4.3 Steady-State Direct Current Circuits with Batteries and Resistors Only

Series and Parallel Resistors

In series, current is the same through each resistor and voltages add: R_eq = R1 + R2 + ... In parallel, voltage is the same across each resistor and currents add: 1/R_eq = 1/R1 + 1/R2 + ... Combining resistors lets you simplify complex circuits into a single equivalent resistance for analysis.

Kirchhoff's Laws

Kirchhoff's Voltage Law (KVL): the sum of voltage changes around any closed loop is zero (energy conservation). Kirchhoff's Current Law (KCL): the sum of currents into a junction equals the sum out (charge conservation). Together they let you solve for unknown currents and voltages in multi-loop circuits with batteries and resistors.

Real Batteries and Power

A real battery has internal resistance r, so terminal voltage is V = EMF − Ir. Power dissipated by a resistor is P = I²R = V²/R = IV. In steady state, capacitors (if present) are fully charged and carry no current, so branches with only capacitors act as open circuits.

Key terms

Series resistor combination

R_eq = R1 + R2 + ... ; same current flows through each resistor, voltages add.

Parallel resistor combination

1/R_eq = 1/R1 + 1/R2 + ... ; same voltage across each resistor, currents add.

Kirchhoff's Voltage Law (KVL)

The sum of voltage changes around any closed loop equals zero; based on conservation of energy.

Kirchhoff's Current Law (KCL)

The current entering a junction equals the current leaving it; based on conservation of charge.

Terminal voltage of a real battery

V = EMF − Ir, where r is the internal resistance and I is the current supplied.

Try an AP-style question

A 12 V battery with 1 Ω internal resistance powers a 5 Ω load. What is the current?

1. 12 A
2. 2.4 A
3. 2 A
4. 1 A