Resistance and Combinations of Resistors: Quick Revision

A brief note on Resistance and Combinations of Resistors. Read it before a test. Try each question at the end before you open its answer.

The big ideas

• $R = \rho\,\dfrac{L}{A}$: longer means more resistance, thicker means less. The resistivity $\rho$ depends only on the material and the temperature.
• Series: one path; the same current; the voltages add; $R_s = R_1 + R_2 + \dots$, larger than the largest.
• Parallel: many paths; the same voltage; the currents add; $\dfrac{1}{R_p} = \dfrac{1}{R_1} + \dfrac{1}{R_2} + \dots$, smaller than the smallest.
• Mixed circuits: solve each parallel group first, then add the series parts.
• Homes are wired in parallel: full voltage for each, separate switches.

The facts at a glance

A worked example

A $14\ \Omega$ and a $35\ \Omega$ resistor are in parallel, and the pair is in series with $6\ \Omega$, on a $20\ \text{V}$ battery. Find the current from the battery.

1. The pair. $\dfrac{14\ \Omega \times 35\ \Omega}{14\ \Omega + 35\ \Omega} = 10\ \Omega$.
2. The total. $10\ \Omega + 6\ \Omega = 16\ \Omega$.
3. The current. $I = \dfrac{20\ \text{V}}{16\ \Omega} = 1.25\ \text{A}$.

Practice with new numbers

A program to try

The program finds the series and the parallel resistance of any list of resistors.

Mistakes to avoid

• Adding resistances that are in parallel. Only series resistances add.
• Forgetting to turn $\dfrac{1}{R_p}$ upside down at the end.
• Using square millimetres in $R = \rho\dfrac{L}{A}$. Use square metres.
• 'Resistivity changes with length.' Resistance does; resistivity does not.

Check yourself

1. A wire $3\ \text{m}$ long has $2.5\ \Omega$. What is the resistance of $4.5\ \text{m}$ of the same wire?

2. A $12\ \Omega$ and a $6\ \Omega$ resistor are in series on $9\ \text{V}$. What current flows?

3. A $15\ \Omega$ and a $30\ \Omega$ resistor are in parallel. What is their combined resistance?

4. Which gives more resistance: two resistors in series, or the same two in parallel?