DC Circuits
Ohm's law and networks
Ohm's law V = IR holds for a resistive element at constant temperature. Series: the same current flows through each element, voltages add, R_total = R₁ + R₂ + .... Parallel: the same voltage appears across each branch, currents add, 1/R_total = 1/R₁ + 1/R₂ + ..., and the total is always less than the smallest branch.
Kirchhoff's laws solve networks that simple combination rules cannot: current into a node equals current out (KCL), and the algebraic sum of voltages around a closed loop is zero (KVL).
Useful tools
Voltage divider (unloaded): V_out = V_in x R₂/(R₁ + R₂). Two-branch current divider: I₁ = I_total x R₂/(R₁ + R₂). The Wheatstone bridge is balanced when R₁/R₂ = R₃/R₄ and no current flows through the detector - the basis of strain-gauge and temperature-sensing circuits. Thevenin's theorem reduces any linear network to one emf in series with one resistance; superposition handles several sources, one at a time.
Traps
- Adding a resistor in parallel always lowers the total resistance.
- A voltmeter of finite resistance drags down the voltage it measures, especially in high-resistance circuits.
- Kirchhoff signs must be consistent with the chosen loop direction.
- A balanced bridge has no detector current, not zero supply current.
Worked example
6 Ω and 3 Ω in parallel: R = (6 x 3)/(6 + 3) = 2 Ω. Add 4 Ω in series: total 6 Ω. With 12 V applied, I = 2 A and the parallel pair drops 4 V.