Exercises: Parallel Circuits

Work through each section in order. Show all steps: find R_eq, then branch currents, then verify with KCL.

Grade 11·21 problems·~30 min·OpenStax Physics (High School)·section·sec-19-3

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Recall / Warm-Up

Which of the following is the defining property of a parallel circuit?

Components are connected between the same two nodes, so all branches have the same voltage.

All components carry the same current.

The equivalent resistance is the sum of the individual resistances.

Adding more branches increases the total resistance.

In a parallel circuit with three branches, the light bulb in branch 2 burns out (open circuit). What happens to the other two branches?

They are unaffected — each branch operates independently.

All three go out — an open circuit in one branch kills the others.

The remaining two branches dim because the total resistance increased.

The remaining two branches get more current because the total resistance decreased.

Kirchhoff's Current Law (KCL) states that at any junction in a circuit:

The total current entering the junction equals the total current leaving the junction.

The sum of all voltage drops around any closed loop equals the source voltage.

KCL applies only to parallel circuits; KVL applies only to series circuits.

The current through each branch is equal at a junction.

Fluency Practice

Two resistors, $R_1 = 6\ \Omega$ and $R_2 = 12\ \Omega$ , are connected in parallel. Calculate the equivalent resistance using the product-over-sum shortcut.

Three resistors — $R_1 = 20\ \Omega$ , $R_2 = 30\ \Omega$ , $R_3 = 60\ \Omega$ — are connected in parallel. Calculate the equivalent resistance.

Which of the following must always be true for the equivalent resistance of any parallel combination of resistors?

The equivalent resistance is less than the smallest individual resistor.

The equivalent resistance equals the average of the individual resistances.

The equivalent resistance increases as more resistors are added in parallel.

The equivalent resistance equals the sum of the reciprocals of each resistance.

Three resistors ( $R_1 = 20\ \Omega$ , $R_2 = 30\ \Omega$ , $R_3 = 60\ \Omega$ ) are connected in parallel to a 12 V source. What is the current through $R_2$ ?

Using the same parallel circuit ( $R_1 = 20\ \Omega$ , $R_2 = 30\ \Omega$ , $R_3 = 60\ \Omega$ , $V = 12\ \text{V}$ ), the branch currents are $I_1 = 0.60\ \text{A}$ , $I_2 = 0.40\ \text{A}$ , $I_3 = 0.20\ \text{A}$ . What is the total current from the source?

$1.20\ \text{A}$ — the sum of all branch currents.

$0.40\ \text{A}$ — the average of the three branch currents.

$0.20\ \text{A}$ — the current through the smallest branch.

$0.60\ \text{A}$ — the current through the largest branch.

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