where \(V\) is the voltage across the resistor, \(I\) is the current flowing through it, and \(R\) is the resistance.
2. **Series Resistors**: When resistors are connected end-to-end, they are said to be in series. For resistors in series, the total resistance (\(R_{\text{total}}\)) is the sum of the individual resistances.
\[R_{\text{total}} = R_1 + R_2 + \ldots + R_n\]
3. **Parallel Resistors**: When resistors are connected across the same two points, they are said to be in parallel. For resistors in parallel, the reciprocal of the total resistance (\(R_{\text{total}}\)) is the sum of the reciprocals of the individual resistances.
\[\frac{1}{R_{\text{total}}} = \frac{1}{R_1} + \frac{1}{R_2} + \ldots + \frac{1}{R_n}\]
4. **Equivalent Resistance**: A complex network of resistors can often be simplified into an equivalent resistance. This equivalent resistance represents the entire network as a single resistor.
- For series combinations, the equivalent resistance is simply the sum of the resistances.
- For parallel combinations, the equivalent resistance is found using the reciprocal formula mentioned above.
5. **Voltage and Current Division**: In a network of resistors, the voltage and current distribution across each resistor can be determined using principles of voltage division and current division, which are derived from Ohm's law and the series/parallel rules.
6. **Power Dissipation**: The power dissipated by a resistor can be calculated using either \(P = IV\) or \(P = \frac{V^2}{R}\), where \(P\) is power, \(I\) is current, \(V\) is voltage, and \(R\) is resistance.
7. **Kirchhoff's Laws**: Kirchhoff's laws, particularly Kirchhoff's voltage law (KVL) and Kirchhoff's current law (KCL), are used to analyze more complex circuits where simple series and parallel combinations aren't sufficient.
By applying these principles systematically, one can analyze and calculate various properties of networks of resistors, such as total resistance, current flow, voltage distribution, and power dissipation.