Week 5 Lec. 1

Resistors in Series

Series Connection of Resistors

When resistors are connected in series, the same current flows through each resistor, but the voltage across each resistor may differ depending on its resistance. The total resistance is the sum of individual resistances.

• Current: $I_1 = I_2 = I_{eq}$ (same through all resistors)

• Voltage: $V_1 + V_2 + V_3 = V_{eq}$ (sum of voltages across each resistor equals total voltage)

• Equivalent Resistance: $R_{eq} = R_1 + R_2 + R_3$

• Formula for resistors of different lengths: $R_{eq} = \rho \frac{L_1 + L_2 + L_3}{A}$

Example: If $R_1 = 1\Omega$, $R_2 = 10\Omega$, and $V_0 = 22$ V, then $R_{12} = 11\Omega$, $I_{12} = \frac{22}{11} = 2$ A, $V_1 = 2$ V, $V_2 = 20$ V.

Resistors in Parallel

Parallel Connection of Resistors

In a parallel arrangement, the voltage across each resistor is the same, but the current divides among the resistors according to their resistance values. The equivalent resistance is always less than the smallest individual resistance.

• Voltage: $V_1 = V_2 = V_3$ (same across all resistors)

• Current: $I_{total} = I_1 + I_2 + I_3$ (sum of currents through each branch)

• Equivalent Resistance: $\frac{1}{R_{eq}} = \frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3}$

Example: For $R_2 = 20\Omega$, $R_3 = 30\Omega$, $E = 60$ V, $R_{23} = 12\Omega$, $I_{23} = \frac{60}{12} = 5$ A.

Series vs. Parallel Arrangements

Comparison of Equivalent Resistance

Parallel arrangements always have a smaller equivalent resistance than series arrangements for the same resistors. This is because parallel paths provide more ways for current to flow.

• Series: $R_{eq} = R_1 + R_2 + R_3$

• Parallel: $\frac{1}{R_{eq}} = \frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3}$

Brightness in Parallel Circuits

Light Bulb Example

When identical bulbs are connected in parallel, the bulbs in parallel branches (B and C) are brighter than the bulb in the series branch (A), because they receive the full voltage and more current flows through them.

Current in Parallel Circuits

Effect of Adding More Resistors

As more resistors are added in parallel, the total resistance decreases, allowing more current to flow through the circuit.

• Key Point: Adding parallel resistors decreases the equivalent resistance.

Circuit Elements

Branches, Junctions, and Loops

Understanding the structure of circuits is essential for analysis. A branch is a single circuit element, a junction is where branches meet, and a loop is any closed path in the circuit.

• Branch: A single element with two ends (e.g., resistor, capacitor)

• Junction (Node): Point where three or more branches meet

• Loop: Any closed path in the circuit

Kirchhoff’s Laws

Kirchhoff’s Junction Rule

The sum of currents entering a junction equals the sum of currents leaving the junction. This is a statement of conservation of charge.

• Mathematical Form: $\sum I_{in} = \sum I_{out}$

• Physical Meaning: Charge cannot accumulate at a node.

Kirchhoff’s Junction Rule Example

Given currents of 5 A and 9 A entering a junction and 2 A leaving, the current in the fourth wire must be 12 A leaving to satisfy the junction rule.

Kirchhoff’s Loop Rule

The algebraic sum of the changes in potential around any closed loop is zero. This is a statement of conservation of energy in electrical circuits.

• Mathematical Form: $\sum V = 0$ around a closed loop

• Application: Used to analyze circuits with multiple loops and voltage sources

Example: Moving clockwise around a loop with two batteries and two resistors:

• $+\varepsilon_1 - IR_1 - IR_2 - \varepsilon_2 = 0$

Sign Conventions in Loop Rule

When traversing a circuit:

• Voltage gains (battery from - to +): positive sign

• Voltage drops (battery from + to - or across resistors in current direction): negative sign

• Direction of current is arbitrary; negative result means actual current flows opposite to chosen direction

Kirchhoff Circuit Analysis Tips

Steps for Solving Circuits

• Sketch the circuit diagram

• Simplify using equivalent resistors

• Label currents with assumed directions

• Apply Junction Rule to label currents

• Choose independent loops

• Apply Loop Rule to each loop

• Solve the resulting simultaneous linear equations

Additional info: These concepts are foundational for analyzing direct-current circuits, as covered in Chapter 26 of college physics.