Series and Parallel Circuits

Lesson goal: analyse current, voltage, and equivalent resistance in series, parallel, and combined circuits.

Students should read the circuit structure before substituting into equations.

Series resistors:

$$R_s = R_1 + R_2 + ...$$

Parallel resistors:

$$\frac{1}{R_p} = \frac{1}{R_1} + \frac{1}{R_2} + ...$$

Kirchhoff's laws:

$$\sum I_\text{in} = \sum I_\text{out}$$

$$\sum \Delta V_\text{loop} = 0$$

Circuit type	Current	Voltage	Resistance
series	same through components	divides across components	adds
parallel	splits between branches	same across branches	reciprocal sum; lower than smallest branch

1. In a series circuit, current is:

   • A. the same through each component
   • B. zero after the first resistor
   • C. larger after each resistor
   • D. unrelated to the circuit

   Answer: A.

2. In a parallel circuit, voltage across each branch is:

   • A. the same
   • B. always zero
   • C. split equally regardless of resistance
   • D. impossible to measure

   Answer: A.

3. An ammeter should be connected:

   • A. in series
   • B. in parallel only
   • C. across the battery only
   • D. not in a circuit

   Answer: A.

A 12 V source is connected to a 3 ohm resistor in series with a parallel pair of 6 ohm and 12 ohm resistors.

1. Parallel section:

   $$R_p = \left(\frac{1}{6}+\frac{1}{12}\right)^{-1}=4\ \Omega$$

2. Total resistance:

   $$R_\text{eq} = 3 + 4 = 7\ \Omega$$

3. Total current:

   $$I = \frac{12}{7}=1.71\ \text{A}$$

Final answer: total current is $1.71\ \text{A}$.

Prompt: explain why a parallel equivalent resistance is lower than any single branch resistance.