Electric Circuits: Current, Resistance, and Circuit Analysis Ch 28

Electric Current and Conductivity

Conduction Electrons in Metals

In metallic solids, the outer-shell electrons are not bound to individual atoms but are free to move throughout the material. This forms an 'electron gas' responsible for electrical conductivity.

• Ions (metal atoms minus valence electrons) occupy fixed positions in the lattice.

• Conduction electrons are bound to the solid as a whole, not to any particular atom, and are free to move around.

• The metal as a whole remains electrically neutral.

Drift Velocity and Current

When an electric field is applied, conduction electrons drift slowly opposite to the field direction, creating an electric current.

• Drift velocity ($v_d$) is the average velocity of electrons due to the electric field, typically much slower than their random thermal motion.

• Current ($I$) is defined as the amount of charge passing through a cross-section of the wire per second.

• Formula: $I = -e n_e v_d A$ where $e$ is the electron charge, $n_e$ is the number density of electrons, $v_d$ is drift velocity, and $A$ is cross-sectional area.

• Because electrons are negatively charged, conventional current direction is opposite to electron flow.

• Unit: 1 ampere (A) = 1 coulomb/second (C/s).

Basic Circuit Elements and Diagrams

Simple Circuit Representation

Electric circuits are composed of sources (batteries), loads (resistors, bulbs), and connecting wires. Circuit diagrams use standardized symbols to represent these elements.

• Source: Provides a fixed potential difference (voltage).

• Load: Consumes electrical energy (e.g., resistor, bulb).

• Wire: Conducts current between elements.

Circuit Symbols

Standardized symbols are used in circuit diagrams to simplify analysis and communication.

• Battery: Source of EMF

• Wire: Conductive path

• Resistor: Limits current

• Bulb: Converts electrical energy to light

• Junction: Point where wires meet

• Capacitor: Stores charge

• Switch: Opens/closes circuit

Series and Parallel Circuits

Series Circuits

In a series circuit, all elements are connected end-to-end, so the same current flows through each element.

• Current is the same through all components.

• Voltage divides among the elements according to their resistance.

• Example: Two identical bulbs in series will have equal brightness.

Parallel Circuits

In a parallel circuit, elements are connected across the same two points, so each receives the same voltage.

• Voltage is the same across all parallel branches.

• Current divides among the branches according to their resistance.

• Example: Bulbs in parallel are brighter than in series because each receives full battery voltage.

Analyzing Circuit Diagrams

Identifying Circuit Connections

Circuit diagrams may look different but represent the same electrical connections. The key is to identify which elements are in series or parallel.

• Series: Current must pass through each element sequentially.

• Parallel: Elements are connected across the same two points.

Kirchhoff's Laws

Kirchhoff's Junction Law

The sum of currents entering a junction equals the sum of currents leaving the junction, reflecting conservation of charge.

• Mathematically: $\sum I_{in} = \sum I_{out}$

Kirchhoff's Loop Law

The sum of potential differences (voltage) around any closed loop in a circuit is zero.

• Mathematically: $\sum \Delta V = 0$

• Used to analyze complex circuits with multiple loops and branches.

Ohm's Law and Resistance

Ohm's Law

Ohm's law relates the current through a conductor to the voltage across it and its resistance.

• Formula: $\Delta V = IR$

• Resistance ($R$) is measured in ohms ($\Omega$).

Equivalent Resistance

To simplify circuit analysis, resistors in series and parallel can be replaced by a single equivalent resistor.

• Series: $R_{eq} = R_1 + R_2 + \ldots$

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

Power in Electric Circuits

Power Dissipation

Power in a circuit element is the rate at which electrical energy is converted to other forms (e.g., heat, light).

• Formula: $P = VI$

• Alternative forms: $P = I^2 R = \frac{V^2}{R}$

• Unit: watt (W)

RC Circuits

Charging and Discharging a Capacitor

RC circuits consist of a resistor and capacitor in series or parallel. The charging and discharging of the capacitor follow exponential laws.

• Charge on capacitor: $Q = Q_0 e^{-t/\tau}$ (discharging), $Q = C \varepsilon (1 - e^{-t/\tau})$ (charging)

• Current: $I = -\frac{dQ}{dt} = \frac{Q_0}{RC} e^{-t/\tau}$

• Time constant: $\tau = RC$

Grounding in Circuits

Purpose of Grounding

Grounding provides a reference point for circuit potential and enhances safety by preventing unwanted voltage differences between the circuit and the earth.

• Protects users and equipment from electrical faults.

• Ensures stable operation of electrical systems.

Summary Table: Circuit Elements and Their Functions

Element	Symbol	Function
Battery	—	Provides EMF (voltage)
Wire	—	Conducts current
Resistor	—	Limits current, dissipates energy
Bulb	—	Converts electrical energy to light
Junction	—	Connects multiple wires
Capacitor	—	Stores electrical energy
Switch	—	Opens/closes circuit