Parallel Circuits

In a parallel circuit, components are connected across each other, providing multiple paths for current.

Rules for parallel circuits

Potential difference: The p.d. across each branch is the same. Each branch is connected directly across the source.

$V_{\text{total}} = V_1 = V_2 = V_3$

Current: The total current from the source equals the sum of the currents in each branch.

$I_{\text{total}} = I_1 + I_2 + I_3$

Resistance (Supplement): The combined resistance is found using:

$\frac{1}{R_{\text{total}}} = \frac{1}{R_1} + \frac{1}{R_2}$

The total resistance in parallel is always less than the smallest individual resistance.

Worked example

Two resistors, 6.0 $\Omega$ and 12 $\Omega$, are connected in parallel across a 12 V battery. Calculate the total resistance and the total current.

$\frac{1}{R_{\text{total}}} = \frac{1}{6.0} + \frac{1}{12} = \frac{2}{12} + \frac{1}{12} = \frac{3}{12}$

$R_{\text{total}} = \frac{12}{3} = 4.0\text{ }\Omega$

$I_{\text{total}} = \frac{V}{R} = \frac{12}{4.0} = 3.0\text{ A}$

Check: $I_1 = 12/6 = 2.0$ A; $I_2 = 12/12 = 1.0$ A; total = 3.0 A. Correct.

Common errors and how to correct them

Adding resistances directly in parallel (treating them as series). In parallel, use the reciprocal formula. The total resistance is less than the smallest branch.

Forgetting to take the reciprocal at the end. After calculating $1/R_{\text{total}}$, you must flip it to find $R_{\text{total}}$.

Getting a parallel resistance larger than any individual resistor. This is always wrong. Check the reciprocal calculation.

How to apply this in an exam

State the rules: same p.d. across each branch, currents add up, use the reciprocal formula for resistance. Verify your answer by checking that total resistance is less than the smallest individual resistor.