Chapter 25: Current, Resistance, and Electromotive Force – Study Notes

Chapter 25: Current, Resistance, and Electromotive Force

Introduction to Electric Circuits

Electric circuits are fundamental to modern technology, enabling the controlled flow of electric charges to power devices such as computers, televisions, and industrial systems. In a simple circuit, such as a flashlight, the current that exits the battery is equal to the current entering the bulb, but the energy of the charges decreases as they pass through circuit elements like light bulbs.

• Electric current is the flow of electric charge through a conductor.

• Energy is transferred from the source (e.g., battery) to circuit elements (e.g., bulbs), where it is converted to other forms such as light and heat.

Motion of Electrons in an Electric Field

Random and Drift Motion

Electrons in a conductor exhibit two types of motion: random thermal motion and net drift due to an applied electric field.

• Without an electric field: Electrons move randomly at high speeds (~$10^6$ m/s), resulting in no net current.

• With an electric field: Electrons still move randomly, but there is a slow net motion ("drift speed") in the direction opposite to the electric field, typically around ~$10^{-4}$ m/s.

Drift velocity is the average velocity of charge carriers due to the electric field.

Electric Current

Definition and Formula

Electric current is defined as the rate at which charge flows through a surface.

• Current (I): $I = \frac{dQ}{dt}$

• SI unit: ampere (A), where $1\,\mathrm{A} = 1\,\mathrm{C/s}$

• Named after André-Marie Ampère (1775–1836).

Direction of Current Flow

• Conventional current is defined as the flow of positive charges, even though in metals, the actual charge carriers are electrons (negative charges).

• In diagrams, current direction is shown as the direction positive charges would move.

Types of Charge Carriers

• In metals: charge carriers are electrons.

• In ionic solutions: both positive and negative ions can carry current.

• Total current is the sum of the currents due to all types of charge carriers.

Current Density

Definition and Formula

Current density is a vector quantity that describes the amount of current flowing per unit area.

• Current density (J): $\vec{J} = n q \vec{v}_d$

• Where $n$ is the number density of charge carriers, $q$ is the charge per carrier, and $\vec{v}_d$ is the drift velocity.

• SI unit: $\mathrm{A/m^2}$

• Current density is always in the direction of the electric field, regardless of the sign of the charge carriers.

Ohm's Law and Resistivity

Ohm's Law

Ohm's Law relates the current density in a material to the applied electric field.

• Ohm's Law (microscopic form): $\vec{J} = \sigma \vec{E}$

• Where $\sigma$ is the electrical conductivity of the material.

• Alternatively, $\vec{E} = \rho \vec{J}$, where $\rho$ is the resistivity.

Resistivity and Conductivity

• Resistivity ($\rho$): $\rho = \frac{E}{J}$

• SI unit: $\Omega \cdot \mathrm{m}$

• Conductivity ($\sigma$): $\sigma = \frac{1}{\rho}$

• SI unit: Siemens per meter (S/m)

Resistivity and Temperature

For most metals, resistivity increases with temperature. For semiconductors, resistivity decreases with temperature.

• Temperature dependence: $\rho(T) = \rho_0 [1 + \alpha (T - T_0)]$

• $\rho_0$ is the resistivity at reference temperature $T_0$.

• $\alpha$ is the temperature coefficient of resistivity.

Table: Approximate Temperature Coefficients of Resistivity

Material	Temperature Coefficient $\alpha$ (°C$^{-1}$)
Copper	0.00393
Aluminum	0.00429
Lead	0.0039
Manganin	0.00002
Mercury	0.00089
Nichrome	0.0001
Silver	0.0038
Tungsten	0.0045

Superconductivity

• Some materials exhibit superconductivity: below a critical temperature $T_c$, their resistivity drops to zero.

• Current in a superconducting circuit can persist indefinitely without an applied voltage.

Resistance and Ohmic Devices

Resistance

• Resistance (R): $R = \rho \frac{L}{A}$

• Where $L$ is the length and $A$ is the cross-sectional area of the conductor.

• SI unit: ohm ($\Omega$)

Ohm's Law (Macroscopic Form)

• $V = IR$

• For ohmic materials, current is proportional to voltage at constant temperature.

• Non-ohmic devices (e.g., diodes) do not follow a linear $I$-$V$ relationship.

Resistor Color Codes

Resistors are often color-coded to indicate their resistance and tolerance values.

• Each color band represents a digit or multiplier.

• Common tolerances: ±1%, ±5%, ±10%.

Electromotive Force (emf) and Circuits

Definition of emf

• Electromotive force (emf, $\mathcal{E}$): The energy per unit charge supplied by a source to move charge from lower to higher potential.

• SI unit: volt (V)

• Example: A 1.5 V battery does 1.5 J of work per coulomb of charge.

Internal Resistance

• Real sources of emf have internal resistance ($r$), which reduces the terminal voltage when current flows.

• Terminal voltage: $V = \mathcal{E} - Ir$

Circuit Symbols

• Longer line: higher potential (positive terminal)

• Shorter line: lower potential (negative terminal)

• Other symbols: resistor, voltmeter, ammeter, source of emf with internal resistance

Energy and Power in Electric Circuits

Power in Circuits

• Power delivered to or extracted from a circuit element: $P = VI$

• For a resistor: $P = I^2R = \frac{V^2}{R}$

• For a source with emf and internal resistance: $P = \mathcal{E}I - I^2r$

Summary Table: Key Equations

Quantity	Equation	SI Unit
Current	$I = \frac{dQ}{dt}$	A (ampere)
Current Density	$\vec{J} = n q \vec{v}_d$	A/m2
Ohm's Law	$V = IR$	V (volt)
Resistance	$R = \rho \frac{L}{A}$	$\Omega$ (ohm)
Resistivity	$\rho = \frac{E}{J}$	$\Omega \cdot$ m
Power	$P = VI$	W (watt)
Terminal Voltage	$V = \mathcal{E} - Ir$	V (volt)

Examples and Applications

• Example: Calculating drift speed, current density, and resistance for a given wire using the above formulas.

• Application: Understanding how resistivity changes with temperature is crucial for designing electrical components that operate reliably under varying thermal conditions.

Additional info: Some context and formulas have been expanded for clarity and completeness based on standard college physics curriculum.