Electric Circuits: Quick Revision

A brief note on Electric Circuits. Read it before a test. Try each question at the end before you open its answer.

The big ideas

• Ohm's law: $V = IR$. The resistance of a wire is $R = \rho\dfrac{L}{A}$; power is $P = VI = I^2R = \dfrac{V^2}{R}$.
• In series the same current flows and the resistances add. In parallel the same voltage is across every branch and the reciprocals add.
• Kirchhoff: the current into a junction equals the current out; round any loop the rises and falls of potential add to zero.
• A real cell: $V = E - Ir$. A load takes the most power when $R = r$.
• A balanced Wheatstone bridge: $\dfrac{P}{Q} = \dfrac{R}{S}$.

The facts at a glance

• Series: $R = R_1 + R_2 + \ldots$
• Parallel: $\dfrac{1}{R} = \dfrac{1}{R_1} + \dfrac{1}{R_2} + \ldots$
• Junction rule: $\sum I_{\text{in}} = \sum I_{\text{out}}$
• Loop rule: $\sum \Delta V = 0$
• Terminal voltage: $V = E - Ir$
• Most power to a load: $R = r$, $P_{\text{max}} = \dfrac{E^2}{4r}$

A worked example

A cell of emf $10\ \text{V}$ and internal resistance $0.5\ \Omega$ feeds a $3\ \Omega$ and a $6\ \Omega$ resistor in parallel. Find the current from the cell, its terminal voltage, and the current in the $3\ \Omega$ resistor.

1. The pair. $\dfrac{3\ \Omega \times 6\ \Omega}{3\ \Omega + 6\ \Omega} = 2\ \Omega$.
2. The current. $I = \dfrac{10\ \text{V}}{2\ \Omega + 0.5\ \Omega} = 4\ \text{A}$.
3. The terminal voltage. $V = 10\ \text{V} - 4\ \text{A} \times 0.5\ \Omega = 8\ \text{V}$.
4. The $3\ \Omega$ branch. $I_3 = \dfrac{8\ \text{V}}{3\ \Omega} = 2.67\ \text{A}$.

Practice with new numbers

A program to try

Mistakes to avoid

• Adding resistors in parallel. The total is always smaller than the smallest branch.
• Forgetting the internal resistance in $I = \dfrac{E}{R + r}$.
• Fearing a negative current. It only means the current flows against your arrow.
• Taking the emf as the terminal voltage while a current flows.

Check yourself

1. Two cells, each $1.5\ \text{V}$ with $0.5\ \Omega$ inside, are joined in series to a $5\ \Omega$ lamp. What current flows?

2. A current of $7\ \text{A}$ flows into a junction, and two wires take current out. One carries $2.5\ \text{A}$. What does the other carry?

3. A Wheatstone bridge has $P = 50\ \Omega$, $Q = 75\ \Omega$ and $R = 40\ \Omega$. What $S$ balances it?

4. A battery has an internal resistance of $0.8\ \Omega$. Which load takes the most power from it?