Task Sets: Parallel Circuits - OpenStax Physics (High School)

A

Recall / Warm-Up

1.

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

A.

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

B.

All components carry the same current.

C.

The equivalent resistance is the sum of the individual resistances.

D.

Adding more branches increases the total resistance.

2.

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

A.

They are unaffected — each branch operates independently.

B.

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

C.

The remaining two branches dim because the total resistance increased.

D.

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

3.

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

A.

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

B.

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

C.

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

D.

The current through each branch is equal at a junction.

B

Fluency Practice

1.

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.

2.

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

3.

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

A.

The equivalent resistance is less than the smallest individual resistor.

B.

The equivalent resistance equals the average of the individual resistances.

C.

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

D.

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

4.

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 $R_2$ ?

5.

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?

A.

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

B.

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

C.

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

D.

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

C

Mixed Practice

D

Word Problems

1.

Three resistors are connected in parallel to a 12 V battery: $R_1 = 20\ \Omega$ , $R_2 = 30\ \Omega$ , $R_3 = 60\ \Omega$ .

1.

Calculate the equivalent resistance of the parallel combination.

2.

Calculate the total current from the battery.

3.

Calculate the current through $R_1$ , and verify KCL: does $I_1 + I_2 + I_3 = I_{total}$ ? (Enter just the value of $I_1$ in amperes.)

2.

A parallel circuit has three branches. $R_1 = 15\ \Omega$ and $R_2 = 30\ \Omega$ are known. The total current from the 12 V source is $I_{total} = 1.8\ \text{A}$ .

Find the resistance of the unknown branch $R_3$ .

E

Error Analysis

1.

Student's reasoning:

"A parallel circuit has branches A and B. Each is connected to 12 V and carries 2 A. Now a third branch C is added (also 6 Ω). The source must split its current among three branches instead of two — so branches A and B each now carry $2/3 \approx 1.33\ \text{A}$ instead of 2 A."

A student describes how adding branches to a parallel circuit affects each existing branch. Read their reasoning and find the error.

A.

The student is correct — source current is divided equally among parallel branches.

B.

The student is wrong. Each branch still has 12 V across it — its current doesn't change. Branch C adds its own current. The total source current increases; individual branch currents do not decrease.

C.

The student is right that branch currents decrease, but the math should use the new equivalent resistance, not a simple division by 3.

D.

The student is right — adding more parallel branches always reduces the current through each existing branch.

2.

Student's work:

"For parallel resistors, I average the two values: $R_{eq} = (R_1 + R_2)/2 = (8 + 12)/2 = 10\ \Omega$ ."

"I'll check: is $R_{eq} = 10\ \Omega$ less than the smallest resistor $R_1 = 8\ \Omega$ ? No, $10 > 8$ — but that must just mean this particular case is an exception."

A student calculates the equivalent resistance for two parallel resistors, $R_1 = 8\ \Omega$ and $R_2 = 12\ \Omega$ . Read their work and find the error.

A.

The student is correct — the average is a valid shortcut for two parallel resistors.

B.

The student used the wrong formula. The correct formula is $R_{eq} = R_1 R_2 / (R_1 + R_2) = 96/20 = 4.8\ \Omega$ . Their own check shows the error: $R_{eq}$ must always be less than the smallest branch resistance.

C.

The student's formula is wrong, but their check is also flawed — $R_{eq}$ for parallel resistors can sometimes exceed the smallest branch.

D.

The check is correct — if $R_{eq} > R_1$ , it means the calculation is right and the rule doesn't apply here.

F

Challenge

1.

Two resistors are connected in parallel. Their equivalent resistance is $R_{eq} = 4\ \Omega$ and one resistor has $R_1 = 12\ \Omega$ . Find the resistance of the other resistor, $R_2$ .

2.