Ohm's law is the cornerstone of circuit analysis, expressing the linear relationship between voltage, current, and resistance in a resistive element. Mathematically, \(V = IR\), where \(V\) is the voltage across the component in volts, \(I\) is the current flowing through it in amps, and \(R\) is the resistance in ohms (\(\Omega\)), a unit named after Georg Simon Ohm. The law can be rearranged into two useful forms: \(I = V/R\) for computing current from voltage and resistance, and \(R = V/I\) for computing resistance from measured voltage and current.
Electrical power, measured in watts, combines voltage and current through the relationship \(P = VI\). By substituting Ohm's law into this expression, two additional forms emerge that are often more convenient in circuit calculations: \(P = I^2 R\) when current and resistance are known, and \(P = V^2 / R\) when voltage and resistance are known. These equations allow engineers to determine how much energy a device dissipates or delivers in any resistive portion of a circuit.
Kirchhoff's two laws extend Ohm's law to networks with multiple components. Kirchhoff's Current Law (KCL) states that the sum of currents entering any node equals the sum of currents leaving that node (\(\sum I_{in} = \sum I_{out}\)), a direct consequence of conservation of charge. Kirchhoff's Voltage Law (KVL) states that the sum of voltage drops around any closed loop equals zero (\(\sum V = 0\)), reflecting conservation of energy, since the total energy gained by charges traversing a loop must equal the total energy lost. Together, KCL, KVL, and Ohm's law form the mathematical foundation for analyzing even the most complex linear circuits.