Chapter 22: Current and Resistance – Study Notes

Current and Resistance

Introduction to Current and Resistance

This chapter introduces the fundamental concepts of electric current and resistance, which are essential for understanding the behavior of electrical circuits. The discussion is grounded in two key physical principles: conservation of charge and conservation of energy. The chapter aims to provide both conceptual and quantitative descriptions of current, electric potential difference, and resistance.

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

• Resistance quantifies how much a material opposes the flow of electric current.

• These concepts are foundational for analyzing circuits, which will be explored further in subsequent chapters.

Model of Current

Physical Basis of Electric Current

Electric current is defined as the motion of charges through a material, typically a metal wire. The flow of current can be illustrated using a parallel-plate capacitor connected by a wire.

• When a wire connects two oppositely charged plates, the excess charge rapidly redistributes, discharging the plates and reaching electrostatic equilibrium.

• Current-carrying wires exhibit observable effects, such as deflecting a compass needle (due to magnetic fields), heating up, and causing lightbulb filaments to glow.

• The brightness of a glowing wire increases with increasing current.

Example: A lightbulb filament glows because it is part of the connecting wire carrying current between capacitor plates.

Charge Carriers and Electric Field

Nature of Charge Carriers in Metals

In metals, the charge carriers are conduction electrons, which are free to move throughout the solid. The metal as a whole remains electrically neutral.

• Conduction electrons are not bound to any particular atom and can move freely.

• When an electric field is applied, it exerts a force on the electrons, causing them to accelerate.

• Collisions between electrons and the metal atoms slow the electrons down, converting kinetic energy into thermal energy and heating the metal.

• The motion of electrons will cease unless the electric field is maintained by a potential difference.

Drift Velocity: In a constant electric field, electrons acquire a net average velocity opposite to the field direction, known as drift velocity.

Conservation of Charge and Current

Principles Governing Current Flow

The principle of conservation of charge states that charge cannot be created or destroyed, only moved. This leads to the conservation of current in circuits.

• The current entering a circuit element (e.g., a lightbulb) equals the current leaving it.

• At a junction where wires branch, the sum of input currents equals the sum of output currents.

• This is formalized in Kirchhoff’s Junction Law:

$\sum I_{in} = \sum I_{out}$

Example: In a simple circuit, the current is conserved at every point along the wire.

Defining Electric Current

Quantitative Definition and Units

Electric current is defined as the rate at which charge flows through a cross-section of a wire.

• Current ($I$) is given by:

$I = \frac{\Delta q}{\Delta t}$

• The SI unit of current is the ampere (A), where $1~\text{A} = 1~\text{C}/\text{s}$.

• By convention, current is considered as the flow of positive charge (conventional current direction).

Example: A laptop battery supplying a 1.0 A current for 6.0 hours delivers a total charge:

$q = I \Delta t = (1.0~\text{A})(6.0~\text{h} \times 3600~\text{s/h}) = 21600~\text{C}$

Batteries and Electromotive Force (emf)

Role of Batteries in Circuits

Batteries maintain a continuous flow of charge by creating a constant potential difference between their terminals through internal chemical processes.

• Electromotive force (emf) is the potential difference maintained by a battery or similar device.

• Emf is measured in volts (V).

• Current in a wire connecting battery terminals is driven by the emf.

Example: A battery with emf of 1.5 V maintains a potential difference of 1.5 V across its terminals when no current flows.

Potential Difference and Current

Relationship Between Voltage and Current

The current in a wire is determined by the potential difference across it and the properties of the wire. The proportionality constant is called resistance (R).

• Ohm’s Law relates current, potential difference, and resistance:

$I = \frac{\Delta V_{\text{wire}}}{R}$

or equivalently,

$\Delta V = IR$

• The unit of resistance is the ohm (Ω), where $1~\Omega = 1~\text{V}/\text{A}$.

• Conductance (G) is the reciprocal of resistance:

$G = \frac{1}{R}$

and

$I = G \Delta V$

• The unit of conductance is the siemens (S), where $1~\text{S} = 1~\text{A}/\text{V}$.

Resistivity and Conductivity

Material Properties Affecting Resistance

Resistivity (ρ) is a material property that quantifies how strongly a material opposes the flow of electric current. Conductivity (σ) is its reciprocal.

• Good conductors have low resistivity and high conductivity.

• Poor conductors (insulators) have high resistivity and low conductivity.

• Resistivity increases with temperature for most metals.

• The resistance of a wire of length $L$ and cross-sectional area $A$ is:

$R = \rho \frac{L}{A}$

• Conductivity is given by:

$\sigma = \frac{1}{\rho}$

• Units: Resistivity ($\Omega \cdot m$), Conductivity (S/m).

Example: Measuring the resistivity of a plant leaf using electrodes and a voltage source.

Ohm’s Law and Resistor Circuits

Ohmic and Non-Ohmic Materials

Ohm’s Law states that the current through a conductor is directly proportional to the potential difference across it, provided the material is ohmic.

• Ohmic materials: metals, resistors (linear relationship between $I$ and $V$).

• Non-ohmic devices: batteries, capacitors, diodes (nonlinear $I$-$V$ relationship).

• Ideal wires are assumed to have zero resistance ($R = 0~\Omega$).

• Insulators have very high resistance (often $> 10^{10}~\Omega$).

For a resistor:

$\Delta V = IR$

The electric field inside a resistor carrying current is uniform and points from high to low potential.

$E = \frac{\Delta V_{\text{resistor}}}{L}$

Energy and Power in Circuits

Energy Transfer and Power Dissipation

As charges move through a circuit, energy is transferred from the battery (chemical energy) to the charges (electric potential energy), and then dissipated as thermal energy in resistors.

• The rate at which energy is transferred is called power (P):

$P = I \Delta V$

• For a resistor obeying Ohm’s Law, power can also be written as:

$P = I^2 R$

$P = \frac{(\Delta V)^2}{R}$

• SI unit of power is the watt (W), where $1~\text{W} = 1~\text{J}/\text{s}$.

Example: A flashlight bulb with a current of 0.40 A and voltage of 3.0 V consumes:

$P = (0.40~\text{A})(3.0~\text{V}) = 1.2~\text{W}$

Energy consumed in 5 minutes:

$E = P \times t = (1.2~\text{W})(5~\text{min} \times 60~\text{s/min}) = 360~\text{J}$

Summary Table: Key Quantities in Current and Resistance

Quantity	Symbol	Unit	Definition/Formula
Current	I	A (ampere)	$I = \frac{\Delta q}{\Delta t}$
Resistance	R	Ω (ohm)	$R = \frac{\Delta V}{I}$, $R = \rho \frac{L}{A}$
Conductance	G	S (siemens)	$G = \frac{1}{R}$
Resistivity	ρ	$\Omega \cdot m$	Material property
Conductivity	σ	S/m	$\sigma = \frac{1}{\rho}$
Power	P	W (watt)	$P = I \Delta V$, $P = I^2 R$, $P = \frac{(\Delta V)^2}{R}$

Additional info: Some examples and applications were expanded for clarity, and equations were formatted for academic completeness.