Summary of Direct Current Electric Circuits

## Introduction Electric circuits — Circuit analysis focuses on calculating currents, voltages and resistances in series, parallel and mixed circuits. This material helps you apply Ohm's law, combine resistors, and use charge and current relations to solve quantitative problems. > **Definition:** Potential difference (voltage) is the energy transferred per unit charge when charge moves between two points. Resistance is the opposition to the flow of electric charge, measured in ohms (\(\Omega\)). ## Key concepts broken down ### Ohm's law Ohm's law links voltage, current and resistance: $$V = I \times R$$ This lets you calculate any one quantity when the other two are known. ### Current and charge - Current \(I\) is the rate of flow of charge: \(I = \dfrac{Q}{\Delta t}\). - Charge moved in time \(\Delta t\) is \(Q = I\,\Delta t\). Example: A current of \(2\,\mathrm{A}\) for \(6\) minutes (\(360\,\mathrm{s}\)) moves $$Q = (2)(360) = 720\,\mathrm{C}$$ (Use the total current that passes through the ammeter for branches where the ammeter reads the main current.) ### Resistors in series - Currents are the same through each resistor in series. - Voltages add: \(V_{T} = V_{1} + V_{2} + \dots\). - Resistances add: $$R_{T} = R_{1} + R_{2} + \dots$$ Example: Two identical resistors in series share the battery voltage equally so each gets half the terminal voltage. ### Resistors in parallel - Voltage across each branch is the same. - Currents divide: \(I_{T} = I_{1} + I_{2} + \dots\). - Total resistance for multiple parallel resistors: $$\dfrac{1}{R_{T}} = \dfrac{1}{R_{1}} + \dfrac{1}{R_{2}} + \dfrac{1}{R_{3}} + \dots$$ For two resistors only one convenient formula is often used, but the reciprocal-sum form always works. Example: Three resistors \(3\,\Omega\), \(4\,\Omega\), \(6\,\Omega\) in parallel give $$\dfrac{1}{R_{T}} = \dfrac{1}{3} + \dfrac{1}{4} + \dfrac{1}{6} = \dfrac{1}{0.75}$$ so $$R_{T} = 1.33\,\Omega$$ ### Mixed circuits (series + parallel) - Reduce parallel groups first to an equivalent resistance, then add series resistances to find the total resistance. - Use the total voltage to find total current: $$I_{T} = \dfrac{V_{T}}{R_{T}}$$ - Use the equivalent resistances to find voltages across and currents through components. Worked example (cells and mixed resistors): - Five cells each \(1.5\,\mathrm{V}\) give \(V_{T} = (1.5)(5) = 7.5\,\mathrm{V}\). - Parallel group equivalent \(R_{P} = 1.33\,\Omega\); add series resistor \(R_{s} = 2.67\,\Omega\) to get \(R_{T} = 4.00\,\Omega\). - Total current: $$I_{T} = \dfrac{7.5}{4.0} = 1.875\,\mathrm{A}$$ - Voltage across the series resistor portion of \(2.67\,\Omega\): $$V = I_{T} \times 2.67 = (1.875)(2.67) = 5.01\,\mathrm{V}$$ - Charge through the ammeter in 6 minutes (\(360\,\mathrm{s}\)): $$Q = I_{T}\,\Delta t = (1.875)(360) = 675\,\mathrm{C}$$ ### Using voltmeters and ammeters - A voltmeter measures potential difference and is connected in parallel across the component. - An ammeter measures current and is placed in series with the branch where current is measured. - If the ammeter is in the main line before a branching point it reads the total current. ### Bulbs and brightness (practical reasoning) - Identical bulbs: brightness depends on power, which depends on voltage and current. - Series: same current, power differences come from voltage division. - Parallel: same voltage across each bulb, so identical bulbs glow equally bright. - Adding a parallel branch reduces total resistance and can make some bulbs brighter by increasing total current from the battery. ## Practical examples and question-solving tips 1. To find the current in a circuit: - Find equivalent resistance \(R_{T}\). - Use \(I_{T} = \dfrac{V_{T}}{R_{T}}\). 2. To find voltage across a series resistor: use \(V = I_{T}R\). 3. To find current through a parallel resistor: first find voltage across the parallel group, then \(I = \dfrac{V}{R_{branch}}\). 4. W